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Volume 37, Issue 5
An Arbitrary Order Mixed Finite Element Method with Boundary Value Correction for the Darcy Flow on Curved Domains

Yongli Hou & Yanqiu Wang

DOI: 10.4208/cicp.OA-2024-0317

Commun. Comput. Phys., 37 (2025), pp. 1227-1249.

Published online: 2025-05

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  • Abstract

We propose a boundary value correction method for the Brezzi-Douglas-Marini mixed finite element discretization of the Darcy flow with non-homogeneous Neumann boundary condition on 2D curved domains. The discretization is defined on a body-fitted triangular mesh, i.e. the boundary nodes of the mesh lie on the curved physical boundary. However, the boundary edges of the triangular mesh, which are straight, may not coincide with the curved physical boundary. A boundary value correction technique is then designed to transform the Neumann boundary condition from the physical boundary to the boundary of the triangular mesh. One advantage of the boundary value correction method is that it avoids using curved mesh elements and thus reduces the complexity of implementation. We prove that the proposed method reaches optimal convergence for arbitrary order discretizations. Supporting numerical results are presented.

  • Keywords

Mixed finite element method, Neumann boundary condition, curved domain, boundary value correction method.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-37-1227, author = {Hou , Yongli and Wang , Yanqiu}, title = {An Arbitrary Order Mixed Finite Element Method with Boundary Value Correction for the Darcy Flow on Curved Domains}, journal = {Communications in Computational Physics}, year = {2025}, volume = {37}, number = {5}, pages = {1227--1249}, abstract = {

We propose a boundary value correction method for the Brezzi-Douglas-Marini mixed finite element discretization of the Darcy flow with non-homogeneous Neumann boundary condition on 2D curved domains. The discretization is defined on a body-fitted triangular mesh, i.e. the boundary nodes of the mesh lie on the curved physical boundary. However, the boundary edges of the triangular mesh, which are straight, may not coincide with the curved physical boundary. A boundary value correction technique is then designed to transform the Neumann boundary condition from the physical boundary to the boundary of the triangular mesh. One advantage of the boundary value correction method is that it avoids using curved mesh elements and thus reduces the complexity of implementation. We prove that the proposed method reaches optimal convergence for arbitrary order discretizations. Supporting numerical results are presented.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2024-0317}, url = {http://global-sci.org/intro/article_detail/cicp/24092.html} }
TY - JOUR T1 - An Arbitrary Order Mixed Finite Element Method with Boundary Value Correction for the Darcy Flow on Curved Domains AU - Hou , Yongli AU - Wang , Yanqiu JO - Communications in Computational Physics VL - 5 SP - 1227 EP - 1249 PY - 2025 DA - 2025/05 SN - 37 DO - http://doi.org/10.4208/cicp.OA-2024-0317 UR - https://global-sci.org/intro/article_detail/cicp/24092.html KW - Mixed finite element method, Neumann boundary condition, curved domain, boundary value correction method. AB -

We propose a boundary value correction method for the Brezzi-Douglas-Marini mixed finite element discretization of the Darcy flow with non-homogeneous Neumann boundary condition on 2D curved domains. The discretization is defined on a body-fitted triangular mesh, i.e. the boundary nodes of the mesh lie on the curved physical boundary. However, the boundary edges of the triangular mesh, which are straight, may not coincide with the curved physical boundary. A boundary value correction technique is then designed to transform the Neumann boundary condition from the physical boundary to the boundary of the triangular mesh. One advantage of the boundary value correction method is that it avoids using curved mesh elements and thus reduces the complexity of implementation. We prove that the proposed method reaches optimal convergence for arbitrary order discretizations. Supporting numerical results are presented.

Hou , Yongli and Wang , Yanqiu. (2025). An Arbitrary Order Mixed Finite Element Method with Boundary Value Correction for the Darcy Flow on Curved Domains. Communications in Computational Physics. 37 (5). 1227-1249. doi:10.4208/cicp.OA-2024-0317
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