On the implementation of the eXtended Finite Element Method (XFEM) for interface problems

TL;DR

This paper details the implementation of the XFEM in deal.II for interface problems, focusing on quadrature, shape functions, boundary conditions, and demonstrating its effectiveness through numerical examples.

Contribution

It provides a comprehensive implementation of XFEM in deal.II, including handling of quadrature, shape functions, and interface conditions, with numerical verification of optimal convergence.

Findings

Successful implementation of XFEM in deal.II for interface problems

Demonstration of the method's effectiveness on example problems

Numerical verification of optimal convergence

Abstract

The eXtended Finite Element Method (XFEM) is used to solve interface problems with an unfitted mesh. We present an implementation of the XFEM in the FEM-library deal.II. The main parts of the implementation are (i) the appropriate quadrature rule; (ii) the shape functions for the extended part of the finite element formulation; (iii) the boundary and interface conditions. We show how to handle the XFEM formulation providing a code that demonstrates the solution of two exemplary interface problems for a strong and a weak discontinuity respectively. In the weak discontinuity case, the loss of conformity due to the blending effect and its remedy are discussed. Furthermore, the optimal convergence of the presented unfitted method is numerically verified.