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Journal articles on the topic 'Raviart-Thomas element'

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Published: 6 September 2023

Last updated: 7 September 2023

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1

Bartels, Sören, and Zhangxian Wang. "Orthogonality relations of Crouzeix–Raviart and Raviart–Thomas finite element spaces." Numerische Mathematik 148, no. 1 (May 2021): 127–39. http://dx.doi.org/10.1007/s00211-021-01199-3.

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AbstractIdentities that relate projections of Raviart–Thomas finite element vector fields to discrete gradients of Crouzeix–Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix–Ravaiart space onto elementwise constant functions are deduced.

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2

Kobayashi, Kenta, and Takuya Tsuchiya. "Error analysis of Crouzeix–Raviart and Raviart–Thomas finite element methods." Japan Journal of Industrial and Applied Mathematics 35, no. 3 (September 6, 2018): 1191–211. http://dx.doi.org/10.1007/s13160-018-0325-9.

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3

Lu, Zuliang, Yanping Chen, and Weishan Zheng. "A Posteriori Error Estimates of Lowest Order Raviart-Thomas Mixed Finite Element Methods for Bilinear Optimal Control Problems." East Asian Journal on Applied Mathematics 2, no. 2 (May 2012): 108–25. http://dx.doi.org/10.4208/eajam.130212.300312a.

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AbstractA Raviart-Thomas mixed finite element discretization for general bilinear optimal control problems is discussed. The state and co-state are approximated by lowest order Raviart-Thomas mixed finite element spaces, and the control is discretized by piecewise constant functions. A posteriori error estimates are derived for both the coupled state and the control solutions, and the error estimators can be used to construct more efficient adaptive finite element approximations for bilinear optimal control problems. An adaptive algorithm to guide the mesh refinement is also provided. Finally, we present a numerical example to demonstrate our theoretical results.

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4

Carstensen, Carsten. "Explicit Error Estimates for Courant, Crouzeix-Raviart and Raviart-Thomas Finite Element Methods." Journal of Computational Mathematics 30, no. 4 (June 2012): 337–53. http://dx.doi.org/10.4208/jcm.1108-m3677.

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5

VOHRALÍK, MARTIN, and BARBARA I. WOHLMUTH. "MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS." Mathematical Models and Methods in Applied Sciences 23, no. 05 (February 21, 2013): 803–38. http://dx.doi.org/10.1142/s0218202512500613.

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In this paper, we study the mixed finite element method for linear diffusion problems. We focus on the lowest-order Raviart–Thomas case. For simplicial meshes, we propose several new approaches to reduce the original indefinite saddle point systems for the flux and potential unknowns to (positive definite) systems for one potential unknown per element. Our construction principle is closely related to that of the so-called multi-point flux-approximation method and leads to local flux expressions. We present a set of numerical examples illustrating the influence of the elimination process on the structure and on the condition number of the reduced matrix. We also discuss different versions of the discrete maximum principle in the lowest-order Raviart–Thomas method. Finally, we recall mixed finite element methods on general polygonal meshes and show that they are a special type of the mimetic finite difference, mixed finite volume, and hybrid finite volume family.

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6

Dubois, Francois, Isabelle Greff, and Charles Pierre. "Raviart–Thomas finite elements of Petrov–Galerkin type." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 5 (August 6, 2019): 1553–76. http://dx.doi.org/10.1051/m2an/2019020.

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Finite volume methods are widely used, in particular because they allow an explicit and local computation of a discrete gradient. This computation is only based on the values of a given scalar field. In this contribution, we wish to achieve the same goal in a mixed finite element context of Petrov–Galerkin type so as to ensure a local computation of the gradient at the interfaces of the elements. The shape functions are the Raviart–Thomas finite elements. Our purpose is to define test functions that are in duality with these shape functions: precisely, the shape and test functions will be asked to satisfy some orthogonality property. This paradigm is addressed for the discrete solution of the Poisson problem. The general theory of Babuška brings necessary and sufficient stability conditions for a Petrov–Galerkin mixed problem to be convergent. In order to ensure stability, we propose specific constraints for the dual test functions. With this choice, we prove that the mixed Petrov–Galerkin scheme is identical to the four point finite volume scheme of Herbin, and to the mass lumping approach developed by Baranger, Maitre and Oudin. Convergence is proven with the usual techniques of mixed finite elements.

