Volume 10, Issue 4
Anisotropic Mesh Adaptation for 3D Anisotropic Diffusion Problems with Application to Fractured Reservoir Simulation

Numer. Math. Theor. Meth. Appl., 10 (2017), pp. 913-940.

Published online: 2017-11

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• Abstract

Anisotropic mesh adaptation is studied for linear finite element solution of 3D anisotropic diffusion problems. The $\mathbb{M}$-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 to improve the satisfaction of the maximum principle.

• Keywords

Finite element method, anisotropic mesh adaptation, three dimensional, anisotropic diffusion, discrete maximum principle, petroleum engineering.

65M60, 65M50

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@Article{NMTMA-10-913, author = {}, title = {Anisotropic Mesh Adaptation for 3D Anisotropic Diffusion Problems with Application to Fractured Reservoir Simulation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2017}, volume = {10}, number = {4}, pages = {913--940}, abstract = {

Anisotropic mesh adaptation is studied for linear finite element solution of 3D anisotropic diffusion problems. The $\mathbb{M}$-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 to improve the satisfaction of the maximum principle.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2017.m1625}, url = {http://global-sci.org/intro/article_detail/nmtma/10462.html} }
TY - JOUR T1 - Anisotropic Mesh Adaptation for 3D Anisotropic Diffusion Problems with Application to Fractured Reservoir Simulation JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 913 EP - 940 PY - 2017 DA - 2017/11 SN - 10 DO - http://doi.org/10.4208/nmtma.2017.m1625 UR - https://global-sci.org/intro/article_detail/nmtma/10462.html KW - Finite element method, anisotropic mesh adaptation, three dimensional, anisotropic diffusion, discrete maximum principle, petroleum engineering. AB -

Anisotropic mesh adaptation is studied for linear finite element solution of 3D anisotropic diffusion problems. The $\mathbb{M}$-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 to improve the satisfaction of the maximum principle.

Xianping Li & Weizhang Huang. (2020). Anisotropic Mesh Adaptation for 3D Anisotropic Diffusion Problems with Application to Fractured Reservoir Simulation. Numerical Mathematics: Theory, Methods and Applications. 10 (4). 913-940. doi:10.4208/nmtma.2017.m1625
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