A simple method for matrix-valued coefficient elliptic equations with sharp-edged interfaces

Keywords: Traditional finite element method Elliptic equation with matrix-valued coefficients Body-fitting grids Sharp-edged interface Symmetric positive definite a b s t r a c t The traditional finite element method has a number of nice properties, and thus it is highly desired for matrix-valued coefficient elliptic equations with sharp-edged interfaces. However , its efficient implementation with body-fitting grids for such problems is highly non-trivial. In this paper, we propose a simple finite element method with body-fitting grids based on semi-Cartesian grid, which makes its implementation fairly straightforward, even for complicated geometry. All possible situations that the interface cuts the grid are considered. Further, the symmetry and positive definiteness of the resulting matrix are ensured for positive definite coefficients. Numerical experiments show that it is second order accurate in the L 1 norm and numerically very stable. Elliptic interface problems arise naturally in a wide variety of applications, especially problems with multiphysics/mul-tiscale features. Therefore, its efficient numerical solution is of immense practical interest, and has received considerable interest in the past decades. In the literature, there are a large number of methods for solving such problems, e.g., immersed interface method (IIM), boundary condition capturing method (BCCM), matched interface and boundary method (MIBM), immersed finite element method (IFEM), nontraditional finite element method (NFEM) and finite element method using a Cartesian grid with added nodes (FEMCGAN), which we briefly review below. The IIM, developed in [1], incorporates the interface conditions for the solution and conormal derivative, i.e., ½uŠ – 0 and b @u @n  à – 0, directly into the finite difference stencil. The resulting linear system is sparse, but neither symmetric nor positive definite, which calls for fast iterative methods. In [2], a fast iterative method was developed for constant coefficient problems with the interface conditions ½uŠ ¼ 0 and b @u @n  à – 0, where non-body-fitting Cartesian grids are used and interfaces are not necessarily aligned with element edges. Numerical experiments indicate that the conforming and nonconforming versions respectively achieve second-order and first-order accuracy in the L 1 norm. For an updated overview of the method and its diverse applications, we refer to [3]. The BCCM [4] uses the ghost fluid method [5] to capture the boundary conditions. It is robust and simple to implement, and admits rigorous convergence proof via a variational interpretation [6]. It has been sped up by a multigrid method [7]. The method works …