An Arbitrarily High Order Unfitted Finite Element Method for Elliptic Interface Problems with Automatic Mesh Generation

TL;DR

This paper introduces a high-order unfitted finite element method with automatic mesh generation for elliptic interface problems, improving mesh quality and stability on Cartesian grids with complex interfaces.

Contribution

It develops a reliable algorithm for automatic mesh refinement and new basis functions to control condition numbers, enabling high-order accuracy for interface problems.

Findings

Method achieves high-order accuracy on complex interfaces.

Automatic mesh merging improves computational efficiency.

Numerical tests demonstrate competitive performance.

Abstract

We consider the reliable implementation of high-order unfitted finite element methods on Cartesian meshes with hanging nodes for elliptic interface problems. We construct a reliable algorithm to merge small interface elements with their surrounding elements to automatically generate the finite element mesh whose elements are large with respect to both domains. We propose new basis functions for the interface elements to control the growth of the condition number of the stiffness matrix in terms of the finite element approximation order, the number of elements of the mesh, and the interface deviation which quantifies the mesh resolution of the geometry of the interface. Numerical examples are presented to illustrate the competitive performance of the method.