An Arbitrary Order Mixed Finite Element Method with Boundary Value Correction for the Darcy Flow on Curved Domains (Communications in Computational Physics, 2025)
We propose a boundary value correction method for the Brezzi-Douglas-Marini mixed finite element discretization of the Darcy flow with non-homogeneousNeumann boundary condition on 2D curved domains. The discretization is defined ona body-fitted triangular mesh, i.e. the boundary nodes of the mesh lie on the curvedphysical boundary. However, the boundary edges of the triangular mesh, which arestraight, may not coincide with the curved physical boundary. A boundary valuecorrection technique is then designed to transform the Neumann boundary conditionfrom the physical boundary to the boundary of the triangular mesh. One advantage ofthe boundary value correction method is that it avoids using curved mesh elementsand thus reduces the complexity of implementation. We prove that the proposedmethod reaches optimal convergence for arbitrary order discretizations. Supportingnumerical results are presented.