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7

Braess, D., and R. Verfürth. "A Posteriori Error Estimators for the Raviart–Thomas Element." SIAM Journal on Numerical Analysis 33, no. 6 (December 1996): 2431–44. http://dx.doi.org/10.1137/s0036142994264079.

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8

Huang, Peiqi, Jinru Chen, and Mingchao Cai. "A Mortar Method Using Nonconforming and Mixed Finite Elements for the Coupled Stokes-Darcy Model." Advances in Applied Mathematics and Mechanics 9, no. 3 (January 17, 2017): 596–620. http://dx.doi.org/10.4208/aamm.2016.m1397.

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AbstractIn this work, we study numerical methods for a coupled fluid-porous media flow model. The model consists of Stokes equations and Darcy's equations in two neighboring subdomains, coupling together through certain interface conditions. The weak form for the coupled model is of saddle point type. A mortar finite element method is proposed to approximate the weak form of the coupled problem. In our method, nonconforming Crouzeix-Raviart elements are applied in the fluid subdomain and the lowest order Raviart-Thomas elements are applied in the porous media subdomain; Meshes in different subdomains are allowed to be nonmatching on the common interface; Interface conditions are weakly imposed via adding constraint in the definition of the finite element space. The well-posedness of the discrete problem and the optimal error estimate for the proposed method are established. Numerical experiments are also given to confirm the theoretical results.

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9

Lashuk, I. V., and P. S. Vassilevski. "Element agglomeration coarse Raviart-Thomas spaces with improved approximation properties." Numerical Linear Algebra with Applications 19, no. 2 (January 13, 2012): 414–26. http://dx.doi.org/10.1002/nla.1819.

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10

Lu, Zuliang. "Adaptive Mixed Finite Element Methods for Parabolic Optimal Control Problems." Mathematical Problems in Engineering 2011 (2011): 1–21. http://dx.doi.org/10.1155/2011/217493.

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We will investigate the adaptive mixed finite element methods for parabolic optimal control problems. The state and the costate are approximated by the lowest-order Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise constant elements. We derive a posteriori error estimates of the mixed finite element solutions for optimal control problems. Such a posteriori error estimates can be used to construct more efficient and reliable adaptive mixed finite element method for the optimal control problems. Next we introduce an adaptive algorithm to guide the mesh refinement. A numerical example is given to demonstrate our theoretical results.

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11

Fortin, Michel, and Abdellatif Serghini Mounim. "Mixed and Hybrid Finite Element Methods for Convection-Diffusion Problems and Their Relationships with Finite Volume: The Multi-Dimensional Case." Journal of Mathematics Research 9, no. 1 (January 9, 2017): 68. http://dx.doi.org/10.5539/jmr.v9n1p68.

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We introduced in (Fortin & Serghini Mounim, 2005) a new method which allows us to extend the connection between the finite volume and dual mixed hybrid (DMH) methods to advection-diffusion problems in the one-dimensional case. In the present work we propose to extend the results of (Fortin & Serghini Mounim, 2005) to multidimensional hyperbolic and parabolic problems. The numerical approximation is achieved using the Raviart-Thomas (Raviart & Thomas, 1977) finite elements of lowest degree on triangular or rectangular partitions. We show the link with numerous finite volume schemes by use of appropriate numerical integrations. This will permit a better understanding of these finite volume schemes and the large number of DMH results available could carry out their analysis in a unified fashion. Furthermore, a stabilized method is proposed. We end with some discussion on possible extensions of our schemes.

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12

Lu, Zuliang, and Xiao Huang. "A Priori Error Estimates of Mixed Finite Element Methods for General Linear Hyperbolic Convex Optimal Control Problems." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/547490.

