Dissertations / Theses on the topic 'Galerkin methods'
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Elfverson, Daniel. "On discontinuous Galerkin multiscale methods." Licentiate thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-200260.
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In this thesis a new multiscale method, the discontinuous Galerkin multiscale method, is proposed. The method uses localized fine scale computations to correct a global coarse scale equation and thereby takes the fine scale features into account. We show a priori error bounds for convection dominated diffusion-convection-reaction problems with variable coefficients. We also present a posteriori error bound in the case of no convection or reaction and present an adaptive algorithm which tunes the method parameters automatically. We also present extensive numerical experiments which verify our analytical findings.
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Dogan, Abdulkadir. "Petrov-Galerkin finite element methods." Thesis, Bangor University, 1997. https://research.bangor.ac.uk/portal/en/theses/petrovgalerkin-finite-element-methods(4d767fc7-4ad1-402a-9e6e-fd440b722406).html.
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Harbrecht, Helmut, and Reinhold Schneider. "Adaptive Wavelet Galerkin BEM." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600559.
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The wavelet Galerkin scheme for the fast solution of boundary integral equations produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that stays proportional to the number of unknowns. In this paper we present an adaptive version of the scheme which preserves the super-convergence of the Galerkin method.
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Casoni, Rero Eva. "Shock capturing for discontinuous Galerkin methods." Doctoral thesis, Universitat Politècnica de Catalunya, 2011. http://hdl.handle.net/10803/51571.
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Aquesta tesi doctoral proposa formulacions de Galerkin Discontinu (DG) d’alt ordre per la captura de shocks, obtenint alhora
solucions altament precises per problemes de flux compressible.
En les últimes dècades, la investigació en els mètodes de DG ha estat en constant creixement. L'èxit dels mètodes DG en
problemes hiperbòlics ha conduit el seu desenvolupament en lleis de conservació no lineals i problemes de convecció dominant.
Entre els avantatges dels mètodes DG, destaquen la seva estabilitat inherent i les propietats locals de conservació. D'altra banda,
els mètodes DG estan especialment dissenyats per l’ús aproximacions d'ordre superior. De fet, en els últims anys s'ha demostrat
que la resolució de problemes de convecció dominant ja no es restringeix només a elements d'ordre inferior. De fet, es necessiten
models numèrics d'alta precisió per aconseguir prediccions altament fiables dins la dinàmica de fluids computacional (CFD). En
aquest context es presenten i discuteixen dos tècniques de captura de shocks.
En primer lloc, es presenta una tècnica novedosa i senzilla basada en la introducció d'una nova base de funcions de forma. Aquesta
base té la capacitat de canviar a nivell local entre una interpolació contínua o discontínua, depenent de la suavitat de la funció que
es vol aproximar. En presència de xocs, les discontinuïtats introduïdes dins l’element permeten incloure l'estabilització necessària
gràcies a l’ús dels fluxos numèrics, i alhora exploten les propietats intrínsiques del mètodes DG. En conseqüència, es poden utilitzar
malles grolleres amb elements d’ordre superior. Amb aquestes discretitzacions i, utilitzant el mètode proposats, els xocs queden
continguts a l’interior de l’element i per tant, és possible evitar l’ús de tècniques de refinament adaptatiu de la malla, alhora que es
manté la localitat i compacitat dels esquemes DG.
En segon lloc, es proposa una tècnica clàssica i, aparentment simple: la introducció de la viscositat artificial. Primerament es realitza
un estudi detallat per al cas unidimensional. S’obté una viscositat d’alta precisió que escala segons el valor hk amb 1 ≤ k ≤ p i essent
h la mida de l’element. En conseqüència, s’obté un xoc amb amplitud del mateix ordre. Seguidament, l'estudi de la viscositat
unidimensional obtenida s'extén al cas multidimensional per a malles triangulars. L'extensió es basa en la projecció de la viscositat
unidimensional en unes determinades direccions espacials dins l’element. Es demostra de manera consistent que la viscositat
introduïda és, com a molt, del mateix ordre que la resolució donada per la discretització espacial, és a dir, h/p. El mètode és
especialment eficient per aproximacions de Galerkin discontinu d’alt ordre, per exemple p≥ 3.
Les dues metodologies es validen mitjançant una àmplia selecció d’exemples numèrics. En alguns exemples, els mètodes
proposats permeten una reducció en el nombre de graus de llibertat necessaris per capturar xocs acuradament de fins i tot un ordre
de magnitud, en comparació amb mètodes estàndar de refinament adaptatiu amb aproximacions de baix ordre.This thesis proposes shock-capturing methods for high-order Discontinuous Galerkin (DG) formulations providing highly accurate solutions for compressible flows. In the last decades, research in DG methods has been very active. The success of DG in hyperbolic problems has driven many studies for nonlinear conservation laws and convection-dominated problems. Among all the advantages of DG, their inherent stability and local conservation properties are relevant. Moreover, DG methods are naturally suited for high-order approximations. Actually, in recent years it has been shown that convection-dominated problems are no longer restricted to low-order elements. In fact, highly accurate numerical models for High-Fidelity predictions in CFD are necessary. Under this rationale, two shock-capturing techniques are presented and discussed. First, a novel and simple technique based on on the introduction of a new basis of shape functions is presented. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization thanks to the numerical fluxes, thus exploiting DG inherent properties. Large high-order elements can therefore be used and shocks are captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Second, a classical and, apparently simple, technique is advocated: the introduction of artificial viscosity. First, a one-dimensional study is perfomed. Viscosity of the order O(hk) with 1≤ k≤ p is obtained, hence inducing a shock width of the same order. Second, the study extends the accurate one-dimensional viscosity to triangular multidimensional meshes. The extension is based on the projection of the one-dimensional viscosity into some characteristic spatial directions within the elements. It is consistently shown that the introduced viscosity scales, at most, withthe DG resolutions length scales, h/p. The method is especially reliable for highorder DG approximations, say p≥3. A wide range of different numerical tests validate both methodologies. In some examples the proposed methods allow to reduce by an order of magnitude the number of degrees of freedom necessary to accurately capture the shocks, compared to standard low order h-adaptive approaches.
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Murdoch, Thomas. "Galerkin methods for nonlinear elliptic equations." Thesis, University of Oxford, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329932.
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Dong, Zhaonan. "Discontinuous Galerkin methods on polytopic meshes." Thesis, University of Leicester, 2017. http://hdl.handle.net/2381/39140.
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This thesis is concerned with the analysis and implementation of the hp-version interior penalty discontinuous Galerkin finite element method (DGFEM) on computational meshes consisting of general polygonal/polyhedral (polytopic) elements. Two model problems are considered: general advection-diffusion-reaction boundary value problems and time dependent parabolic problems. New hp-version a priori error bounds are derived based on a specific choice of the interior penalty parameter which allows for edge/face-degeneration as well as an arbitrary number of faces and hanging nodes per element. The proposed method employs elemental polynomial bases of total degree p (Pp- bases) defined in the physical coordinate system, without requiring mapping from a given reference or canonical frame. A series of numerical experiments highlighting the performance of the proposed DGFEM are presented. In particular, we study the competitiveness of the p-version DGFEM employing a Pp-basis on both polytopic and tensor-product elements with a (standard) DGFEM and FEM employing a (mapped) Qp-basis. Moreover, a careful theoretical analysis of optimal convergence rate in p for Pp-basis is derived for several commonly used projectors, which leads to sharp bounds of exponential convergence with respect to degrees of freedom (dof) for the Pp-basis.
