Academic literature on the topic 'Spatial H-2 norm'
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Journal articles on the topic "Spatial H-2 norm"
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Ern, Alexandre, Iain Smears, and Martin Vohralík. "Equilibrated flux a posteriori error estimates in $L^2(H^1)$-norms for high-order discretizations of parabolic problems." IMA Journal of Numerical Analysis 39, no. 3 (June 25, 2018): 1158–79. http://dx.doi.org/10.1093/imanum/dry035.
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Abstract We consider the a posteriori error analysis of fully discrete approximations of parabolic problems based on conforming $hp$-finite element methods in space and an arbitrary order discontinuous Galerkin method in time. Using an equilibrated flux reconstruction we present a posteriori error estimates yielding guaranteed upper bounds on the $L^2(H^1)$-norm of the error, without unknown constants and without restrictions on the spatial and temporal meshes. It is known from the literature that the analysis of the efficiency of the estimators represents a significant challenge for $L^2(H^1)$-norm estimates. Here we show that the estimator is bounded by the $L^2(H^1)$-norm of the error plus the temporal jumps under the one-sided parabolic condition $h^2 \lesssim \tau $. This result improves on earlier works that required stronger two-sided hypotheses such as $h \simeq \tau $ or $h^2\simeq \tau $; instead, our result now encompasses practically relevant cases for computations and allows for locally refined spatial meshes. The constants in our bounds are robust with respect to the mesh and time-step sizes, the spatial polynomial degrees and the refinement and coarsening between time steps, thereby removing any transition condition.
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Qin, Yifan, Xiaocheng Yang, Yunzhu Ren, Yinghong Xu, and Wahidullah Niazi. "A Newton Linearized Crank-Nicolson Method for the Nonlinear Space Fractional Sobolev Equation." Journal of Function Spaces 2021 (April 26, 2021): 1–11. http://dx.doi.org/10.1155/2021/9979791.
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In this paper, one class of finite difference scheme is proposed to solve nonlinear space fractional Sobolev equation based on the Crank-Nicolson (CN) method. Firstly, a fractional centered finite difference method in space and the CN method in time are utilized to discretize the original equation. Next, the existence, uniqueness, stability, and convergence of the numerical method are analyzed at length, and the convergence orders are proved to be O τ 2 + h 2 in the sense of l 2 -norm, H α / 2 -norm, and l ∞ -norm. Finally, the extensive numerical examples are carried out to verify our theoretical results and show the effectiveness of our algorithm in simulating spatial fractional Sobolev equation.
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Zhao, Jie, Hong Li, Zhichao Fang, and Yang Liu. "A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids." Mathematics 7, no. 7 (July 5, 2019): 600. http://dx.doi.org/10.3390/math7070600.
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In this article, the time-fractional reaction-diffusion equations are solved by using a mixed finite volume element (MFVE) method and the L 1 -formula of approximating the Caputo fractional derivative. The existence, uniqueness and unconditional stability analysis for the fully discrete MFVE scheme are given. A priori error estimates for the scalar unknown variable (in L 2 ( Ω ) -norm) and the vector-valued auxiliary variable (in ( L 2 ( Ω ) ) 2 -norm and H ( div , Ω ) -norm) are derived. Finally, two numerical examples in one-dimensional and two-dimensional spatial regions are given to examine the feasibility and effectiveness.
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Dong, Gang, Zhichang Guo, and Wenjuan Yao. "Numerical methods for time-fractional convection-diffusion problems with high-order accuracy." Open Mathematics 19, no. 1 (January 1, 2021): 782–802. http://dx.doi.org/10.1515/math-2021-0036.
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Abstract In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.
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Li, Hou-Biao, Ming-Yan Song, Er-Jie Zhong, and Xian-Ming Gu. "Numerical Gradient Schemes for Heat Equations Based on the Collocation Polynomial and Hermite Interpolation." Mathematics 7, no. 1 (January 17, 2019): 93. http://dx.doi.org/10.3390/math7010093.
