L 2 error estimates and superconvergence of the finite volume element methods on quadrilateral meshes

Lv, Junliang; Li, Yonghai

doi:10.1007/s10444-011-9215-2

Cited by 32 publications

(28 citation statements)

References 20 publications

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“…Since 1999, Li and his colleagues have studied the bilinear FV schemes over quadrilateral meshes and obtained an optimal L 2 estimate under very weak mesh conditions(cf. [23,25,26]). The L 2 estimate for high order schemes is usually obtained with the help of a superconvergence argument; see, e.g., [39,40].…”

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confidence: 94%

$L^2$ 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: 94%

$L^2$ 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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“…Using the approximation property, we obtain E 32 ≤ Ch 2 u 3,p v 1,q so that 12) noting that (Π * h v − v)| ∂ Ω = 0. Let τ be an interior edge, that is, a common edge of two adjacent element K and K ′ .…”

Section: Fig2 Rectangular Elementmentioning

confidence: 93%

“…Later, Lv and Li in [12] extended result (1.1) to the isoparametric bilinear FVE on quadrilateral meshes under the h 2 -uniform mesh condition. Recently, Zhang and Zou in [20] also derived some superconvergence results for the bi-complete k-order FVE on rectangular meshes, and in the case of bilinear FVE (k = 1), their result is…”

Section: Introductionmentioning

confidence: 99%

“…The main feature of FVE method is that it inherits some physical conservation laws of original problems locally, which are very desirable in practical applications. During the last decades, many research works have been presented for FVE methods solving various partial differential equations, see [1,2,3,4,5,6,7,8,9,10,12,13,14,15,17,18,20] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

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The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems

Zhang

Tang

2016

Numer. Math. Theory Methods Appl.

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We study the gradient superconvergence of bilinear finite volume element (FVE) solving the elliptic problems. First, a superclose weak estimate is established for the bilinear form of the FVE method. Then, we prove that the gradient approximation of the FVE solution has the superconvergence property: maxwhere ∇u h (P) denotes the average gradient on elements containing point P and S is the set of optimal stress points composed of the mesh points, the midpoints of edges and elements.

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“…Moreover, the superconvergence of the FVM solution in an average gradient norm has been also obtained. See, for example, [13], [26], [28]. For the linear elliptic and parabolic problems, Chou et al [16] show the superconvergence estimates in the L p -norm for the error between the FVM solution and the corresponding FEM solution and between their gradients.…”

Section: ∞mentioning

confidence: 99%

The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

Zhang

2015

Appl Math

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L 2 error estimates and superconvergence of the finite volume element methods on quadrilateral meshes

Cited by 32 publications

References 20 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

The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems

The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

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