Trace Inverse Estimates and Kernel-based Nitsche 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 L2 error bounds of orders hs−1 and hs for piecewise Hs solutions, 2≤ s≤ m, where h is the maximum subdomain fill distance and m is the kernel Sobolev order. The L2 bound additionally requires piecewise H2 regularity of the homogeneous adjoint interface problem. For kernels Φm(ε ·) with 0<ε≤1, we derive a scale-explicit condition-number bound in the kernel-translate basis and an O(h−2) bound in an L2-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.
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BibTeX
@misc{ling2026trace,
title = {{Trace Inverse Estimates and Kernel-based Nitsche Galerkin Methods for Elliptic Interface Problems}},
author = {Leevan Ling and Shu Ma and Weifeng Qiu and Yizhong Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27537},
eprint = {2610.27537},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27537},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Trace Inverse Estimates and Kernel-based Nitsche Galerkin Methods for Elliptic Interface Problems
AU - Leevan Ling
AU - Shu Ma
AU - Weifeng Qiu
AU - Yizhong Sun
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27537
UR - https://arxiv.org/abs/2610.27537
AB - 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≤ s≤ 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 Φ_m(ε ·) with 0<ε≤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.
ER -