Superconvergence of Immersed Finite Element Methods for Interface Problems

TL;DR

This paper investigates superconvergence properties of immersed finite element methods for 1D interface problems, demonstrating that these methods retain superconvergence without mesh alignment, even with low solution regularity.

Contribution

It shows that immersed finite element methods preserve superconvergence properties without requiring mesh-interface alignment, unlike classical methods.

Findings

Superconvergence occurs at roots of generalized orthogonal polynomials.

Immersed finite element solutions inherit superconvergence properties from standard FEM.

Superconvergence is achieved despite low regularity of the solution.

Abstract

In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions.