A direct method for accurate solution and gradient computations for elliptic interface problems

Abstract

Recently, an augmented IIM (Li et al. SIAM J. Numer. Anal. 55, 570–597 2017) has been developed and analyzed for some interface problems. The augmented IIM can provide not only a second-order accurate solution in the entire domain but also second-order accurate gradients from each side of the interface. In the augmented IIM, the PDE is reformulated and an augmented variable of co-dimension one is introduced and solved through a Schur complement system. Nevertheless, the augmented IIM is somewhat difficult to implement and understand for non-experts in the area. In this paper, a direct method (thus simpler than augmented IIM) without using the augmented variable is proposed for elliptic interface problems with a variable coefficient that can have a finite jump across an interface. The resulting finite difference scheme is the standard five-point central scheme at regular grid points, while it is a compact nine-point scheme at irregular grid points. The computed solution is second-order accurate and will be used to recover the gradient from each side of the interface to second-order accuracy. The discrete Green’s function is constructed and used to prove second-order convergence of both the solution and gradient for the one-dimensional algorithm with a piecewise constant coefficient. For the two-dimensional algorithm with a piecewise constant coefficient, we numerically prove second-order convergence of the solution by applying the discrete elliptic maximum principle using an optimization solver, while the second-order convergence of the gradient and the same conclusion for more general problems will be only demonstrated in the numerical examples.