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The aim of this work is to investigate the discretization of general linear hyperbolic convex optimal control problems by using the mixed finite element methods. The state and costate are approximated by thekorder (k≥0) Raviart-Thomas mixed finite elements and the control is approximated by piecewise polynomials of orderk. By applying the elliptic projection operators and Gronwall’s lemma, we derive a priori error estimates of optimal order for both the coupled state and the control approximation.

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13

Spiridonov, Denis, Jian Huang, Maria Vasilyeva, Yunqing Huang, and Eric T. Chung. "Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model." Mathematics 7, no. 12 (December 10, 2019): 1212. http://dx.doi.org/10.3390/math7121212.

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In this paper, the solution of the Darcy-Forchheimer model in high contrast heterogeneous media is studied. This problem is solved by a mixed finite element method (MFEM) on a fine grid (the reference solution), where the pressure is approximated by piecewise constant elements; meanwhile, the velocity is discretized by the lowest order Raviart-Thomas elements. The solution on a coarse grid is performed by using the mixed generalized multiscale finite element method (mixed GMsFEM). The nonlinear equation can be solved by the well known Picard iteration. Several numerical experiments are presented in a two-dimensional heterogeneous domain to show the good applicability of the proposed multiscale method.

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14

Ishizaka, Hiroki, Kenta Kobayashi, and Takuya Tsuchiya. "Crouzeix–Raviart and Raviart–Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition." Japan Journal of Industrial and Applied Mathematics 38, no. 2 (February 9, 2021): 645–75. http://dx.doi.org/10.1007/s13160-020-00455-7.

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15

Bochev, Pavel B., and Denis Ridzal. "Rehabilitation of the Lowest-Order Raviart–Thomas Element on Quadrilateral Grids." SIAM Journal on Numerical Analysis 47, no. 1 (January 2009): 487–507. http://dx.doi.org/10.1137/070704265.

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16

Hua, Yuchun, and Yuelong Tang. "Superconvergence of Semidiscrete Splitting Positive Definite Mixed Finite Elements for Hyperbolic Optimal Control Problems." Advances in Mathematical Physics 2022 (January 6, 2022): 1–10. http://dx.doi.org/10.1155/2022/3520668.

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In this paper, we consider semidiscrete splitting positive definite mixed finite element methods for optimal control problems governed by hyperbolic equations with integral constraints. The state and costate are approximated by the lowest order Raviart-Thomas mixed rectangular finite element, and the control is approximated by piecewise constant functions. We derive some convergence and superconvergence results for the control, the state and the adjoint state. A numerical example is provided to demonstrate our theoretical results.

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17

Hernández, E. "Approximation of the Vibration Modes of a Plate and Shells Coupled With a Fluid." Journal of Applied Mechanics 73, no. 6 (January 5, 2006): 1005–10. http://dx.doi.org/10.1115/1.2173675.

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We consider a method to compute the vibration modes of an elastic thin structure (shell or plate) in contact with a compressible fluid. For the structure, the classical Naghdi equations, based on the Reissner–Mindlin hypothesis, are considered and its approximation using the mixed interpolation of tensorial component 4 finite element method. The fluid equations are discretized by using Raviart–Thomas elements, and a non-conforming coupling is used on the fluid-solid interface. Numerical experiments are reported, assessing the efficiency of this coupled scheme.

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18

Manickam, K., and P. Prakash. "Mixed Finite Element Methods for Fourth Order Elliptic Optimal Control Problems." Numerical Mathematics: Theory, Methods and Applications 9, no. 4 (November 2016): 528–48. http://dx.doi.org/10.4208/nmtma.2016.m1405.

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AbstractIn this paper, a priori error estimates are derived for the mixed finite element discretization of optimal control problems governed by fourth order elliptic partial differential equations. The state and co-state are discretized by Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. The error estimates derived for the state variable as well as those for the control variable seem to be new. We illustrate with a numerical example to confirm our theoretical results.

