Error estimates for a partially penalized immersed finite element method for elasticity interface problems

¹ Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
² Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong, PR China

^* Corresponding author: tlin@vt.edu

Received: 29 November 2018
Accepted: 28 June 2019

Abstract

This article is about the error analysis for a partially penalized immersed finite element (PPIFE) method designed to solve linear planar-elasticity problems whose Lamé parameters are piecewise constants with an interface-independent mesh. The bilinear form in this method contains penalties to handle the discontinuity in the global immersed finite element (IFE) functions across interface edges. We establish a stress trace inequality for IFE functions on interface elements, we employ a patch idea to derive an optimal error bound for the stress of the IFE interpolation on interface edges, and we design a suitable energy norm by which the bilinear form in this PPIFE method is coercive. These key ingredients enable us to prove that this PPIFE method converges optimally in both an energy norm and the usual L² norm under the standard piecewise H²-regularity assumption for the exact solution. Features of the proposed PPIFE method are demonstrated with numerical examples.