Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes
Abstract
We develop an arbitrary-degree geometry-conforming finite element (GC-FE) framework for two-dimensional elliptic boundary value and interface problems on curved domains. Using the Frenet--Serret transformation, curved-boundary and interface-fitted segments are represented exactly, while polynomials in Frenet coordinates generate generally nonpolynomial local shape functions in physical coordinates. For interface-unfitted meshes, GC-FE spaces on curved-boundary elements are coupled with geometry-conforming immersed finite element (GC-IFE) spaces on interface-cut elements, with standard polynomial spaces used elsewhere. We establish optimal approximation, inverse, and trace estimates for the GC-FE spaces. For fitted meshes, we prove well-posedness and optimal error estimates in energy and L2 norms for a symmetric interior penalty discontinuous Galerkin discretization. By retaining the prescribed curves exactly, the method avoids the geometric variational crime associated with curved-geometry approximation and requires no corresponding geometric consistency estimates. Numerical experiments confirm the predicted rates, show global accuracy comparable to nodal isoparametric finite elements and smaller true-interface trace errors in the reported tests, and demonstrate the coupled GC-FE-GC-IFE method on interface-unfitted meshes.
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BibTeX
@misc{lin2026geometryconforming,
title = {{Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes}},
author = {Yuanhui Lin and Tao Lin and Xu Zhang and Minfu Feng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.14484},
eprint = {2608.14484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14484},
}
RIS
TY - UNPB
TI - Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes
AU - Yuanhui Lin
AU - Tao Lin
AU - Xu Zhang
AU - Minfu Feng
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2608.14484
UR - https://arxiv.org/abs/2608.14484
AB - We develop an arbitrary-degree geometry-conforming finite element (GC-FE) framework for two-dimensional elliptic boundary value and interface problems on curved domains. Using the Frenet--Serret transformation, curved-boundary and interface-fitted segments are represented exactly, while polynomials in Frenet coordinates generate generally nonpolynomial local shape functions in physical coordinates. For interface-unfitted meshes, GC-FE spaces on curved-boundary elements are coupled with geometry-conforming immersed finite element (GC-IFE) spaces on interface-cut elements, with standard polynomial spaces used elsewhere. We establish optimal approximation, inverse, and trace estimates for the GC-FE spaces. For fitted meshes, we prove well-posedness and optimal error estimates in energy and L^2 norms for a symmetric interior penalty discontinuous Galerkin discretization. By retaining the prescribed curves exactly, the method avoids the geometric variational crime associated with curved-geometry approximation and requires no corresponding geometric consistency estimates. Numerical experiments confirm the predicted rates, show global accuracy comparable to nodal isoparametric finite elements and smaller true-interface trace errors in the reported tests, and demonstrate the coupled GC-FE-GC-IFE method on interface-unfitted meshes.
ER -