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As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is that it is unconditionally stable and convergent on the order O ( τ 2 + h 4 ) (where τ is the time step size and h is the mesh size), under the maximum norm for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. The convergence order of this kind of method is also O ( τ 2 + h 4 ) under the discrete maximum norm when the spatial step size is twice the one of H-OCD, which accelerates the computational process. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in the time direction. The results of numerical experiments are consistent with the theoretical analysis.
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Guo, Yuling, and Jianguo Huang. "A Robust Finite Element Method for Elastic Vibration Problems." Computational Methods in Applied Mathematics 20, no. 3 (July 1, 2020): 481–500. http://dx.doi.org/10.1515/cmam-2018-0197.
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AbstractA robust finite element method is introduced for solving elastic vibration problems in two dimensions. The temporal discretization is carried out using the {P_{1}}-continuous discontinuous Galerkin (CDG) method, while the spatial discretization is based on the Crouziex–Raviart (CR) element. It is shown after a technical derivation that the error of the displacement (resp. velocity) in the energy norm (resp. {L^{2}} norm) is bounded by {O(h+k)} (resp. {O(h^{2}+k)}), where h and k denote the mesh sizes of the subdivisions in space and time, respectively. Under some regularity assumptions on the exact solution, the error bound is independent of the Lamé coefficients of the elastic material under discussion. A series of numerical results are offered to illustrate numerical performance of the proposed method and some other fully discrete methods for comparison.
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Yadav, Sangita, and Amiya K. Pani. "Superconvergent discontinuous Galerkin methods for nonlinear parabolic initial and boundary value problems." Journal of Numerical Mathematics 27, no. 3 (September 25, 2019): 183–202. http://dx.doi.org/10.1515/jnma-2018-0035.
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Abstract In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L∞(0, T; L2)-norm is of order k + 1. With the help of a suitable post-processing of the semi-discrete potential, it is proved that the resulting post-processed potential converges with order of convergence $\begin{array}{} \displaystyle O\big(\!\sqrt{{}\log(T/h^2)}\,h^{k+2}\big) \end{array}$ in L∞(0, T; L2)-norm. These results extend the HDG analysis of Chabaud and Cockburn [Math. Comp. 81 (2012), 107–129] for the heat equation to non-linear parabolic problems.
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Luo, Zhengyan, Lintao Ma, and Yinghui Zhang. "Optimal decay rates of higher–order derivatives of solutions for the compressible nematic liquid crystal flows in $ \mathbb R^3 $." AIMS Mathematics 7, no. 4 (2022): 6234–58. http://dx.doi.org/10.3934/math.2022347.
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<abstract><p>In this paper, we are concerned with optimal decay rates of higher–order derivatives of the smooth solutions to the $ 3D $ compressible nematic liquid crystal flows. The main novelty of this paper is three–fold: First, under the assumptions that the initial perturbation is small in $ H^N $–norm $ (N\geq3) $ and bounded in $ L^1 $–norm, we show that the highest–order spatial derivatives of density and velocity converge to zero at the $ L^2 $–rates is $ (1+t)^{-\frac{3}{4}-\frac{N }{2 }} $, which are the same as ones of the heat equation, and particularly faster than the $ L^2 $–rate $ (1+t)^{-\frac{1}{4}-\frac{N }{2 }} $ in [J.C. Gao, et al., J. Differential Equations, 261: 2334-2383, 2016]. Second, if the initial data satisfies some additional low frequency assumption, we also establish the lower optimal decay rates of solution as well as its all–order spatial derivatives. Therefore, our decay rates are optimal in this sense. Third, we prove that the lower bound of the time derivatives of density, velocity and macroscopic average converge to zero at the $ L^2 $–rate is $ (1+t)^{-\frac{5}{4}} $. Our method is based on low-frequency and high-frequency decomposition and energy methods.</p></abstract>
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Yang, Hua-Yu, Yun Wang, Ping He, Weishan Zhu, and Long-Long Feng. "The spatial distribution deviation and the power suppression of baryons from dark matter." Monthly Notices of the Royal Astronomical Society 509, no. 1 (October 22, 2021): 1036–47. http://dx.doi.org/10.1093/mnras/stab3062.
