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

Abstract

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

New superconvergent structures developed from the finite volume element method in 1D

“…Combining ( 37)- (40), one can get (28) immediately. Further more, by using the inf-sup condition (57), we have the superconvergence of the derivative (29).…”

“…The finite volume element method (FVEM) [3,5,6,7,11,15,16,20,23,38], which is famous for the local conservation property, has been studied widely for the stability and H 1 estimate [14,15,24,25,34,41,45], L 2 estimate [13,19,21,28,29,31,39], and superconvergence [8,9,12,30,40]. In this paper, we mainly focus on the new superconvergent structures developed from the FVEM in 1D.…”

On the Construction and Analysis of Finite Volume Element Schemes with Optimal $L^2$ Convergence Rate

“…The finite volume element method (FVEM) [1-3, 6, 9-11, 14-16, 18, 19, 21, 26, 29-31, 34, 36] is a type of finite volume method (FVM), which is famous for the local conservation property and has been successfully applied to a broad range of problems. Till now, a lot of progress has been made in understanding the stability and H 1 estimate [8,9,20,22,28,35,37], L 2 estimate [7,13,17,23,24,32], and superconvergence [4,5,25,33] for the FVEM. Most of the existing results talk about the FVE schemes with symmetric dual meshes.…”

“…Nevertheless, not all of these FVE schemes have optimal L 2 convergence rate, including the even-order FVE schemes with uniform dual meshes. Therefore, how to choose the dual meshes is a main issue of the FVE schemes' construction [8,23,25,32,33,35,37]. To the authors' knowledge, current results about the L 2 estimate are mainly focused on sufficient conditions for FVE schemes to have the optimal L 2 convergence rate.…”

Conditioning of the finite volume element method for diffusion problems with general simplicial meshes

“…This type of estimation has so far been carried out only for quasi-uniform or regular meshes too; e.g. see [30,31,33,37].…”