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

TL;DR

This paper introduces an exponentially accurate spectral element method for elliptic interface problems with nonsmooth solutions, utilizing geometric meshes and a logarithmic auxiliary map to effectively handle singularities.

Contribution

The paper proposes a novel nonconforming spectral element method that achieves exponential accuracy for elliptic interface problems with complex singularities, using geometric meshes and a special auxiliary map.

Findings

The method attains exponential accuracy in numerical examples.

It effectively handles interface singularities with geometric meshes.

Solution is computed efficiently via preconditioned conjugate gradient method.

Abstract

The solution of the elliptic partial differential equation has interface singularity at the points which are either the intersections of interfaces or the intersections of interfaces with the boundary of the domain. The singularities that arises in the elliptic interface problems are very complex. In this article, we propose an exponentially accurate nonconforming spectral element method for these problems based on [7, 18]. A geometric mesh is used in the neighbourhood of the singularities and the auxiliary map of the form $z=ln\xi$ is introduced to remove the singularities. The method is essentially a least-squares method and the solution can be obtained by solving the normal equations using the preconditioned conjugate gradient method (PCGM) without computing the mass and stiffness matrices. Numerical examples are presented to show the exponential accuracy of the method.