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

摘要

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.