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19

Zhu, Ailing. "Discontinuous Mixed Covolume Methods for Linear Parabolic Integrodifferential Problems." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/649468.

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The semidiscrete and fully discrete discontinuous mixed covolume schemes for the linear parabolic integrodifferential problems on triangular meshes are proposed. The error analysis of the semidiscrete and fully discrete discontinuous mixed covolume scheme is presented and the optimal order error estimate in discontinuousH(div)and first-order error estimate inL2are obtained with the lowest order Raviart-Thomas mixed element space.

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20

Hou, Tianliang, and Li Li. "Error Estimates of Mixed Methods for Optimal Control Problems Governed by General Elliptic Equations." Advances in Applied Mathematics and Mechanics 8, no. 6 (September 19, 2016): 1050–71. http://dx.doi.org/10.4208/aamm.2014.m807.

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AbstractIn this paper, we investigate the error estimates of mixed finite element methods for optimal control problems governed by general elliptic equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive L2 and H–1-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.

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21

Glowinski, Roland, and Serguei Lapin. "Solution of a Wave Equation by a Mixed Finite Element - Fictitious Domain Method." Computational Methods in Applied Mathematics 4, no. 4 (2004): 431–44. http://dx.doi.org/10.2478/cmam-2004-0024.

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AbstractThe main goal of this article is to investigate the capability of fictitious domain methods to simulate the scattering of linear waves by an obstacle whose shape does not fit the mesh. The space-time discretization relies on a combination of a mixed finite element method µa la Raviart-Thomas with a fairly standard finite difference scheme for the time discretization. The numerical results described in the article point to a good performance of the numerical method investigated here.

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22

Chen, Yanping, Tianliang Hou, and Weishan Zheng. "Error Estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity." Advances in Applied Mathematics and Mechanics 4, no. 06 (December 2012): 751–68. http://dx.doi.org/10.4208/aamm.12-12s05.

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AbstractIn this paper, we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We deriveL2andL∞-error estimates for the control variable. Moreover, using a recovery operator, we also derive some superconvergence results for the control variable. Finally, a numerical example is given to demonstrate the theoretical results.

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23

MING, PINGBING, and ZHONG-CI SHI. "TWO NONCONFORMING QUADRILATERAL ELEMENTS FOR THE REISSNER–MINDLIN PLATE." Mathematical Models and Methods in Applied Sciences 15, no. 10 (October 2005): 1503–17. http://dx.doi.org/10.1142/s0218202505000868.

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We construct two low order nonconforming quadrilateral elements for the Reissner–Mindlin plate. The first one consists of a modified nonconforming rotated Q1 element for one component of the rotation and the standard four-node isoparametric element for the other component as well as for the the approximation of the transverse displacement, a modified rotated Raviart–Thomas interpolation operator is employed as the shear reduction operator. The second differs from the first only in the approximation of the rotation, which employs the modified rotated Q1 element for both components of the rotation, and a jump term accounting the discontinuity of the rotation approximation is included in the variational formulation. Both elements give optimal error bounds uniform in the plate thickness with respect to the energy norm as well as the L2 norm.

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24

Gillette, Andrew, Alexander Rand, and Chandrajit Bajaj. "Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes." Computational Methods in Applied Mathematics 16, no. 4 (October 1, 2016): 667–83. http://dx.doi.org/10.1515/cmam-2016-0019.

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AbstractWe combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart–Thomas, and Brezzi–Douglas–Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties.

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25

Chen, Jinru, and Likang Li. "Preconditioning projection nonconforming element method for the lowest-order Raviart-Thomas mixed triangular element method." Applied Mathematics and Computation 93, no. 1 (July 1998): 31–49. http://dx.doi.org/10.1016/s0096-3003(97)10112-6.

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26

BERMÚDEZ, ALFREDO, PABLO GAMALLO, and RODOLFO RODRÍGUEZ. "AN HEXAHEDRAL FACE ELEMENT METHOD FOR THE DISPLACEMENT FORMULATION OF STRUCTURAL ACOUSTICS PROBLEMS." Journal of Computational Acoustics 09, no. 03 (September 2001): 911–18. http://dx.doi.org/10.1142/s0218396x01000851.

