Higher-order finite volume methods for elliptic boundary value problems

Chen, Zhongying; Wu, Junfeng; Xu, Yuesheng

doi:10.1007/s10444-011-9201-8

Cited by 89 publications

(78 citation statements)

References 30 publications

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“…That is, the analysis must to be performed scheme by scheme. Only recently, general stability frameworks for high order FVMs over one dimensional and two dimensional triangular or quadrilateral meshes have been established in [4,8,31,39,40]. Once the stability has been established, the error analysis in the energy norm (H 1 norm) is then routine.…”

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confidence: 99%

$L^2$ Error Estimates for a Class of Any Order Finite Volume Schemes Over Quadrilateral Meshes

Lin¹,

Yang²,

Zou³

2015

SIAM J. Numer. Anal.

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In this paper, we propose a unified L 2 error estimate for a class of bi-r finite volume (FV) schemes on a quadrilateral mesh for elliptic equations, where r ≥ 1 is arbitrary. The main result is to show that the FV solution possesses the optimal order L 2 error provided that (u, f ) ∈ H r+1 × H r , where u is the exact solution and f is the source term of the elliptic equation. Our analysis includes two basic ideas: (1) By the Aubin-Nistche technique, the L 2 error estimate of an FV scheme can be reduced to the analysis of the difference of bilinear forms and right-hand sides between the FV and its corresponding finite element (FE) equations, respectively; (2) with the help of a special transfer operator from the trial to test space, the difference between the FV and FE equations can be estimated through analyzing the effect of some Gauss quadrature. Numerical experiments are given to demonstrate the proved results.1. Introduction. The finite volume (FV) method (FVM) enjoys great popularity among scientific and engineering computations for its local conservation property, easy implimentation, and other advantages; see, e.g., [14,19,20] and [28,29,30,31,32,33,34,35]. In comparison to its wide applications, the development of the FVM theory (cf. [1,2,3,5,6,10,12,16,17,21,22]) lags far behind, especially for high order schemes.The stability (or inf-sup condition) is critical in the analysis of FVMs. Some earlier literature (see, e.g., [24,22,37,38]) on the analysis of high order FVMs is based on the so-called element-wise stiffness matrix analysis, in which the eigenvalues of the local stiffness matrix are estimated and the techniques are scheme dependent. That is, the analysis must to be performed scheme by scheme. Only recently, general stability frameworks for high order FVMs over one dimensional and two dimensional triangular or quadrilateral meshes have been established in [4,8,31,39,40]. Once the stability has been established, the error analysis in the energy norm (H 1 norm) is then routine.The L 2 norm error estimate is a challenging task, however. To our knowledge, only a few studies on the L 2 estimate of FVMs have been published in the literature and most studies focus on linear schemes. In 1994, Chen [9] proved the optimal convergence rate of L 2 norm error of the linear FV schemes for elliptic equations in two

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confidence: 99%

$L^2$ Error Estimates for a Class of Any Order Finite Volume Schemes Over Quadrilateral Meshes

Lin¹,

Yang²,

Zou³

2015

SIAM J. Numer. Anal.

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“…Remark 2.1 If E * h is chosen as in the Method 1, we can derive a new finite volume method from formulation (11). In the new scheme, the trial function is continuous on the matching part of a grid and is discontinuous on the nonmatching part of it.…”

Section: Finite Volume Methodsmentioning

confidence: 99%

“…and u h ∈ U h be the solutions to problems (1) and (11), respectively. Then, there exists a constant C > 0 independent of h such that…”

Section: Theorem 44 Assume That the Partitionmentioning

confidence: 99%

“…It is known that the finite volume methods [11][12][13][14][15] discretize the integral form of conservation law of differential equation by choosing the finite element space as the trial space. The traditional standard finite element space formed by continuous basis functions requires either matching grids or interpolations for nonmatching grids with hanging nodes.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive Bilinear Element Finite Volume Methods for Second-Order Elliptic Problems on Nonmatching Grids

Chen

Sheng

et al. 2014

J Sci Comput

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In this article, we propose and analyze two kinds of adaptive bilinear element finite volume methods for second-order elliptic problems on nonmatching grids. One of them chooses the piecewise bilinear finite element space as the trial function space, which is continuous on the matching part of a grid and is discontinuous on the nonmatching part of it. The other directly uses discontinuous piecewise bilinear element space for the trial function space. A priori estimations ensure the convergence and a posteriori estimations pave the way for adaptive methods. Several numerical experiments are presented to conform our theoretical results.

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“…For the finite volume method, a cell centered finite volume discretization to Poisson equation using multigrid approach, has been used already in 1998 [14]. For the elliptic boundary value problems the higher order finite volume method have been studied in [5,6].…”

Section: The Adaptive Gridmentioning

confidence: 99%

Adaptive cell-centered finite volume method for diffusion equations on a consistent quadtree grid

2015

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Models applied in image processing are often described by nonlinear PDEs in which a good approximation of gradient plays an important role especially in such cases where irregular finite volume grids are used. In image processing, such a situation can occur during a coarsening based on quadtree grids. We present a construction of a deformed quadtree grid in which the connection of representative points of two adjacent finite volumes is perpendicular to their common boundary enabling us to apply the classical finite volume methods. On the other hand, for such an adjusted grid, the intersection of representative points connection with a finite volume boundary is not a middle point of their common edge and standard methods cannot achieve a good accuracy. In this paper we present a new cell-centered finite volume method to evaluate solution gradients, which results into a solution of a simple linear algebraic system and we prove its unique solvability. Finally we present numerical experiments for the regularized Perona-Malik model in which we applied this new method.

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Higher-order finite volume methods for elliptic boundary value problems

Cited by 89 publications

References 30 publications

$L^2$ Error Estimates for a Class of Any Order Finite Volume Schemes Over Quadrilateral Meshes

$L^2$ Error Estimates for a Class of Any Order Finite Volume Schemes Over Quadrilateral Meshes

Adaptive Bilinear Element Finite Volume Methods for Second-Order Elliptic Problems on Nonmatching Grids

Adaptive cell-centered finite volume method for diffusion equations on a consistent quadtree grid

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