A boundary-corrected weak Galerkin mixed finite method for elliptic interface problems with curved interfaces

TL;DR

The paper develops a boundary-corrected weak Galerkin mixed finite element method for elliptic interface problems with curved interfaces on body-fitted polygonal meshes. By transferring interface data from the curved interface $\Gamma$ to its polygonal approximation $\Gamma_h$ through a Taylor-based boundary value correction, it avoids numerical integration on curved elements while preserving accuracy. The Neumann interface condition is imposed weakly via a penalty, and an additional divergence stabilization enhances stability. The authors prove optimal-energy-norm convergence for arbitrary-order discretizations and validate the theory with numerical experiments on circular and curved interfaces.

Abstract

We propose a boundary-corrected weak Galerkin mixed finite element method for solving elliptic interface problems in 2D domains with curved interfaces. The method is formulated on body-fitted polygonal meshes, where interface edges are straight and may not align exactly with the curved physical interface. To address this discrepancy, a boundary value correction technique is employed to transfer the interface conditions from the physical interface to the approximate interface using a Taylor expansion approach. The Neumann interface condition is then weakly imposed in the variational formulation. This approach eliminates the need for numerical integration on curved elements, thereby reducing implementation complexity. We establish optimal-order convergence in the energy norm for arbitrary-order discretizations. Numerical results are provided to support the theoretical findings.

Figures (5)

• Figure 1: (a). The physical domain $\Omega$ is divided into $\Omega_1$ and $\Omega_2$. (b). The approximation domain $\Omega_{h,1}$ and $\Omega_{h,2}$, with $\Omega_{h,2}$ shaded in yellow.
• Figure 2: For $K\in\mathcal{T}_{h,1}$, there are three possibilities for $\Omega_h^e$. (a). The case for $\Omega_h^e\subset \Omega_1\backslash \Omega_{h,1}$. (b). The case for $\Omega_h^e\subset \Omega_{h,1}\backslash \Omega_1$. (c). The case for $e=\widetilde{e}$ and $\Omega_h^e=\emptyset$.
• Figure 3: The distance vector $\delta_h(\boldsymbol x_h)$ and the unit vector $\boldsymbol{\nu_h}$ to $\Gamma_h$.
• Figure 4: The true boundary $\Gamma$ (blue curve), the approximated boundary $\Gamma_h$ (red lines) and two typical regions $\Omega_h^{e_1,+},\Omega_h^{e_2,-}$ bounded by $\Gamma$ and $\Gamma_h$.
• Figure 5: (a). An example of triangular meshes ($h$=1/4) on domain with circular interface (blue dashed curve). (b). An example of quadrilateral mesh ($h$=1/4) on domain with curved interface $\Gamma$ (blue dashed curve).

Theorems & Definitions (35)

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