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Several finite element methods for the numerical computation of elastoacoustic vibrations are compared. They are applied to two formulations based on different variables to describe the fluid: presssure and displacement potential in one case, and displacements in the other. While the first one is discretized by standard Lagrangean finite elements for both variables, the second one is solved by "face" Raviart-Thomas elements. In each case we consider both tetrahedral and hexahedral meshes. Elastoacoustic eigenmodes have been computed for a test example by means of MATLAB implementations of all these methods. The numerical results allow us to compare all of them in terms of error versus number of degrees of freedom and computing time.

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27

Gatica, Gabriel N., Antonio Márquez, and Manuel A. Sánchez. "Pseudostress-Based Mixed Finite Element Methods for the Stokes Problem in ℝn with Dirichlet Boundary Conditions. I: A Priori Error Analysis." Communications in Computational Physics 12, no. 1 (July 2012): 109–34. http://dx.doi.org/10.4208/cicp.010311.041011a.

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AbstractWe consider a non-standard mixed method for the Stokes problem in ℝn, n Є {2,3}, with Dirichlet boundary conditions, in which, after using the incompressibility condition to eliminate the pressure, the pseudostress tensor σ and the velocity vector u become the only unknowns. Then, we apply the Babuška-Brezzi theory to prove the well-posedness of the corresponding continuous and discrete formulations. In particular, we show that Raviart-Thomas elements of order k≥0 for σ and piecewise polynomials of degree k for u ensure unique solvability and stability of the associated Galerkin scheme. In addition, we introduce and analyze an augmented approach for our pseudostress-velocity formulation. The methodology employed is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition for the velocity, all of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. For instance, Raviart-Thomas elements of order k≥0 for σ and continuous piecewise polynomials of degree k+1 for u become a feasible choice in this case. Finally, extensive numerical experiments illustrating the good performance of the methods and comparing them with other procedures available in the literature, are provided.

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28

Tang, Yuelong, and Yuchun Hua. "A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems." Open Mathematics 20, no. 1 (January 1, 2022): 629–45. http://dx.doi.org/10.1515/math-2022-0053.

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Abstract In this article, we consider fully discrete characteristic mixed finite elements for convection-diffusion optimal control problems. We use the characteristic line method to treat the hyperbolic part of the state equation as a directional derivative. The state and the co-state are discretized by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. Using some proper duality problems, we derive a posteriori error estimates for the scalar functions. Such estimates are not available in the literature. A numerical example is presented to validate the theoretical results.

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29

Chen, Yanping, and Zhuoqing Lin. "A Posteriori Error Estimates of Semidiscrete Mixed Finite Element Methods for Parabolic Optimal Control Problems." East Asian Journal on Applied Mathematics 5, no. 1 (February 2015): 85–108. http://dx.doi.org/10.4208/eajam.010314.110115a.

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AbstractA posteriori error estimates of semidiscrete mixed finite element methods for quadratic optimal control problems involving linear parabolic equations are developed. The state and co-state are discretised by Raviart-Thomas mixed finite element spaces of order k, and the control is approximated by piecewise polynomials of order k (k ≥ 0). We derive our a posteriori error estimates for the state and the control approximations via a mixed elliptic reconstruction method. These estimates seem to be unavailable elsewhere in the literature, although they represent an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

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30

Arnold, Douglas N., and Richard S. Falk. "Analysis of a Linear–Linear Finite Element for the Reissner–Mindlin Plate Model." Mathematical Models and Methods in Applied Sciences 07, no. 02 (March 1997): 217–38. http://dx.doi.org/10.1142/s0218202597000141.