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Kovacs, Denis Christoph. "Inertial Manifolds and Nonlinear Galerkin Methods." Thesis, Virginia Tech, 2005. http://hdl.handle.net/10919/30792.
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Nonlinear Galerkin methods utilize approximate inertial manifolds to reduce the spatial error of the standard Galerkin method. For certain scenarios, where a rough forcing term is used, a simple postprocessing step yields the same improvements that can be observed with nonlinear Galerkin. We show that this improvement is mainly due to the information about the forcing term that is neglected by standard Galerkin. Moreover, we construct a simple postprocessing scheme that uses only this neglected information but gives the same increase in accuracy as nonlinear or postprocessed Galerkin methods.Master of Science
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Harbrecht, Helmut, Ulf Kähler, and Reinhold Schneider. "Wavelet Galerkin BEM on unstructured meshes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601459.
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The present paper is devoted to the fast solution of boundary integral equations on unstructured meshes by the Galerkin scheme. On the given mesh we construct a wavelet basis providing vanishing moments with respect to the traces of polynomials in the space. With this basis at hand, the system matrix in wavelet coordinates can be compressed to $\mathcal{O}(N\log N)$ relevant matrix coefficients, where $N$ denotes the number of unknowns. The compressed system matrix can be computed within suboptimal complexity by using techniques from the fast multipole method or panel clustering. Numerical results prove that we succeeded in developing a fast wavelet Galerkin scheme for solving the considered class of problems.
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Of, Günther, Gregory J. Rodin, Olaf Steinbach, and Matthias Taus. "Coupling Methods for Interior Penalty Discontinuous Galerkin Finite Element Methods and Boundary Element Methods." Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-96885.
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This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for some of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.
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Galbraith, Marshall C. "A Discontinuous Galerkin Chimera Overset Solver." University of Cincinnati / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1384427339.
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Kanschat, Guido. "Discontinuous Galerkin methods for viscous incompressible flow." Wiesbaden : Deutscher Universitäts-Verlag, 2008. http://dx.doi.org/10.1007/978-3-8350-5519-3.
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Dicken, Volker, and Peter Maaß. "Wavelet-Galerkin methods for ill-posed problems." Universität Potsdam, 1995. http://opus.kobv.de/ubp/volltexte/2007/1389/.
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Projection methods based on wavelet functions combine optimal
convergence rates with algorithmic efficiency. The proofs in this paper utilize the approximation properties of wavelets and results from the general theory of regularization methods. Moreover, adaptive strategies can be incorporated
still leading to optimal convergence rates for the resulting algorithms. The so-called wavelet-vaguelette decompositions enable the realization of especially fast algorithms for certain operators.
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Elfverson, Daniel. "Discontinuous Galerkin Multiscale Methods for Elliptic Problems." Thesis, Uppsala universitet, Institutionen för informationsteknologi, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-138960.
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In this paper a continuous Galerkin multiscale method (CGMM) and a discontinuous Galerkin multiscale method (DGMM) are proposed, both based on the variational multiscale method for solving partial differential equations numerically. The solution is decoupled into a coarse and a fine scale contribution, where the fine-scale contribution is computed on patches with localized right hand side. Numerical experiments are presented where exponential decay of the error is observed when increasing the size of the patches for both CGMM and DGMM. DGMM gives much better accuracy when the same size of the patches are used.
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Apel, Thomas, and Gert Lube. "Anisotropic mesh refinement in stabilized Galerkin methods." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800640.
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The numerical solution of the convection-diffusion-reaction problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least squares type accomodates diffusion-dominated as well as convection- and/or reaction- dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is accomplished using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used.
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Kanschat, Guido. "Discontinuous Galerkin methods for viscous incompressible fl." Wiesbaden Dt. Univ.-Verl, 2004. http://dx.doi.org/10.1007/978-3-8350-5519-3.
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Gryngarten, Leandro Damian. "Multi-phase flows using discontinuous Galerkin methods." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/45824.
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This thesis is concerned with the development of numerical techniques to simulate compressible multi-phase flows, in particular a high-accuracy numerical approach with mesh adaptivity. The Discontinuous Galerkin (DG) method was chosen as the framework for this work for being characterized for its high-order of accuracy -thus low numerical diffusion- and being compatible with mesh adaptivity due to its locality. A DG solver named DiGGIT (Discontinuous Galerkin at the Georgia Institute of Technology) has been developed and several aspects of the method have been studied. The Local Discontinuous Galerkin (LDG) method -an extension of DG for equations with high-order derivatives- was extended to solve multiphase flows using Diffused Interface Methods (DIM). This multi-phase model includes the convection of the volume fraction, which is treated as a Hamilton-Jacobi equation. This is the first study, to the author's knowledge, in which the volume fraction of a DIM is solved using the DG and the LDG methods. The formulation is independent of the Equation of State (EOS) and it can differ for each phase. This allows for a more accurate representation of the different fluids by using cubic EOSs, like the Peng-Robinson and the van der Waals models. Surface tension is modeled with a new numerical technique appropriate for LDG. Spurious oscillations due to surface tension are common to all the capturing schemes, and this new approach presents oscillations comparable in magnitude to the most common schemes. The moment limiter (ML) was generalized for non-uniform grids with hanging nodes that result from adaptive mesh refinement (AMR). The effect of characteristic, primitive, or conservative decomposition in the limiting stage was studied. The characteristic option cannot be used with the ML in multi-dimensions. In general, primitive variable decomposition is a better option than with conservative variables, particularly for multiphase flows, since the former type of decomposition reduces the numerical oscillations at material discontinuities. An additional limiting technique was introduced for DIM to preserve positivity while minimizing the numerical diffusion, which is especially important at the interface. The accuracy-preserving total variation diminishing (AP-TVD) marker for ``troubled-cell' detection, which uses an averaged-derivative basis, was modified to use the Legendre polynomial basis. Given that the latest basis is generally used for DG, the new approach avoids transforming to the averaged-derivative basis, what results in a more efficient technique.
Furthermore, a new error estimator was proposed to determine where to refine or coarsen the grid. This estimator was compared against other estimator used in the literature and it showed an improved performance. In order to provide equal order of accuracy in time as in space, the commonly used 3rd-order TVD Runge-Kutta (RK) scheme in the DG method was replaced in some cases by the Spectral Deferred Correction (SDC) technique. High orders in time were shown to only be required when the error in time is significant. For instance, convection-dominated compressible flows require for stability a time step much smaller than is required for accuracy, so in such cases 3rd-order TVD RK resulted to be more efficient than SDC with higher orders.
All these new capabilities were included in DiGGIT and have provided a generalized approach capable of solving sub- and super-critical flows at sub- and super-sonic speeds, using a high-order scheme in space and time, and with AMR.
