Coercivity results of a modified Q 1 -finite volume element scheme for anisotropic diffusion problems
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Coercivity results of a modified Q1 -finite volume element scheme for anisotropic diffusion problems Qi Hong1 · Jiming Wu2
Received: 8 April 2017 / Accepted: 28 September 2017 © Springer Science+Business Media, LLC 2017
Abstract In this paper, we study a so-called modified Q1 -finite volume element scheme that is obtained by employing the trapezoidal rule to approximate the line integrals in the classical Q1 -finite volume element method. A necessary and sufficient condition is obtained for the positive definiteness of a certain element stiffness matrix. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme on arbitrary convex quadrilateral meshes. When the diffusion tensor is an identity matrix, this sufficient condition reduces to a geometric one, covering some standard meshes, such as the traditional h1+γ -parallelogram meshes and some trapezoidal meshes. More interesting is that, this sufficient condition has explicit expression, by which one can easily judge on any diffusion tensor and any mesh with any mesh size h > 0. The H 1 error estimate of the modified Q1 finite volume element scheme is obtained without the traditional h1+γ -parallelogram assumption. Some numerical experiments are carried out to validate the theoretical analysis. Keywords Q1 -finite volume element method · Modified Q1 -finite volume element scheme · Coercivity · H 1 error estimate
Communicated by: Aihui Zhou Jiming Wu
wu [email protected] Qi Hong [email protected] 1
Graduate School of China Academy of Engineering Physics, Beijing, 100088, China
2
Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China
Q. Hong, J. Wu
Mathematics Subject Classification (2010) 65N15 · 65N30
1 Introduction In many applications such as radiation hydrodynamics and reservoir simulation, certain diffusion processes are coupled with some other physical ones. In these cases, accurate and robust solution of heterogeneous and highly anisotropic diffusion problems on possibly severely distorted meshes poses great challenges. In the past decades, many numerical methods such as finite difference method, finite element method and finite volume method have been developed for the solution of such problems. Finite volume method has a long history, the most important feature of which is that it inherits locally some physical conservation laws of original problems, which is desirable in many practical applications. Here we are concerned with a special type of finite volume method, i.e., the finite volume element method (FVEM). Li [14], Li and Zhu [17] utilized finite elements on primary grids and generalized characteristic functions on dual grids to obtain the so-called generalized difference methods by using the Petrov-Galerkin method. Since then, many researchers have contributed to this subject [15, 16]. Bank and Rose [1] remade and analyzed the box method. Further studies in the name of the box method can be found in [12] and [23]. Cai [2], Cai [3], Cai et al. [4], S¨uli [24] studie
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