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An analysis is presented for a recently proposed finite element method for the Reissner–Mindlin plate problem. The method is based on the standard variational principle, uses nonconforming linear elements to approximate the rotations and conforming linear elements to approximate the transverse displacements, and avoids the usual "locking problem" by interpolating the shear stress into a rotated space of lowest order Raviart-Thomas elements. When the plate thickness t = O(h), it is proved that the method gives optimal order error estimates uniform in t. However, the analysis suggests and numerical calculations confirm that the method can produce poor approximations for moderate sized values of the plate thickness. Indeed, for t fixed, the method does not converge as the mesh size h tends to zero.

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Yang, Lei. "A Mixed Element Method for the Desorption-Diffusion-Seepage Model of Gas Flow in Deformable Coalbed Methane Reservoirs." Mathematical Problems in Engineering 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/735931.

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We present a desorption-diffusion-seepage model for the gas flow problem in deformable coalbed methane reservoirs. Effects of fracture systems deformation on permeability have been considered in the proposed model. A mixed finite element method is introduced to solve the gas flow model, in which the coalbed gas pressure and velocity can be approximated simultaneously. Numerical experiments using the lowest order Raviart-Thomas (RT0) mixed element are carried out to describe the dynamic characteristics of gas pressure, velocity, and concentration. Error estimate results indicate that approximation solutions could achieve first-order convergence rates.

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Dond, Asha K., and Amiya K. Pani. "A Priori and A Posteriori Estimates of Conforming and Mixed FEM for a Kirchhoff Equation of Elliptic Type." Computational Methods in Applied Mathematics 17, no. 2 (April 1, 2017): 217–36. http://dx.doi.org/10.1515/cmam-2016-0041.

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AbstractIn this article, a priori and a posteriori estimates of conforming and expanded mixed finite element methods for a Kirchhoff equation of elliptic type are derived. For the expanded mixed finite element method, a variant of Brouwer’s fixed point argument combined with a monotonicity argument yields the well-posedness of the discrete nonlinear system. Further, a use of both Helmholtz decomposition of $L^{2}$-vector valued functions and the discrete Helmholtz decomposition of the Raviart–Thomas finite elements helps in a crucial way to achieve optimal a priori as well as a posteriori error bounds. For both conforming and expanded mixed form, reliable and efficient a posteriori estimators are established. Finally, the numerical experiments are performed to validate the theoretical convergence rates.

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33

Wohlmuth, Barbara I., and Ronald H. W. Hoppe. "A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements." Mathematics of Computation 68, no. 228 (May 19, 1999): 1347–79. http://dx.doi.org/10.1090/s0025-5718-99-01125-4.

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null, M. M. Guo, and D. J. Liu. "The Discrete Raviart-Thomas Mixed Finite Element Method for the $P$-Laplace Equation." International Journal of Numerical Analysis and Modeling 20, no. 3 (June 2023): 313–28. http://dx.doi.org/10.4208/ijnam2023-1012.

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BARRETT, JOHN W., and LEONID PRIGOZHIN. "A QUASI-VARIATIONAL INEQUALITY PROBLEM IN SUPERCONDUCTIVITY." Mathematical Models and Methods in Applied Sciences 20, no. 05 (May 2010): 679–706. http://dx.doi.org/10.1142/s0218202510004404.

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We derive a class of analytical solutions and a dual formulation of a scalar two-space-dimensional quasi-variational inequality problem in applied superconductivity. We approximate this formulation by a fully practical finite element method based on the lowest order Raviart–Thomas element, which yields approximations to both the primal and dual variables (the magnetic and electric fields). We prove the subsequence convergence of this approximation, and hence prove the existence of a solution to both the dual and primal formulations, for strictly star-shaped domains. The effectiveness of the approximation is illustrated by numerical examples with and without this domain restriction.

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Tang, Yuelong. "Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems." AIMS Mathematics 8, no. 5 (2023): 12506–19. http://dx.doi.org/10.3934/math.2023628.

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<abstract><p>In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order $ m = 1 $ Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results.</p></abstract>

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37

Churilova, Maria. "Analysis of marking criteria for mesh adaptation in Cosserat elasticity." MATEC Web of Conferences 245 (2018): 08004. http://dx.doi.org/10.1051/matecconf/201824508004.