Canonical test cases are presented to verify and validate the formulation in one, two, and three dimensions. Finally, the solver is applied to practical applications. Shock-bubble interaction is studied and the effect of the different thermodynamic closures is assessed. Interaction between single-drops and a wall is simulated. Sticking and the onset of splashing are observed. In addition, the solver is used to simulate turbulent flows, where the high-order of accuracy clearly shows its benefits. Finally, the methodology is challenged with the simulation of a liquid jet in cross flow.
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Madhavan, Pravin. "Analysis of discontinuous Galerkin methods on surfaces." Thesis, University of Warwick, 2014. http://wrap.warwick.ac.uk/66157/.
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In this thesis, we extend the discontinuous Galerkin framework to surface partial differential equations. This is done by deriving both a priori and a posteriori error estimates for model elliptic problems posed on compact smooth and oriented surfaces in R3, and investigating both theoretical estimates and several generalisations numerically.
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Hall, Edward John Cumes. "Anisotropic adaptive refinement for discontinuous Galerkin methods." Thesis, University of Leicester, 2007. http://hdl.handle.net/2381/30536.
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We consider both the a priori and a posteriori error analysis and hp-adaptation strategies for discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched polynomial degrees. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution, exploiting duality based arguments.;The a priori error analysis is carried out in two settings. In the first, full orientation of elements is allowed but only (possibly high-order) isotropic polynomial degrees considered; our analysis, therefore, extends previous results, where only finite element spaces comprising piecewise linear polynomials were considered, by utilizing techniques from tensor analysis. In the second case, anisotropic polynomial degrees are allowed, but the elements are assumed to be axiparallel; we thus apply previously known interpolation error results to the goal-oriented setting.;Based on our a posteriori error bound we first design and implement an adaptive anisotropic h-refinement algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement, chosen on a competitive basis requiring the solution of local problems. The superiority of the proposed algorithm in comparison with a standard h-isotropic mesh refinement algorithm and a Hessian based h-anisotropic adaptive procedure is illustrated by a series of numerical experiments. We then describe a fully hp -adaptive algorithm, once again using a competitive refinement approach, which, numerical experiments reveal, offers considerable improvements over both a standard hp-isotropic refinement algorithm and an h-anisotropic/p-isotropic adaptive procedure.
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Sabawi, Younis Abid. "Adaptive discontinuous Galerkin methods for interface problems." Thesis, University of Leicester, 2017. http://hdl.handle.net/2381/39386.
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The aim of this thesis is to derive adaptive methods for discontinuous Galerkin approximations for both elliptic and parabolic interface problems. The derivation of adaptive method, is usually based on a posteriori error estimates. To this end, we present a residual-type a posteriori error estimator for interior penalty discontinuous Galerkin (dG) methods for an elliptic interface problem involving possibly curved interfaces, with flux-balancing interface conditions, e.g., modelling mass transfer of solutes through semi-permeable membranes. The method allows for extremely general curved element shapes employed to resolve the interface geometry exactly. Respective upper and lower bounds of the error in the respective dG-energy norm with respect to the estimator are proven. The a posteriori error bounds are subsequently used to prove a basic a priori convergence result. Moreover, a contraction property for a standard adaptive algorithm utilising these a posteriori bounds, with a bulk refinement criterion is also shown, thereby proving that the a posteriori bounds can lead to a convergent adaptive algorithm subject to some mesh restrictions. This work is also concerned with the derivation of a new L1∞(L2)-norm a posteriori error bound for the fully discrete adaptive approximation for non-linear interface parabolic problems. More specifically, the time discretization uses the backward Euler Galerkin method and the space discretization uses the interior penalty discontinuous Galerkin finite element method. The key idea in our analysis is to adapt the elliptic reconstruction technique, introduced by Makridakis and Nochetto [48], enabling us to use the a posteriori error estimators derived for elliptic interface models and to obtain optimal order in both L1∞(L2) and L1∞(L2) + L2(H¹) norms. The effectiveness of all the error estimators and the proposed algorithms is confirmed through a series of numerical experiments.
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Merton, Simon Richard. "Discontinuous Galerkin methods for computational radiation transport." Thesis, Imperial College London, 2012. http://hdl.handle.net/10044/1/9906.
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This Thesis demonstrates advanced new discretisation technologies that improve the accuracy and stability of the discontinuous Galerkin finite element method applied to the Boltzmann Transport Equation, describing the advective transport of neutral particles such as photons and neutrons within a domain. The discontinuous Galerkin method in its standard form is susceptible to oscillation detrimental to the solution. The discretisation schemes presented in this Thesis enhance the basic form with linear and non-linear Petrov Galerkin methods that remove these oscillations. The new schemes are complemented by an adjoint-based error recovery technique that improves the standard solution when applied to goal-based functional and eigenvalue problems. The chapters in this Thesis have been structured to be submitted individually for journal publication, and are arranged as follows. Chapter 1 outlines the Thesis and contains a brief literature review. Chapter 2 introduces the underlying space-angle discretisation method used in the work, and discusses a series of potential modifications to the standard discontinuous Galerkin method. These differ in how the upwinding is performed on the element boundary, and comprise an upwind-average method, a Petrov-Galerkin method that removes oscillation by adding artificial diffusion internal to an element and a more sophisticated Petrov-Galerkin scheme that adds dissipation in the coupling between each element. These schemes are tested in one-dimension and Taylor analysis of their convergence rate is included. The chapter concludes with selection of one of the schemes to be developed in the next part of the Thesis. Chapter 3 develops the selected method extending it to multi-dimensions. The result is a new discontinuous Petrov-Galerkin method that is residual based and removes unwanted oscillation from the transport solution by adding numerical dissipation internal to an element. The method uses a common length scale in the upwind term for all elements. This is not always satisfactory, however, as it gives the same magnitude and type of dissipation everywhere in the domain. The chapter concludes by recommending some form of non-linearity be included to address this issue. Chapter 4 adds non-linearity to the scheme. This projects the streamline direction, in which the dissipation acts, onto the solution gradient direction. It defines locally the optimal amount of dissipation needed in the discretisation. The non-linear scheme is tested on a variety of steady-state and time-dependent transport problems. Chapters 5, 6 and 7 develop an adjoint-based error measure to complement the scheme in functional and eigenvalue problems. This is done by deriving an approximation to the error in the the bulk functional or eigenvalue, and then removing it from the calculated value in a post-process defect iteration. This is shown to dramatically accelerate mesh convergence of the goal-based functional or eigenvalue. Chapter 8 concludes the Thesis with recommendations for a further plan of work.
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Hadi, Justin. "Discontinuous Galerkin methods for hyperbolic conservation laws." Thesis, Imperial College London, 2012. http://hdl.handle.net/10044/1/10563.
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New numerical methods are developed for single phase compressible gas flow and two phase gas/liquid flow in the framework of the discontinuous Galerkin finite element method (DGFEM) and applied to Riemann problems. A residual based diffusion scheme inspired by the streamline upwind Petrov-Galerkin method (SUPG) of Brookes and Hughes [15] is applied to the Euler equations of gas dynamics and the single pressure incompressible liquid/compressible gas flow system of Toumi and Kumbaro [137]. A Roe [119] based approximate Riemann solver is applied. To minimise unstable overshoots, diffusivity is added in the direction of the gradient of the solution as opposed to the direction of the streamlines in SUPG for the continuous finite element method (CFEM). The methods are tested on Cartesian meshes with scalar advection problems, the computationally challenging Sod shock tube and Lax Riemann problems, explosion problems in gas dynamics and the water faucet test and explosion problems in two phase flow. An extension to two dimensions and comparisons to existing methods are made. A framework for the well posedness of two phase flow equations is posited and virtual mass terms are added to the two phase flow equations of Toumi and Kumbaro to ensure hyperbolicity. A viscous path based Roe solver for DGFEM is applied mirroring the method of Toumi and Kumbaro in a framework for discontinuous solutions.