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ABSTRACT The spatial distribution between dark matter and baryonic matter of the Universe is biased or deviates from each other. In this work, by comparing the results derived from IllustrisTNG and WIGEON simulations, we find that many results obtained from TNG are similar to those from WIGEON data, but differences between the two simulations do exist. For the ratio of density power spectrum between dark matter and baryonic matter, as scales become smaller and smaller, the power spectra for baryons are increasingly suppressed for WIGEON simulations; while for TNG simulations, the suppression stops at $k=15-20\, {h {\rm Mpc}^{-1}}$, and the power spectrum ratios increase when $k\gt 20\, {h {\rm Mpc}^{-1}}$. The suppression of power ratio for WIGEON is also redshift-dependent. From z = 1 to z = 0, the power ratio decreases from about 70 per cent to less than 50 per cent at $k=8\, {h {\rm Mpc}^{-1}}$. For TNG simulation, the suppression of power ratio is enhanced with decreasing redshifts in the scale range $k\gt 4\, {h {\rm Mpc}^{-1}}$, but is nearly unchanged with redshifts in $k\lt 4\, {h {\rm Mpc}^{-1}}$. These results indicate that turbulent heating can also have the consequence to suppress the power ratio between baryons and dark matter. Regarding the power suppression for TNG simulations as the norm, the power suppression by turbulence for WIGEON simulations is roughly estimated to be 45 per cent at $k=2\, {h {\rm Mpc}^{-1}}$, and gradually increases to 69 per cent at $k=8\, {h {\rm Mpc}^{-1}}$, indicating the impact of turbulence on the cosmic baryons are more significant on small scales.
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Bir, Bikram, Deepjyoti Goswami, and Amiya K. Pani. "Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data." Computational Methods in Applied Mathematics 22, no. 2 (February 12, 2022): 297–325. http://dx.doi.org/10.1515/cmam-2022-0012.
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Abstract In this paper, a penalty formulation is proposed and analyzed in both continuous and finite element setups, for the two-dimensional Oldroyd model of order one, when the initial velocity is in 𝐇 0 1 {\mathbf{H}_{0}^{1}} . New regularity results which are valid uniformly in time as t → ∞ {t\to\infty} and in the penalty parameter ε as ε → 0 {\varepsilon\to 0} are derived for the solution of the penalized problem. Then, based on conforming finite elements to discretize the spatial variables and keeping temporal variable continuous, a semidiscrete problem is discussed and a uniform-in-time a priori bound of the discrete velocity in Dirichlet norm is derived with the help of a penalized discrete Stokes operator and a modified uniform Gronwall’s lemma. Further, optimal error estimates for the penalized velocity in 𝐋 2 {\mathbf{L}^{2}} as well in 𝐇 1 {\mathbf{H}^{1}} -norms and for the penalized pressure in L 2 {L^{2}} -norm have been established for the semidiscrete problem with non-smooth data. These error estimates hold uniformly in time under uniqueness assumption and also in the penalty parameter as it goes to zero. Our analysis relies on the suitable use of the inverse of the penalized Stokes operator, penalized Stokes–Volterra projection and judicious application of weighted time estimates with positivity property of the memory term. Finally, several numerical experiments are conducted on benchmark problems which confirm our theoretical findings.
Dissertations / Theses on the topic "Spatial H-2 norm"
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Halim, Dunant. "Vibration Analysis and Control of Smart Structures." Thesis, 2003. http://hdl.handle.net/1959.13/24886.