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The article is devoted to comparison of finite element marking criteria for adaptive mesh refinement while solving plane Cosserat elasticity problems. The goal is to compare the resulting adaptive meshes obtained with different marking strategies. Mesh refinement and error control is done using the functional type a posteriori error majorant. Implemented algorithms use the zero-order Raviart-Thomas approximation on triangular meshes. Four widely used marking criteria are utilized for mesh adaptation. The comparative analysis is presented for two plane-strain problems.

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Brandts, Jan H. "Superconvergence for triangular order Raviart–Thomas mixed finite elements and for triangular standard quadratic finite element methods." Applied Numerical Mathematics 34, no. 1 (June 2000): 39–58. http://dx.doi.org/10.1016/s0168-9274(99)00034-3.

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39

Yang, Huaijun, and Dongyang Shi. "Superconvergence analysis of the lowest order rectangular Raviart–Thomas element for semilinear parabolic equation." Applied Mathematics Letters 105 (July 2020): 106280. http://dx.doi.org/10.1016/j.aml.2020.106280.

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40

Brenner, Susanne C. "A Multigrid Algorithm for the Lowest-Order Raviart–Thomas Mixed Triangular Finite Element Method." SIAM Journal on Numerical Analysis 29, no. 3 (June 1992): 647–78. http://dx.doi.org/10.1137/0729042.

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41

Rostand, V., and D. Y. Le Roux. "Raviart–Thomas and Brezzi–Douglas–Marini finite‐element approximations of the shallow‐water equations." International Journal for Numerical Methods in Fluids 57, no. 8 (July 20, 2008): 951–76. http://dx.doi.org/10.1002/fld.1668.

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42

Diogene Vianney, Pongui ngoma, Nguimbi Germain, and Likibi Pellat Rhoss Beaunheur. "The effect of numerical integration in mixed finite element approximation in the simulation of miscible displacement." International Journal of Applied Mathematical Research 6, no. 2 (April 17, 2017): 44. http://dx.doi.org/10.14419/ijamr.v6i2.7320.

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We consider the effect of numerical integration in finite element procedures applied to a nonlinear system of two coupled partial differential equations describing the miscible displacement of one incompressible fluid by another in a porous meduim. We consider the use of the numerical quadrature scheme for approximating the pressure and velocity by a mixed method using Raviart - Thomas space of index and the concentration by a standard Galerkin method. We also give some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal order estimates are derived when the imposed external flows are smoothly distributed.

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43

CASCON, J. MANUEL, RICARDO H. NOCHETTO, and KUNIBERT G. SIEBERT. "DESIGN AND CONVERGENCE OF AFEM IN H(DIV)." Mathematical Models and Methods in Applied Sciences 17, no. 11 (November 2007): 1849–81. http://dx.doi.org/10.1142/s0218202507002492.

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We design an adaptive finite element method (AFEM) for mixed boundary value problems associated with the differential operator A-∇div in H(div, Ω). For A being a variable coefficient matrix with possible jump discontinuities, we provide a complete a posteriori error analysis which applies to both Raviart–Thomas ℝ𝕋n and Brezzi–Douglas–Marini 𝔹𝔻𝕄n elements of any order n in dimensions d = 2, 3. We prove a strict reduction of the total error between consecutive iterates, namely a contraction property for the sum of energy error and oscillation, the latter being solution-dependent. We present numerical experiments for ℝ𝕋n with n = 0, 1 and 𝔹𝔻𝕄1 which document the performance of AFEM and corroborate as well as extend the theory.

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Capatina, Daniela, and Cuiyu He. "Flux recovery for Cut Finite Element Method and its application in a posteriori error estimation." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 6 (November 2021): 2759–84. http://dx.doi.org/10.1051/m2an/2021071.

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In this article, we aim to recover locally conservative and H(div) conforming fluxes for the linear Cut Finite Element Solution with Nitsche’s method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart–Thomas space is completely local and does not require to solve any mixed problem. The L2-norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we also prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.

