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The solution of the elliptic partial differential equation has interfacesingularity at the points which are either the intersections of interfaces orthe intersections of interfaces with the boundary of the domain. Thesingularities that arises in the elliptic interface problems are very complex.In this article, we propose an exponentially accurate nonconforming spectralelement method for these problems based on [7, 18]. A geometric mesh is used inthe neighbourhood of the singularities and the auxiliary map of the form$z=ln\xi$ is introduced to remove the singularities. The method is essentiallya least-squares method and the solution can be obtained by solving the normalequations using the preconditioned conjugate gradient method (PCGM) withoutcomputing the mass and stiffness matrices. Numerical examples are presented toshow the exponential accuracy of the method.

A study of spectral element method for elliptic interface problems with nonsmooth solutions in $\mathbb{R}^{2}$