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Kanschat, Guido. "Discontinuous Galerkin methods for viscous incompressible flow." Wiesbaden : Dt. Univ.-Verl, 2004. http://www.myilibrary.com?id=134464.
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Hellwig, Friederike. "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods." Doctoral thesis, Humboldt-Universität zu Berlin, 2019. http://dx.doi.org/10.18452/20034.
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Die vorliegende Arbeit "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods" beweist optimale Konvergenzraten für vier diskontinuierliche Petrov-Galerkin (dPG) Finite-Elemente-Methoden für das Poisson-Modell-Problem für genügend feine Anfangstriangulierung. Sie zeigt dazu die Äquivalenz dieser vier Methoden zu zwei anderen Klassen von Methoden, den reduzierten gemischten Methoden und den verallgemeinerten Least-Squares-Methoden. Die erste Klasse benutzt ein gemischtes System aus konformen Courant- und nichtkonformen Crouzeix-Raviart-Finite-Elemente-Funktionen. Die zweite Klasse verallgemeinert die Standard-Least-Squares-Methoden durch eine Mittelpunktsquadratur und Gewichtsfunktionen.
Diese Arbeit verallgemeinert ein Resultat aus [Carstensen, Bringmann, Hellwig, Wriggers 2018], indem die vier dPG-Methoden simultan als Spezialfälle dieser zwei Klassen charakterisiert werden. Sie entwickelt alternative Fehlerschätzer für beide Methoden und beweist deren Zuverlässigkeit und Effizienz.
Ein Hauptresultat der Arbeit ist der Beweis optimaler Konvergenzraten der adaptiven Methoden durch Beweis der Axiome aus [Carstensen, Feischl, Page, Praetorius 2014]. Daraus folgen dann insbesondere die optimalen Konvergenzraten der vier dPG-Methoden.
Numerische Experimente bestätigen diese optimalen Konvergenzraten für beide Klassen von Methoden. Außerdem ergänzen sie die Theorie durch ausführliche Vergleiche beider Methoden untereinander und mit den äquivalenten dPG-Methoden.The thesis "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods" proves optimal convergence rates for four lowest-order discontinuous Petrov-Galerkin methods for the Poisson model problem for a sufficiently small initial mesh-size in two different ways by equivalences to two other non-standard classes of finite element methods, the reduced mixed and the weighted Least-Squares method. The first is a mixed system of equations with first-order conforming Courant and nonconforming Crouzeix-Raviart functions. The second is a generalized Least-Squares formulation with a midpoint quadrature rule and weight functions. The thesis generalizes a result on the primal discontinuous Petrov-Galerkin method from [Carstensen, Bringmann, Hellwig, Wriggers 2018] and characterizes all four discontinuous Petrov-Galerkin methods simultaneously as particular instances of these methods. It establishes alternative reliable and efficient error estimators for both methods. A main accomplishment of this thesis is the proof of optimal convergence rates of the adaptive schemes in the axiomatic framework [Carstensen, Feischl, Page, Praetorius 2014]. The optimal convergence rates of the four discontinuous Petrov-Galerkin methods then follow as special cases from this rate-optimality. Numerical experiments verify the optimal convergence rates of both types of methods for different choices of parameters. Moreover, they complement the theory by a thorough comparison of both methods among each other and with their equivalent discontinuous Petrov-Galerkin schemes.
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Herron, Madonna Geradine. "A novel approach to image derivative approximation using finite element methods." Thesis, University of Ulster, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.364539.
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SINGH, ONKAR DEEP. "ITERATIVE SOLVERS FOR DISCONTINUOUS GALERKIN FINITE ELEMENT METHODS." University of Cincinnati / OhioLINK, 2004. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1093023928.
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Villardi, de Montlaur Adeline de. "High-order discontinuous Galerkin methods for incompressible flows." Doctoral thesis, Universitat Politècnica de Catalunya, 2009. http://hdl.handle.net/10803/5928.
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Aquesta tesi doctoral proposa formulacions de Galerkin discontinu (DG) d'alt ordre per fluxos viscosos incompressibles. Es desenvolupa un nou mètode de DG amb penalti interior (IPM-DG), que condueix a una forma feble simètrica i coerciva pel terme de difusió, i que permet assolir una aproximació espacial d'alt ordre. Aquest mètode s'aplica per resoldre les equacions de Stokes i Navier-Stokes. L'espai d'aproximació de la velocitat es descompon dins de cada element en una part solenoidal i una altra irrotacional, de manera que es pot dividir la forma dèbil IPM-DG en dos problemes desacoblats. El primer permet el càlcul de les velocitats i de les pressions híbrides, mentre que el segon calcula les pressions en l'interior dels elements. Aquest desacoblament permet una reducció important del número de graus de llibertat tant per velocitat com per pressió. S'introdueix també un paràmetre extra de penalti resultant en una formulació DG alternativa per calcular les velocitats solenoidales, on les pressions no apareixen. Les pressions es poden calcular com un post-procés de la solució de les velocitats. Es contemplen altres formulacions DG, com per exemple el mètode Compact Discontinuous Galerkin, i es comparen al mètode IPM-DG.
Es proposen mètodes implícits de Runge-Kutta d'alt ordre per problemes transitoris incompressibles, permetent obtenir esquemes incondicionalment estables i amb alt ordre de precisió temporal. Les equacions de Navier-Stokes incompressibles transitòries s'interpreten com un sistema de Equacions Algebraiques Diferencials, és a dir, un sistema d'equacions diferencials ordinàries corresponent a la equació de conservació del moment, més les restriccions algebraiques corresponent a la condició d'incompressibilitat.
Mitjançant exemples numèrics es mostra l'aplicabilitat de les metodologies proposades i es comparen la seva eficiència i precisió.
This PhD thesis proposes divergence-free Discontinuous Galerkin formulations providing high orders of accuracy for incompressible viscous flows.
A new Interior Penalty Discontinuous Galerkin (IPM-DG) formulation is developed, leading to a symmetric and coercive bilinear weak form for the diffusion term, and achieving high-order spatial approximations. It is applied to the solution of the Stokes and Navier-Stokes equations. The velocity approximation space is decomposed in every element into a solenoidal part and an irrotational part. This allows to split the IPM weak form in two uncoupled problems. The first one solves for velocity and hybrid pressure, and the second one allows the evaluation of pressures in the interior of the elements. This results in an important reduction of the total number of degrees of freedom for both velocity and pressure.
The introduction of an extra penalty parameter leads to an alternative DG formulation for the computation of solenoidal velocities with no presence of pressure terms. Pressure can then be computed as a post-process of the velocity solution. Other DG formulations, such as the Compact Discontinuous Galerkin method, are contemplated and compared to IPM-DG.
