Trace Inverse Estimates and Kernel-based Nitsche Galerkin Methods for Elliptic Interface Problems

Abstract

We establish trace inverse estimates for products of independently generated Sobolev-kernel trial spaces on Lipschitz subdomains and use them to analyze symmetric Nitsche Galerkin methods for elliptic interface problems. The estimates control one-sided gradient traces by subdomain gradients, the interface jump, and an exterior boundary trace. Their conormal form provides the flux control required by the Nitsche formulation without imposing value constraints on the trial spaces, including configurations with an interior inclusion. For quasi-uniform center families with comparable fill distances and a fixed kernel scale, we obtain coercivity and well-posedness for sufficiently large penalty parameters independent of refinement. We prove energy and $L^2$ error bounds of orders $h^{s-1}$ and $h^s$ for piecewise $H^s$ solutions, $2\le s\le m$, where $h$ is the maximum fill distance and $m$ is the kernel Sobolev order. The $L^2$ bound additionally requires piecewise $H^2$ regularity of the homogeneous adjoint interface problem. For scaled kernels ${\Phi }_{m,\epsilon }$ with $0<\epsilon \le 1$, we also derive a scale-explicit condition-number bound in the kernel-translate basis. The analysis assumes exact geometry and integration. Planar experiments examine convergence and support scaling, whereas a three-dimensional mixed-boundary calculation illustrates the method.