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Anisotropic Mesh Adaptation for 3D Anisotropic Diffusion Problems with Application to Fractured Reservoir Simulation

Part of:Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  12 September 2017

Xianping Li and
Weizhang Huang
Xianping Li*
Affiliation:
Department of Mathematics and Statistics, University of Missouri-Kansas City, Kansas City, MO 64110, U.S.A
Weizhang Huang*
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045, U.S.A
*
*Corresponding author. Email addresses:lixianp@umkc.edu (X. P. Li), whuang@ku.edu (W. Z. Huang)
*Corresponding author. Email addresses:lixianp@umkc.edu (X. P. Li), whuang@ku.edu (W. Z. Huang)

Abstract

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Anisotropic mesh adaptation is studied for linear finite element solution of 3D anisotropic diffusion problems. The 𝕄-uniform mesh approach is used, where an anisotropic adaptive mesh is generated as a uniform one in the metric specified by a tensor. In addition to mesh adaptation, preservation of the maximum principle is also studied. Some new sufficient conditions for maximum principle preservation are developed, and a mesh quality measure is defined to server as a good indicator. Four different metric tensors are investigated: one is the identity matrix, one focuses on minimizing an error bound, another one on preservation of the maximum principle, while the fourth combines both. Numerical examples show that these metric tensors serve their purposes. Particularly, the fourth leads to meshes that improve the satisfaction of the maximum principle by the finite element solution while concentrating elements in regions where the error is large. Application of the anisotropic mesh adaptation to fractured reservoir simulation in petroleum engineering is also investigated, where unphysical solutions can occur and mesh adaptation can help improving the satisfaction of the maximum principle.

Keywords

Finite element methodanisotropic mesh adaptationthree dimensionalanisotropic diffusiondiscrete maximum principlepetroleum engineering

MSC classification

Secondary:65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods65M50: Mesh generation and refinement

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017, pp. 913 - 940
Copyright
Copyright © Global-Science Press 2017 

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