High-order Implicit Runge-Kutta methods are then proposed to solve transient incompressible problems, allowing to obtain unconditionally stable schemes with high orders of accuracy in time. For this purpose, the unsteady incompressible Navier-Stokes equations are interpreted as a system of Differential Algebraic Equations, that is, a system of ordinary differential equations corresponding to the conservation of momentum equation, plus algebraic constraints corresponding to the incompressibility condition.
Numerical examples demonstrate the applicability of the proposed methodologies and compare their efficiency and accuracy.
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Schmidtmann, Olaf, Fred Feudel, and Norbert Seehafer. "Nonlinear Galerkin methods for the 3D magnetohydrodynamic equations." Universität Potsdam, 1997. http://opus.kobv.de/ubp/volltexte/2007/1443/.
Full textAbstract:
The usage of nonlinear Galerkin methods for the numerical solution of
partial differential equations is demonstrated by treating an example. We desribe the implementation of a nonlinear Galerkin method based on an approximate inertial manifold for the 3D magnetohydrodynamic equations and compare its efficiency with the linear Galerkin approximation. Special bifurcation points, time-averaged values of energy and enstrophy as well as Kaplan-Yorke dimensions are calculated for both schemes in order to estimate the number of modes necessary to correctly describe the behavior of the exact solutions.
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Altmann, Christoph [Verfasser]. "Explicit Discontinuous Galerkin Methods for Magnetohydrodynamics / Christoph Altmann." München : Verlag Dr. Hut, 2012. http://d-nb.info/1029400253/34.
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Priestley, A. "Lagrange and characteristic Galerkin methods for evolutionary problems." Thesis, University of Oxford, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.376942.
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Virtanen, Juha Mikael. "Adaptive discontinuous Galerkin methods for fourth order problems." Thesis, University of Leicester, 2010. http://hdl.handle.net/2381/10091.
Full textAbstract:
This work is concerned with the derivation of adaptive methods for discontinuous Galerkin approximations of linear fourth order elliptic and parabolic partial differential equations. Adaptive methods are usually based on a posteriori error estimates. To this end, a new residual-based a posteriori error estimator for discontinuous Galerkin approximations to the biharmonic equation with essential boundary conditions is presented. The estimator is shown to be both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm, under minimal regularity assumptions. The reliability bound is based on a new recovery operator, which maps discontinuous finite element spaces to conforming finite element spaces (of two polynomial degrees higher), consisting of triangular or quadrilateral Hsieh-Clough-Tocher macroelements. The efficiency bound is based on bubble function techniques. The performance of the estimator within an h-adaptive mesh refinement procedure is validated through a series of numerical examples, verifying also its asymptotic exactness. Some remarks on the question of proof of convergence of adaptive algorithms for discontinuous Galerkin for fourth order elliptic problems are also presented. Furthermore, we derive a new energy-norm a posteriori error bound for an implicit Euler time-stepping method combined with spatial discontinuous Galerkin scheme for linear fourth order parabolic problems. A key tool in the analysis is the elliptic reconstruction technique. A new challenge, compared to the case of conforming finite element methods for parabolic problems, is the control of the evolution of the error due to non-conformity. Based on the error estimators, we derive an adaptive numerical method and discuss its practical implementation and illustrate its performance in a series of numerical experiments.
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Mukhamedov, Farukh. "High performance computing for the discontinuous Galerkin methods." Thesis, Brunel University, 2018. http://bura.brunel.ac.uk/handle/2438/16769.
Full textAbstract:
Discontinuous Galerkin methods form a class of numerical methods to find a solution of partial differential equations by combining features of finite element and finite volume methods. Methods are defined using a weak form of a particular model problem, allowing for discontinuities in the discrete trial and test spaces. Using a discontinuous discrete space mesh provides proper flexibility and a compact discretisation pattern, allowing a multidomain and multiphysics simulation. Discontinuous Galerkin methods with a higher approximation polynomial order, the socalled p-version, performs better in terms of convergence rate, compared with the low order h-version with smaller element sizes and bigger mesh. However, the condition number of the Galerkin system grows subsequently. This causes surge in the amount of required storage, computational complexity and in the time required for computation. We use the following three approaches to keep the advantages and eliminate the disadvantages. The first approach will be a specific choice of basis functions which we call C1 polynomials. These ensure that the majority of integrals over the edge of the mesh elements disappears. This reduces the total number of non-zero elements in the resulting system. This decreases the computational complexity without loss in precision. This approach does not affect the number of iterations required by chosen Conjugate Gradients method when compared to the other choice of basis functions. It actually decreases the total number of algebraic operations performed. The second approach is the introduction of suitable preconditioners. In our case, the Additive two-layer Schwarz method, developed in [4], for the iterative Conjugate Gradients method is considered. This directly affects the spectral condition number of the system matrix and decreases the number of iterations required for the computation. This approach, however, increases the total number of algebraic operations and might require more operational time. To tackle the rise in the number of algebraic operations, we introduced a modified Additive two-layer non-overlapping Schwarz method with a Multigrid process. This using a fixed low-order approximation polynomial degree on a coarse grid. We show that this approach is spectrally equivalent to the first preconditioner, and requires less time for computation. The third approach is a development of an efficient mathematical framework for distributed data structure. This allows a high performance, massively parallel, implementation of the discontinuous Galerkin method. We demonstrate that it is possible to exploit properties of the system matrix and C1 polynomials as basis functions to optimize the parallel structures. The previously mentioned parallel data structure allows us to parallelize at the same time both the matrix-vector multiplication routines for the Conjugate Gradients method, as well as the preconditioner routines on the solver level. This minimizes the transfer ratio amongst the distributed system. Finally, we combined all three approaches and created a framework, which allowed us to successfully implement all of the above.
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Schlachter, Louisa [Verfasser]. "Stochastic Galerkin Methods in Hyperbolic Equations / Louisa Schlachter." München : Verlag Dr. Hut, 2021. http://d-nb.info/1232847550/34.
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Metcalfe, Stephen Arthur. "Adaptive discontinuous Galerkin methods for nonlinear parabolic problems." Thesis, University of Leicester, 2015. http://hdl.handle.net/2381/32041.
Full textAbstract:
This work is devoted to the study of a posteriori error estimation and adaptivity in parabolic problems with a particular focus on spatial discontinuous Galerkin (dG) discretisations. We begin by deriving an a posteriori error estimator for a linear non-stationary convection-diffusion problem that is discretised with a backward Euler dG method. An adaptive algorithm is then proposed to utilise the error estimator. The effectiveness of both the error estimator and the proposed algorithm is shown through a series of numerical experiments. Moving on to nonlinear problems, we investigate the numerical approximation of blow-up. To begin this study, we first look at the numerical approximation of blow-up in nonlinear ODEs through standard time stepping schemes. We then derive an a posteriori error estimator for an implicit-explicit (IMEX) dG discretisation of a semilinear parabolic PDE with quadratic nonlinearity. An adaptive algorithm is proposed that uses the error estimator to approach the blow-up time. The adaptive algorithm is then applied in a series of test cases to gauge the effectiveness of the error estimator. Finally, we consider the adaptive numerical approximation of a nonlinear interface problem that is used to model the mass transfer of solutes through semi-permiable membranes. An a posteriori error estimator is proposed for the IMEX dG discretisation of the model and its effectiveness tested through a series of numerical experiments.
