A unified analysis of a class of quadratic finite volume element schemes on triangular meshes
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A unified analysis of a class of quadratic finite volume element schemes on triangular meshes Yanhui Zhou1 · Jiming Wu1 Received: 21 December 2019 / Accepted: 13 July 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper presents a general framework for the coercivity analysis of a class of quadratic finite volume element (FVE) schemes on triangular meshes for solving elliptic boundary value problems. This class of schemes covers all the existing quadratic schemes of Lagrange type. With the help of a new mapping from the trial function space to the test function space, we find that each element matrix can be decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between the FVE and FEM on the element boundary, while the third part can be expressed as the tensor product of two vectors. Thanks to this decomposition, we obtain a sufficient condition to guarantee the existence, uniqueness, and coercivity result of the FVE solution on triangular meshes. Moreover, based on this sufficient condition, some minimum angle conditions with simple, analytic, and computable expressions can be derived and they depend only on the constructive parameters of the schemes. As a byproduct, some existing coercivity results are improved. Finally, an optimal H 1 error estimate is proved by the standard techniques. Keywords Quadratic finite volume element schemes · Triangular meshes · Coercivity result · Minimum angle condition · Optimal H 1 error estimate Mathematics Subject Classification (2010) 65N08 · 65N12 · 35J25
Communicated by: Aihui Zhou Jiming Wu
wu [email protected] Yanhui Zhou [email protected] 1
Institute of Applied Physics and Computational Mathematics, Beijing, 100088, People’s Republic of China
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Adv Comput Math
(2020) 46:71
1 Introduction The finite volume method (FVM) is one of the major numerical methods for solving partial differential equations (c.f. [2, 22, 28, 29, 32]), since it preserves the local conservation law. The finite volume element method (FVEM) is one type of FVM, and its mathematical progress can be found in [23, 26, 44] and the references cited therein. The coercivity result is one of the most challenging works for the analysis of FVEMs, especially for the high-order schemes. For the linear FVEM on triangular meshes, its element stiffness matrix can be regarded as a small perturbation of linear FEM for variable coefficient, then the coercivity result can be proved (c.f. [1, 4, 18, 19, 40]), and the error estimates were presented in [1, 4, 12, 13, 18, 19, 38, 40] for incomplete references. Recently, [16, 17, 47] studied the adaptive linear FVEM on triangular meshes, and [35] studied the conditioning of linear FVEM on arbitrary simplicial meshes. Unlike the linear scheme, the existing quadratic scheme is constructed by two parameters α and β, where α ∈ (0, 1/2) on the element boundary and β ∈ (0, 2/3) in the interior of eleme
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