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This thesis represents the work that has been done by the author in the area of vibration analysis and control of smart structures during his PhD candidature. The research was concentrated on flexible structures, using piezoelectric materials as actuators and sensors. The thesis consists of four major parts. The first part (Chapter 2) is the modelling of piezoelectric laminate structures using modal analysis and finite element methods. The second part (Chapter 4) involves the model correction of pointwise and spatial models of resonant systems. The model correction solution compensates for the errors associated with the truncation of high frequency modes. The third part (Chapter 5) is the optimal placement methodology for general actuators and sensors. In particular, optimal placement of piezoelectric actuators and sensors over a thin plate are considered and implemented in the laboratory. The last part (Chapters 6 to 8) deals with vibration control of smart structures. Several different approaches for vibration control are considered. Vibration control using resonant, spatial H-2 and H-infinity control is proposed and implemented on real systems experimentally. It is possible, for certain modes, to obtain the very satisfactory result of up to 30 dB vibration reduction.PhD Doctorate
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Halim, Dunant. "Vibration Analysis and Control of Smart Structures." 2003. http://hdl.handle.net/1959.13/24886.
Full textAbstract:
This thesis represents the work that has been done by the author in the area of vibration analysis and control of smart structures during his PhD candidature. The research was concentrated on flexible structures, using piezoelectric materials as actuators and sensors. The thesis consists of four major parts. The first part (Chapter 2) is the modelling of piezoelectric laminate structures using modal analysis and finite element methods. The second part (Chapter 4) involves the model correction of pointwise and spatial models of resonant systems. The model correction solution compensates for the errors associated with the truncation of high frequency modes. The third part (Chapter 5) is the optimal placement methodology for general actuators and sensors. In particular, optimal placement of piezoelectric actuators and sensors over a thin plate are considered and implemented in the laboratory. The last part (Chapters 6 to 8) deals with vibration control of smart structures. Several different approaches for vibration control are considered. Vibration control using resonant, spatial H-2 and H-infinity control is proposed and implemented on real systems experimentally. It is possible, for certain modes, to obtain the very satisfactory result of up to 30 dB vibration reduction.PhD Doctorate
Conference papers on the topic "Spatial H-2 norm"
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Sadr, R., Z. Zheng, M. Yoda, and A. T. Conlisk. "An Experimental and Modeling Study of Electroosmotic Bulk and Near-Wall Flows in Two-Dimensional Micro- and Nanochannels." In ASME 2003 International Mechanical Engineering Congress and Exposition. ASMEDC, 2003. http://dx.doi.org/10.1115/imece2003-42917.
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Electrokinetically driven flow of electrolyte solutions through micro- and nanochannels is of interest in microelectromechanical systems (MEMS) and nanotechnology applications. In this work, fully developed and steady electroosmotic flow (EOF) of dilute sodium tetraborate and sodium chloride aqueous solutions in a rectangular channel where the channel hight h is comparable to its width W is examined. EOF is also studied under conditions of electric double layer (EDL) overlap, or λ/h ∼ O(1), where λ is the Debye thickness, for very dilute solutions. The initial experimental data and model results are in very good agreement for dilute sodium tetraborate solutions. The experimental work uses the new nano-particle image velocimetry (nPIV) technique. Evanescent waves from the total internal reflection of light with a wavelength of 488 nm at a refractive index interface is used to illuminate 100 nm neutrally buoyant fluorescent particles in the near-wall region of the flow. The images of these tracer particles over time are processed to obtain the two components of the velocity field parallel to the wall in fully developed EOF of sodium tetraborate at concentrations up to 2 mM in fused quartz rectangular channels with height h up to 10 microns. The spatial resolution of these velocity field data along the dimension normal to the wall is about 100 nm, and the data are obtained within a distance of approximately 100 nm of the wall based upon the 1/e intensity point, or penetration depth. A set of equations modeling EOF in a long channel are solved where h/L << 1, and L is the lengthscale along the flow direction. Unlike most previous models, this work does not use the Debye-Huckel approximation, nor does it assume symmetric boundary conditions. For the case where λ/h << 1, analytical solutions for the velocity, potential and mole fractions are obtained using an asymptotic perturbation approach.