Convergence of the Immersed Interface Method in Linear Elasticity

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.

💡 Research Summary

The manuscript investigates the convergence properties of the Immersed Interface Method (IIM) when applied to linear elasticity problems that involve forces distributed continuously along an internal interface. The authors consider a bounded, simply‑connected Lipschitz domain Ω ⊂ ℝ^d (d = 2 or 3) that contains a closed polyhedral curve (in 2‑D) or surface (in 3‑D) denoted by Γ. On Γ a normal‑directed force density Q(x) n(x) is prescribed, where Q is a smooth scalar function and n is the outward unit normal. In the continuous formulation the body force is expressed as an integral over Γ involving the Dirac delta distribution: \