A three-dimensional high-order geometry-aligned immersed finite element method for elliptic interface problems

Abstract

This paper presents a three-dimensional high-order geometry-aligned immersed finite element (IFE) method for elliptic interface problems with curved interfaces. The framework supports two alternative geometric pathways: direct evaluation from a sufficiently regular global surface parameterization, and high-order element-wise Legendre--Monge reconstruction from surface samples and, when available, consistently oriented normal data. In the latter case, the local height function is approximated by scaled Legendre polynomials and converted into a signed implicit representation. Since the IFE basis functions are constructed within the same Legendre polynomial space, the geometric description and physical finite-element approximation share a unified mathematical foundation, which is the extension of the classical isoparametric concept. This consistent local reconstruction and its implicit representation mitigate error coupling between geometric approximation and finite-element discretization, while facilitating both IFE basis-function construction and numerical integration. A purely algebraic procedure is developed to evaluate higher-order normal derivatives of the Laplacian using the chain rule and metric tensor variations, without explicit symbolic expansions. Numerical integration on interface elements is realized using the Saye-type dimension-reduction scheme. Only one-dimensional root-finding and one-dimensional Gaussian quadrature are required, which greatly lowers the integration complexity. Numerical experiments confirm optimal convergence rates of the present method.