Convergence of the Immersed Interface Method in Linear Elasticity

TL;DR

This paper proves that the immersed interface method's solution error in linear elasticity is proportional to the quadrature error, with convergence demonstrated in both bounded and unbounded domains.

Contribution

It establishes a theoretical link between quadrature approximation errors and solution accuracy in the immersed interface method for elasticity problems.

Findings

Solution error is of the same order as quadrature error.

Convergence demonstrated in ${f L}^2$-norm on curves and manifolds.

Numerical experiments confirm theoretical results.

Abstract

We consider an open, bounded, simply connected (Lipschitz) domain in $\mathbb{R}^d$, which contains a closed polyhedral surface or polygonal contour, referred to as the interface. From this interface, forces are exerted in the normal direction. The forces are continuously distributed over the interface, resulting in an integral expression. This features an important characteristic of the immersed interface method. Since the integral cannot be resolved exactly, one relies on numerical quadrature rules to approximate the integral. Therefore, we consider two different linear elasticity problems with forces over a curve or surface (interface) that is located within the (open) domain of computation: (1) The force is defined by an integral over the interface; (2) The force is defined by a quadrature approximation of the integral over the interface. We prove that the ${\bf L}^2$-norm of the difference between the solutions from the two elasticity problems is of the same order as the error of quadrature. The results are demonstrated for both bounded and unbounded domains. The proof that we establish relies on the use of: (i) fundamental solutions for linear elasticity, exhibiting singular behaviors (in particular around points of action) and not being in ${\bf H}^1$, and (ii) on the use of singularity removal principle and the Extended Trace Theorem. Convergence is demonstrated in the ${\bf L}^2$-norm on curves and manifolds. We show some numerical experiments on the basis of fundamental solutions with a Midpoint quadrature rule in an unbounded and a bounded domain. We note that the error that we estimate is for the exact solutions and not for finite element solutions. Hence in the numerical finite element-based simulations, the numerical results contain an additional error due to the finite element approach.