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Wu, Wei. "Petrov-Galerkin methods for parabolic convection-diffusion problems." Thesis, University of Oxford, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.670384.
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Garcia, Bautista Javier. "Multiwavelet-based hp-adaptation for discontinuous Galerkin methods." Thesis, Ecole centrale de Nantes, 2022. http://www.theses.fr/2022ECDN0046.
Full textAbstract:
L’objectif principal de cette thèse est de développer une méthode hp-adaptative efficace en termes de coût et précision pour les schémas Galerkin discontinus appliqués aux équations de Navier-Stokes, en combinant flexibilité de l’adaptation a posteriori et précision de l’adaptation multi-résolution. Les performances de l’algorithme d’adaptation hp sont illustrées sur plusieurs cas d’écoulements stationnaires en une et deux dimensions. La première direction de recherche emploie une nouvelle méthodologie basée sur les multiondelettes pour estimer l’erreur de discrétisation de la solution numérique dans le contexte de simulations avec adaptation h. Les résultats démontrent clairement la viabilité de cette méthode pour atteindre un gain de calcul significatif par rapport àun raffinement de maillage uniforme. La deuxième voie de recherche aborde l’analyse et le développement d’une nouvelle stratégied’adaptation hp basée sur la décroissance du spectre des multi-ondelettes comme critère adaptation hp. Cette stratégie permet de discriminer avec succès les régions caractérisées par une grande régularité de celles contenant des phénomènes discontinus. De manière remarquable, l’algorithme d’adaptation hp est capable d’atteindre une haute précision caractéristique des solutions numériques d’ordre élevé tout en évitant les oscillations indésirables en adoptant des approximations d’ordre réduit à proximité des singularitésThe main objective of the present thesis is to devise, construct and validate computationally efficient hp-adaptive discontinuous Galerkin schemes of the Navier-Stokes equations by bringing together the flexibility of a posteriori error driven adaptation and the accuracy of multiresolution-based adaptation. The performance of the hp-algorithm is illustrated by several steady flows in one and two dimensions.The first research direction employs a new multiwavelet-based methodology to estimate the discretization error of the numerical solution in the context of h-adaptive simulations. The results certainly demonstrate the viability of h-refinement to reach a significant computational gain with respect to uniformly refined grids. The second line of investigation addresses the analysis and development of a new hp-adaptive strategy based on the decay of the multiwavelet spectrum to drive hp-adaptive simulations. The strategy successfully discriminates between regions characterized by high regularity and discontinuous phenomena and their vicinity. Remarkably, the developed hp-adaptation algorithm is able to achieve the high accuracy characteristic of high-order numerical solutions while avoiding unwanted oscillations by adopting low-order approximations in the proximity of singularities
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Dobrev, Veselin Asenov. "Preconditioning of discontinuous Galerkin methods for second order elliptic problems." [College Station, Tex. : Texas A&M University, 2007. http://hdl.handle.net/1969.1/ETD-TAMU-2531.
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Akyazi, Fatma Dilay. "Element-free Galerkin Method For Plane Stress Problems." Master's thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/12611685/index.pdf.
Full textAbstract:
In this study, the Element-Free Galerkin (EFG) method has been used for the analysis of plane stress problems. A computer program has been developed by using FORTRAN language. The moving least squares (MLS) approximation has been used in generating shape functions. The results obtained by the EFG method have been compared with analytical solution and the numerical results obtained by MSC. Patran/Nastran. The comparisons show that the mesh free method gives more accurate results than the finite element approximation with less computational effort.
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Yucel, Hamdullah. "Adaptive Discontinuous Galerkin Methods For Convectiondominated Optimal Control Problems." Phd thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614523/index.pdf.
Full textAbstract:
Many real-life applications such as the shape optimization of technological devices, the identification
of parameters in environmental processes and flow control problems lead to optimization
problems governed by systems of convection diusion partial dierential equations
(PDEs). When convection dominates diusion, the solutions of these PDEs typically exhibit
layers on small regions where the solution has large gradients. Hence, it requires special numerical
techniques, which take into account the structure of the convection. The integration
of discretization and optimization is important for the overall eciency of the solution process.
Discontinuous Galerkin (DG) methods became recently as an alternative to the finite
dierence, finite volume and continuous finite element methods for solving wave dominated
problems like convection diusion equations since they possess higher accuracy.
This thesis will focus on analysis and application of DG methods for linear-quadratic convection
dominated optimal control problems. Because of the inconsistencies of the standard stabilized
methods such as streamline upwind Petrov Galerkin (SUPG) on convection diusion
optimal control problems, the discretize-then-optimize and the optimize-then-discretize do not commute. However, the upwind symmetric interior penalty Galerkin (SIPG) method leads to
the same discrete optimality systems. The other DG methods such as nonsymmetric interior
penalty Galerkin (NIPG) and incomplete interior penalty Galerkin (IIPG) method also yield
the same discrete optimality systems when penalization constant is taken large enough. We
will study a posteriori error estimates of the upwind SIPG method for the distributed unconstrained
and control constrained optimal control problems. In convection dominated optimal
control problems with boundary and/or interior layers, the oscillations are propagated downwind
and upwind direction in the interior domain, due the opposite sign of convection terms in
state and adjoint equations. Hence, we will use residual based a posteriori error estimators to
reduce these oscillations around the boundary and/or interior layers. Finally, theoretical analysis
will be confirmed by several numerical examples with and without control constraints
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Jakobsson, Håkan. "Discontinous Galerkin Methods for Coupled Flow and Transport problems." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-51335.
Full textAbstract:
We investigate the use of discontinuous Galerkin finite element methods in a mul- tiphysics setting involving coupled flow and transport in porous media. We solve an elliptic equation for the fluid pressure using Nitsche’s method and an approxima- tion, Σ, of the exact convection field σ will be constructed by interpolation onto the lowest-order Raviart-Thomas space of functions. We sequentially solve the transport equation, with the convection field Σ, for the fluid saturation by use of the lowest order discontinuous Galerkin method. We also supply numerical evidence of the importance of local conservation in this setting, and furthermore propose a line of argument indicating that if Σ is constructed using conservative fluxes, the modeling error σ − Σ may not have a great impact on the total error in certain quantities of interest.
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Wei, Xiaoxi. "Mixed discontinuous Galerkin finite element methods for incompressible magnetohydrodynamics." Thesis, University of British Columbia, 2011. http://hdl.handle.net/2429/34789.
Full textAbstract:
We develop and analyze mixed discontinuous Galerkin finite element methods for the numerical approximation
of incompressible magnetohydrodynamics problems.
Incompressible magnetohydrodynamics is the area of physics that is
concerned with the behaviour of electrically conducting,
resistive, incompressible and viscous fluids in the presence of
electromagnetic fields. It is modelled
by a system of nonlinear partial differential equations, which couples the
Navier-Stokes equations with the Maxwell equations.
