Nonconforming Least-Squares Method for Elliptic Partial Differential Equations with Smooth Interfaces

In this paper a numerical method based on least-squares approximation is proposed for elliptic interface problems in two dimensions, where the interface is smooth. The underlying method is spectral element method. In the least-squares formulation a functional is minimized as defined in (4.1). The jump in the solution and its normal derivative across the interface are enforced (in an appropriate Sobolev norm) in the functional. The solution is obtained by solving the normal equations using preconditioned conjugate gradient method. Essentially the method is nonconforming, so a block diagonal matrix is constructed as a preconditioner based on the stability estimate where each diagonal block is decoupled. A conforming solution is obtained by making a set of corrections to the nonconforming solution as in [24] and an error estimate in H-norm is given which shows the exponential convergence of the proposed method. MSC: 65M70,65N35,65Y05,74B05