An arbitrary order mixed finite element method with boundary value correction for the Darcy flow on curved domains

TL;DR

This paper introduces a boundary value correction technique for mixed finite element methods solving Darcy flow on curved domains, enabling optimal convergence without curved mesh elements.

Contribution

It presents a novel boundary value correction method that simplifies implementation and achieves optimal convergence for mixed finite element discretizations on curved domains.

Findings

Achieves optimal convergence rates for arbitrary order discretizations.

Avoids the complexity of curved mesh elements.

Supports numerical validation of the method.

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. Key words: mixed finite element method, Neumann boundary condition, curved domain, boundary value correction method.