In the first part of this thesis, we introduce an interior penalty discontinuous
Galerkin method for the numerical approximation of a linearized incompressible magnetohydrodynamics
problem. The fluid
unknowns are discretized with the discontinuous
℘k-℘k-1 element pair, whereas the magnetic
variables are approximated by discontinuous ℘k-℘k+1 elements. Under minimal regularity
assumptions, we carry out a complete a priori error analysis and
prove that the energy norm error is optimally convergent in the
mesh size in general polyhedral domains, thus guaranteeing the
numerical resolution of the strongest magnetic singularities in
non-convex domains.
In the second part of this thesis, we propose and analyze a new mixed discontinuous Galerkin finite element method for the approximation of a fully nonlinear
incompressible magnetohydrodynamics model. The velocity field is
now discretized by divergence-conforming Brezzi-Douglas-Marini
elements, and the magnetic field by curl-conforming
Nédélec elements. In addition to correctly capturing magnetic
singularities, the method yields exactly divergence-free
velocity approximations, and is thus energy-stable. We show that the energy norm error is
convergent in the mesh size in possibly non-convex polyhedra, and
derive slightly suboptimal a priori error estimates under minimal regularity and small data
assumptions.
Finally, in the third part of this thesis, we present two extensions of our discretization techniques
to time-dependent incompressible
magnetohydrodynamics problems and to Stokes problems with nonstandard
boundary conditions.
All our discretizations and theoretical results are computationally
validated through comprehensive sets of numerical experiments.
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Georgoulis, Emmanuil H. "Discontinuous Galerkin methods on shape-regular and anisotropic meshes." Thesis, University of Oxford, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.270366.
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Hill, Antony A. "Convection in porous media and Legendre, Chebyshev Galerkin methods." Thesis, Durham University, 2005. http://etheses.dur.ac.uk/2356/.
Full textAbstract:
The subject of thermal convection in fluid and porous media is investigated, coupled with the development of efficient spectral finite element methods to improve on the more commonly used techniques for these types of problems. Convection induced by the selective absorption of radiation in a porous medium is investigated in the first four chapters. For the Darcy and Brinkman models for fluid flow the thresholds of the linear and nonlinear theories are shown to be extremely close, demonstrating that the linear theory is accurate enough to predict the onset of convective motion. The exploration of a quadratically modelled internal heat source is discussed next. It is shown that the linear and nonlinear thresholds are close unless the quadratic term becomes dominant over the linear term. Developing a double- diffusive model yields a critical parameter for which no oscillatory convection occurs when it is exceeded. This is an unobserved phenomenon in the present literature. Thermal convection in a linearly viscous fluid in a finite box is also explored. It is demonstrated that the linear and nonlinear thresholds do not coincide, which contradicts results by Georgescu & Mansutti [25]. Legendre and Chebyshev polynomial based spectral methods are also developed for the evaluation of eigenvalues and eigenfunctions inherent in stability analysis in porous media, drawing on the experience of the implementation of the well established techniques in the previous work. These generate sparse matrices, where the standard homogeneous boundary conditions for both porous and fluid media problems are contained within the method.
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Jensen, Max. "Discontinuous Galerkin methods for Friedrichs systems with irregular solutions." Thesis, University of Oxford, 2005. http://sro.sussex.ac.uk/45497/.
Full textAbstract:
This work is concerned with the numerical solution of Friedrichs systems by discontinuous Galerkin finite element methods (DGFEMs). Friedrichs systems are boundary value problems with symmetric, positive, linear first-order partial differential operators and allow the unified treatment of a wide range of elliptic, parabolic, hyperbolic and mixed-type equations. We do not assume that the exact solution of a Friedrichs system belongs to a Sobolev space, but only require that it is contained in the associated graph space, which amounts to differentiability in the characteristic direction. We show that the numerical approximations to the solution of a Friedrichs system by the DGFEM converge in the energy norm under hierarchical h- and p- refinement. We introduce a new compatibility condition for the boundary data, from which we can deduce, for instance, the validity of the integration-by-parts formula. Consequently, we can admit domains with corners and allow changes of the inertial type of the boundary, which corresponds in special cases to the componentwise transition from in- to outflow boundaries. To establish the convergence result we consider in equal parts the theory of graph spaces, Friedrichs systems and DGFEMs. Based on the density of smooth functions in graph spaces over Lipschitz domains, we study trace and extension operators and also investigate the eigensystem associated with the differential operator. We pay particular attention to regularity properties of the traces, that limit the applicability of energy integral methods, which are the theoretical underpinning of Friedrichs systems. We provide a general framework for Friedrichs systems which incorporates a wide range of singular boundary conditions. Assuming the aforementioned compatibility condition we deduce well-posedness of admissible Friedrichs systems and the stability of the DGFEM. In a separate study we prove hp-optimality of least-squares stabilised DGFEMs.
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Frean, Daniel. "Discontinuous Galerkin methods : exploiting superconvergence for improved time-stepping." Thesis, University of East Anglia, 2017. https://ueaeprints.uea.ac.uk/66543/.
Full textAbstract:
The discontinuous Galerkin (DG) methods are one of the most extensively researched classes of numerical methods for solving partial dfferential equations that display convective or diffusive qualities and have been popularly adopted by the scientific and engineering communities as a method capable of achieving arbitrary orders of accuracy in space. The choice of numerical flux function plays a pivotal role in the successful construction of DG methods and has an intrinsic effect on the superconvergence properties. As an inherent property of the spatial discretisation, superconvergence can only be retained in the solution through a sensitive pairing with a time integrator. The results of the literature and of this work suggest that an improved pairing between the spatial and temporal discretisations is both desirable and possible. We perform analysis of three different but related manifestations of superconvergence: the local, super-accurate points themselves; the subsequent global extraction via the Smoothness-Increasing Accuracy-Conserving (SIAC) filters; and the spectral properties that quantify, in terms of dispersion and dissipation errors, how accurately waves are convected. In order to explore the effect of the numerical flux function on superconvergence, we consider a generalisation of the “natural" upwind choice for a Method of Lines solution to the linear advection equation: the upwind-biased flux. We prove that the method is locally superconvergent at roots of a linear combination of the left- and right-Radau polynomials dependent on the value of a flux parameter and that the use of SIAC filters is still able to draw out the superconvergence information and create a globally smooth and superconvergent solution. In exploring the coupling of DG with a time integrator, we introduce a new scheme to a class of multi-stage multi-derivative methods, following recent incorporation of local DG technologies to recover superconvergence and achieve improved wave propagation properties.
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Bennison, Tom. "Adaptive discontinuous Galerkin methods for the neutron transport equation." Thesis, University of Nottingham, 2014. http://eprints.nottingham.ac.uk/28944/.
Full textAbstract:
In this thesis we study the neutron transport (Boltzmann transport equation) which is used to model the movement of neutrons inside a nuclear reactor. More specifically we consider the mono-energetic, time independent neutron transport equation. The neutron transport equation has predominantly been solved numerically by employing low order discretisation methods, particularly in the case of the angular domain. We proceed by surveying the advantages and disadvantages of common numerical methods developed for the numerical solution of the neutron transport equation before explaining our choice of using a discontinuous Galerkin (DG) discretisation for both the spatial and angular domain. The bulk of the thesis describes an arbitrary order in both angle and space solver for the neutron transport equation. We discuss some implementation issues, including the use of an ordered solver to facilitate the solution of the linear systems resulting from the discretisation. The resulting solver is benchmarked using both source and critical eigenvalue computations. In the pseudo three--dimensional case we employ our solver for the computation of the critical eigenvalue for three industrial benchmark problems. We then employ the Dual Weighted Residual (DWR) approach to adaptivity to derive and implement error indicators for both two--dimensional and pseudo three--dimensional neutron transport source problems. Finally, we present some preliminary results on the use of a DWR indicator for the eigenvalue problem.
