Error estimates in $L^2$, $H^1$ and $L^\infty$ in covolume methods
for elliptic and parabolic problems: A unified approach

We study spatially semidiscrete and fully discrete finite volume element approximations of the heat equation with homogeneous Dirichlet boundary conditions in a plane polygonal domain with one reentrant corner. We show that, as a result of the singularity in the solution near the reentrant corner, the convergence rate is reduced from optimal second order, similarly to what was shown for the finite element method in the earlier work [2]. Optimal order convergence may be restored by mesh refinement near the corners of the domain.

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“…[3,4,[6][7][8], and then proceed with our error bounds. We shall first rewrite the Petrov-Galerkin method (1.3) as a Galerkin method.…”

Section: The Semidiscrete Finite Volume Methods For the Parabolicmentioning

confidence: 99%

Parabolic finite volume element equations in nonconvex polygonal domains

Chatzipantelidis¹,

Lazarov²,

Thomée³

2008

Numerical Methods Partial

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“…Finite volume methods for elliptic boundary value problems have been proposed and analyzed under a variety of different names: box methods, covolume methods, diamond cell methods, integral finite difference methods and finite volume element methods, see Bank and Rose, 1987;Cai, 1991;Suli, 1991;Lazarov, Michev and Vassilevsky, 1996;Viozat et al, 1998;Chatzipantelidis, 1999;Chou and Li, 2000;Hermeline, 2000;Eymard, Galluoet and Herbin, 2000;Ewing, Lin and Lin, 2002 elements in a straightforward way. A piecewise constant test space is then constructed using…”

Section: Discretization Of Elliptic Problemsmentioning

confidence: 99%

Finite Volume Methods: Foundation and Analysis

Barth

Herbin

Ohlberger

2017

Encyclopedia of Computational Mechanics Second Edition

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Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form.This article reviews elements of the foundation and analysis of modern finite volume methods.The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, nonoscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of dffision terms and the extension to systems of nonlinear conservation laws.

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“…The following two lemmas have been proved in [10], where Lemma 3.1 indicates that the bilinear form a h (·, I * h ·) is continuous and coercive on S h , while Lemma 3.2 shows that a h (·, I * h ·) is generally unsymmetric but not too far away from being symmetric. Lemma 3.1.…”

Section: Error Analysis Of the Finite Volume Element Schemementioning

confidence: 99%

“…A reason for this might be that the analysis for the nonlinear term is often very involved. For the linear case, a unified approach is presented in [10] to derive error estimates in the L 2 , H 1 , and L ∞ norms by connecting FVE methods with finite element (FE) methods. Error estimates and superconvergence results in the L p norm (2 ≤ p < ∞) are obtained in [11].…”

Section: Introductionmentioning

confidence: 99%

Postprocessing of a finite volume element method for semilinear parabolic problems

Yang¹,

Bi²,

Liu³

2009

ESAIM: M2AN

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Abstract.In this paper, we study a postprocessing procedure for improving accuracy of the finite volume element approximations of semilinear parabolic problems. The procedure amounts to solve a source problem on a coarser grid and then solve a linear elliptic problem on a finer grid after the time evolution is finished. We derive error estimates in the L 2 and H 1 norms for the standard finite volume element scheme and an improved error estimate in the H 1 norm. Numerical results demonstrate the accuracy and efficiency of the procedure.Mathematics Subject Classification. 65N30, 65N15.

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Error estimates in $L^2$, $H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: A unified approach

Cited by 185 publications

References 26 publications

Parabolic finite volume element equations in nonconvex polygonal domains

Parabolic finite volume element equations in nonconvex polygonal domains

Finite Volume Methods: Foundation and Analysis

Postprocessing of a finite volume element method for semilinear parabolic problems

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