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

Abstract

Kernel Galerkin methods allow meshfree trial spaces to be generated independently in each material subdomain and refined through their center sets, without generating an approximation mesh. We establish trace inverse estimates for products of Sobolev-kernel spaces on Lipschitz subdomains and use them to analyze symmetric Nitsche discretizations of linear elliptic interface problems. The estimates control gradient traces on subdomain boundaries by subdomain gradients, the interface jump, and an exterior boundary trace. Their conormal form supplies the flux control required for stability directly from the kernel and center assumptions, without a separate discrete flux hypothesis. For quasi-uniform center families with comparable fill distances and a fixed kernel scale, we obtain 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 subdomain 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 kernels ${\Phi }_m(\epsilon \cdot )$ with $0<\epsilon \le 1$, we derive a scale-explicit condition-number bound in the kernel-translate basis and an $O(h^{-2})$ bound in an $L^2$-orthonormal basis. The analysis assumes exact geometry and integration. Planar experiments examine convergence and kernel scaling, while a three-dimensional heat-conduction problem in a curved composite illustrates the method.