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Arranz, Carreño Aurelio. "Discontinuous Galerkin methods for elasticity and crack propagation problems." Thesis, University of Oxford, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.558688.
Full textAbstract:
A new set of numerical methods for predictive modelling of quasistatic and dynamic crack propagation in brittle materials for aerospace applications based upon the Symmetric Interior Penalty Galerkin Method (SIPG) and the Nonsymmetric Interior Penalty Galerkin Method (NIPG) is proposed. This new approach to solving problems in Computational Fracture Mechanics belongs to the family of Discontinuous Galerkin Finite Element Meth- ods (DGFEMs) and draws on the qualities of both the classical Finite Element Methods (FEMs) and Finite Volume Methods (FVMs) in order to overcome the shortcomings and limitations which both methods exhibit when trying to address the transition from contin- uum to discontinuum during material fracture. This thesis focuses mainly on the numerical linear algebra aspects associated with the two above DGFEM methods when applied to crack propagation problems. A new precondi- tioning technique for the iterative solution of linear systems of equations is introduced and its qualities are presented by means of numerical examples. A coupling technique with pre- conditioners for conforming approximations is also introduced. Several improvements and extensions to the original technique are presented to make crack propagation simulations with the SIPG viable. Results for both quasistatic and dynamic fracture propagation are presented. Practical aspects ofthe implementation are also discussed, revealing the important issues that still have to be addressed and that constitute paths for further research.
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Wang, Siyang. "Finite Difference and Discontinuous Galerkin Methods for Wave Equations." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-320614.
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Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.
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Naddei, Fabio. "Adaptive Large Eddy Simulations based on discontinuous Galerkin methods." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLX060/document.
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L'objectif principal de ce travail est d'améliorer la précision et l'efficacité des modèles LES au moyen des méthodes Galerkine discontinues (DG). Deux thématiques principales ont été étudiées: les stratégies d'adaptation spatiale et les modèles LES pour les méthodes d'ordre élevé.Concernant le premier thème, dans le cadre des méthodes DG la résolution spatiale peut être efficacement adaptée en modifiant localement soit le maillage (adaptation-h) soit le degré polynômial de la solution (adaptation-p). L'adaptation automatique de la résolution nécessite l'estimation des erreurs pour analyser la qualité de la solution locale et les exigences de résolution. L'efficacité de différentes stratégies de la littérature est comparée en effectuant des simulations h- et p-adaptatives.Sur la base de cette étude comparative, des algorithmes statiques et dynamiques p-adaptatifs pour la simulation des écoulements instationnaires sont ensuite développés et analysés. Les simulations numériques réalisées montrent que les algorithmes proposés peuvent réduire le coût de calcul des simulations des écoulements transitoires et statistiquement stationnaires.Un nouvel estimateur d'erreur est ensuite proposé. Il est local, car n'exige que des informations de l'élément et de ses voisins directs, et peut être calculé en cours de simulation pour un coût limité. Il est démontré que l'algorithme statique p-adaptatif basé sur cet estimateur d'erreur peut être utilisé pour améliorer la précision des simulations LES sur des écoulements turbulents statistiquement stationnaires.Concernant le second thème, une nouvelle méthode, consistante avec la discrétisation DG, est développée pour l'analyse a-priori des modèles DG-LES à partir des données DNS. Elle permet d'identifier le transfert d'énergie idéal entre les échelles résolues et non résolues. Cette méthode est appliquée à l'analyse de l'approche VMS (Variational Multiscale). Il est démontré que pour les résolutions fines, l'approche DG-VMS est capable de reproduire le transfert d'énergie idéal. Cependant, pour les résolutions grossières, typique de la LES à nombres de Reynolds élevés, un meilleur accord peut être obtenu en utilisant un modèle mixte Smagorinsky-VMSThe main goal of this work is to improve the accuracy and computational efficiency of Large Eddy Simulations (LES) by means of discontinuous Galerkin (DG) methods. To this end, two main research topics have been investigated: resolution adaptation strategies and LES models for high-order methods.As regards the first topic, in the framework of DG methods the spatial resolution can be efficiently adapted by modifying either the local mesh size (h-adaptation) or the degree of the polynomial representation of the solution (p-adaptation).The automatic resolution adaptation requires the definition of an error estimation strategy to analyse the local solution quality and resolution requirements.The efficiency of several strategies derived from the literature are compared by performing p- and h-adaptive simulations. Based on this comparative study a suitable error indicator for the adaptive scale-resolving simulations is selected.Both static and dynamic p-adaptive algorithms for the simulation of unsteady flows are then developed and analysed. It is demonstrated by numerical simulations that the proposed algorithms can provide a reduction of the computational cost for the simulation of both transient and statistically steady flows.A novel error estimation strategy is then introduced. It is local, requiring only information from the element and direct neighbours, and can be computed at run-time with limited overhead. It is shown that the static p-adaptive algorithm based on this error estimator can be employed to improve the accuracy for LES of statistically steady turbulent flows.As regards the second topic, a novel framework consistent with the DG discretization is developed for the a-priori analysis of DG-LES models from DNS databases. It allows to identify the ideal energy transfer mechanism between resolved and unresolved scales.This approach is applied for the analysis of the DG Variational Multiscale (VMS) approach. It is shown that, for fine resolutions, the DG-VMS approach is able to replicate the ideal energy transfer mechanism.However, for coarse resolutions, typical of LES at high Reynolds numbers, a more accurate agreement is obtained by a mixed Smagorinsky-VMS model
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Thompson, Ross Anthony. "Galerkin Projections Between Finite Element Spaces." Thesis, Virginia Tech, 2015. http://hdl.handle.net/10919/52968.
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Adaptive mesh refinement schemes are used to find accurate low-dimensional approximating spaces when solving elliptic PDEs with Galerkin finite element methods. For nonlinear PDEs, solving the nonlinear problem with Newton's method requires an initial guess of the solution on a refined space, which can be found by interpolating the solution from a previous refinement. Improving the accuracy of the representation of the converged solution computed on a coarse mesh for use as an initial guess on the refined mesh may reduce the number of Newton iterations required for convergence. In this thesis, we present an algorithm to compute an orthogonal L^2 projection between two dimensional finite element spaces constructed from a triangulation of the domain. Furthermore, we present numerical studies that investigate the efficiency of using this algorithm to solve various nonlinear elliptic boundary value problems.Master of Science
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Voonna, Kiran. "Development of discontinuous galerkin method for 1-D inviscid burgers equation." ScholarWorks@UNO, 2003. http://louisdl.louislibraries.org/u?/NOD,75.
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Thesis (M.S.)--University of New Orleans, 2003.Title from electronic submission form. "A thesis ... in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mechanical Engineering"--Thesis t.p. Vita. Includes bibliographical references.