Approximation Capabilities of Immersed Finite Element Spaces for Elasticity Interface Problems

TL;DR

This paper develops and analyzes immersed finite element spaces with various polynomial types for solving elasticity interface problems, demonstrating their optimal approximation capabilities regardless of interface position or material parameters.

Contribution

It introduces a new class of IFE spaces with guaranteed unisolvence and optimal approximation properties for elasticity interface problems.

Findings

Unisolvence of IFE shape functions is guaranteed regardless of interface location.

Optimal approximation capabilities are established through multi-point Taylor expansion.

Boundedness and identities of IFE shape functions are proven.

Abstract

We construct and analyze a group of immersed finite element (IFE) spaces formed by linear, bilinear and rotated Q1 polynomials for solving planar elasticity equation involving interface. The shape functions in these IFE spaces are constructed through a group of approximate jump conditions such that the unisolvence of the bilinear and rotated Q1 IFE shape functions are always guaranteed regardless of the Lam\`e parameters and the interface location. The boundedness property and a group of identities of the proposed IFE shape functions are established. A multi-point Taylor expansion is utilized to show the optimal approximation capabilities for the proposed IFE spaces through the Lagrange type interpolation operators.