Immersed-interface finite element method based on a nonconformal Petrov-Galerkin formulation

Immersed interface methods are becoming increasingly necessary to solve problems of practical importance. To be considered as replacement for conventional, interface-fitted, finite element method (FEM), an immersed-interface finite element method (IIFEM) has to exhibit equivalent accuracy and robustness. The paper develops such an IIFEM based on nonconformal Petrov-Galerkin (ncPG) formulation for two-dimensional linear elasticity. First, a bimaterial patch test (BPT) is developed which, when satisfied, guarantees consistency of an FEM formulation for problems with two materials in a single element. Then, nodal trial functions that satisfy the BPT are derived for triangular elements with immersed interface. These trial functions satisfy the Hadamard interface conditions for elements with immersed interface but are nonconformal on element edges with an immersed interface. Conventional conformal test functions are used in the Petrov-Galerkin formulation. The proposed ncPG-IIFEM is applied to a benchmark multimaterial problem. The solution using ncPG-IIFEM converges optimally and the error in displacement as well as stress are similar to interface-fitted FEM. The ncPG-IIFEM also leads to a well-conditioned linear system with the condition number of the global stiffness matrix similar to interface-fitted FEM. Overall, the proposed ncPG-IIFEM is shown to be straightforward in application, optimal in convergence, and robust.

Automated summary

Immersed interface methods are increasingly necessary for solving practical problems. The paper introduces an immersed-interface finite element method based on nonconformal Petrov-Galerkin formulation, ensuring consistency through a bimaterial patch test and deriving nodal trial functions satisfying Hadamard interface conditions. The method demonstrates optimal convergence and robustness when applied to a benchmark multimaterial problem.