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

TL;DR

Darcy flow on curved 2D domains introduces geometry-induced errors when using polygonal meshes. The authors propose a boundary value correction that transfers Neumann data from the true boundary $\Gamma$ to the mesh boundary $\Gamma_h$, enabling a Brezzi-Douglas-Marini MFEM on $\Omega_h$ with no curved elements and a $\big(\mathrm{div}\,\cdot,\mathrm{div}\,\cdot\big)$ stabilization. The error analysis yields an $O(h^k)$ convergence under $\delta_h \lesssim h^2$ with regularity and parameter choices, including an explicit bound combining interpolation and consistency terms. Numerical experiments on circular and ring domains validate the theory, showing optimal convergence with boundary correction and suboptimal behavior without it. Overall, the method reduces mesh complexity for curved geometries while preserving high-order accuracy and stability in MFEM for Darcy flow.

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.

Figures (4)

• Figure 1: (a) The case when $\Omega_h^e\subset \Omega\backslash \Omega_h$. (b) The case when $\Omega_h^e\subset \Omega_h\backslash \Omega$. (c) The case when $e=\widetilde{e}$ and $\Omega_h^e = \emptyset$.
• Figure 2: The distance $\delta_h(\boldsymbol x_h)$ and the unit vector $\boldsymbol{\nu_h}$ on $\Gamma_h$.
• Figure 3: The true boundary $\Gamma$ (blue curve), the approximated boundary $\Gamma_h$ (red lines) and two typical regions $\Omega_h^{e,+},\Omega_h^{e,-}$ bounded by $\Gamma$ and $\Gamma_h$.
• Figure 4: (a). The mesh with $h=1/2$ on circular domain. (b). The mesh with $h=1/4$ on ring domain.

Theorems & Definitions (33)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Remark 3.1
• Lemma 3.1
• Lemma 3.2
• Lemma 4.1
• Lemma 4.2
• proof