Error estimates for an augmented method for one-dimensional elliptic interface problems

Qian Zhang,Bin Zhang,Zhifeng Weng,Haifeng Ji

doi:10.1186/s13662-018-1771-z

Abstract

Elliptic interface problems have many important scientific and engineering applications. Interface problems are encountered when the computational domain involves multi-materials with different conductivities, densities, or permeability. The solution or its gradient often has a jump across the interface due to discontinuous coefficients or singular sources. In this paper, optimal convergence of an augmented method is derived for one-dimensional interface problems. The dependence of the discontinuous coefficient in the error analysis is also considered. Numerical examples are presented to confirm the theoretical analysis and show that the estimate is sharp.

Highlights

In scientific computation, we often encounter interface problems when multi-materials with different conductivities, densities, or permeability are involved
Since the interface is moving, a new fitted mesh has to be generated at each time step and an interpolation is required to transfer the numerical solutions solved on different meshes
The difficulty is that the interface can pass through the interior of elements of the mesh

Summary

Introduction

We often encounter interface problems when multi-materials with different conductivities, densities, or permeability are involved. Since the interface is moving, a new fitted mesh has to be generated at each time step and an interpolation is required to transfer the numerical solutions solved on different meshes. From this point of view, it would be preferable to use an unfitted mesh in which the interface can be arbitrarily located with respect to the fixed background mesh. The immersed finite element methods (IFEMs) are a class of unfitted mesh methods that modify the basis function on interface elements according to the interface conditions to capture the jumps of the exact solution.

Preliminaries

Conclusion