A High-Order Mixed Finite Element Method for Second Order Elliptic Equations on Curved Domain with Boundary Value Corrections

Abstract

This paper presents a boundary corrections mixed finite element method for second order elliptic equations with the non-homogeneous Neumann boundary condition on curved domains. A key feature of the boundary value corrections is the shift from the true boundary to a surrogate boundary, which avoids numerical integration formula on curved elements. We consider the high-order Raviart-Thomas element $(RT_k)$ of degree $k ≥ 1$ on triangular meshes, achieving an $O(h^{k+1/2})$ convergence in the $L^2$-norm estimate for the velocity field and an $O(h^k)$ convergence in the $H^1$-norm estimate for the pressure. Finally, numerical experiments validate our theoretical results.