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Almonacid, Javier A., Hugo S. Díaz, Gabriel N. Gatica, and Antonio Márquez. "A fully mixed finite element method for the coupling of the Stokes and Darcy–Forchheimer problems." IMA Journal of Numerical Analysis 40, no. 2 (February 1, 2019): 1454–502. http://dx.doi.org/10.1093/imanum/dry099.

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Abstract In this paper we introduce and analyze a fully mixed formulation for the nonlinear problem given by the coupling of the Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. This new approach yields non-Hilbert normed spaces and a twofold saddle point structure for the corresponding operator equation, whose continuous and discrete solvabilities are analyzed by means of a suitable abstract theory developed for this purpose. In particular, feasible choices of finite element subspaces include PEERS of the lowest order for the stress of the fluid, Raviart–Thomas of the lowest order for the Darcy velocity, piecewise constants for the pressures and continuous piecewise linear elements for the vorticity. An a priori error estimates and associated rates of convergence are derived, and several numerical results illustrating the good performance of the method are reported.

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Swager, M. R., and Y. C. Zhou. "Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations." Computational and Mathematical Biophysics 1 (March 20, 2013): 26–41. http://dx.doi.org/10.2478/mlbmb-2013-0001.

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AbstractA general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion processes. Such methods are highly desirable for achieving numerical stability and efficiency. We found that by utilizing the one-to-one correspondence between the continuous piecewise polynomial space of degree k + 1 and the divergencefree vector space of degree k, one can construct high-order two-dimensional exponentially fitted basis functions that are strictly interpolative at a selected node set but are discontinuous on edges in general, spanning nonconforming finite element spaces. First order convergence was proved for the methods constructed from divergence-free Raviart-Thomas space RT 00 at two different node sets.

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Ambartsumyan, Ilona, Eldar Khattatov, Jeonghun J. Lee, and Ivan Yotov. "Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra." Mathematical Models and Methods in Applied Sciences 29, no. 06 (June 15, 2019): 1037–77. http://dx.doi.org/10.1142/s0218202519500167.

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We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

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48

Chen, Yanping, Zuliang Lu, and Yunqing Huang. "Superconvergence of triangular Raviart–Thomas mixed finite element methods for a bilinear constrained optimal control problem." Computers & Mathematics with Applications 66, no. 8 (November 2013): 1498–513. http://dx.doi.org/10.1016/j.camwa.2013.08.019.

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Nie, Cunyun, and Haiyuan Yu. "A Raviart-Thomas mixed finite element scheme for the two-dimensional three-temperature heat conduction problems." International Journal for Numerical Methods in Engineering 111, no. 10 (January 26, 2017): 983–1000. http://dx.doi.org/10.1002/nme.5492.

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50

Bertrand, Fleurianne, Marcel Moldenhauer, and Gerhard Starke. "A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction." Computational Methods in Applied Mathematics 19, no. 3 (July 1, 2019): 663–79. http://dx.doi.org/10.1515/cmam-2018-0004.

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AbstractThe nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie can be used for the displacement approximation and its combination with discontinuous piecewise linear pressure elements is known to constitute a stable combination for incompressible linear elasticity computations. In this contribution, we extend the stress reconstruction procedure and resulting guaranteed a posteriori error estimator developed by Ainsworth, Allendes, Barrenechea and Rankin [2] and by Kim [18] to linear elasticity. In order to get a guaranteed reliability bound with respect to the energy norm involving only known constants, two modifications are carried out: (i) the stress reconstruction in next-to-lowest order Raviart–Thomas spaces is modified in such a way that its anti-symmetric part vanishes in average on each element; (ii) the auxiliary conforming approximation is constructed under the constraint that its divergence coincides with the one for the nonconforming approximation. An important aspect of our construction is that all results hold uniformly in the incompressible limit. Global efficiency is also shown and the effectiveness is illustrated by adaptive computations involving different Lamé parameters including the incompressible limit case.

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