A stabilized $P_1$ immersed finite element method for the interface elasticity problems
This work addresses the challenge of solving interface elasticity problems with non-matching meshes, providing a method that is both accurate and robust for heterogeneous materials.
The authors develop a stabilized P1 immersed finite element method for planar elasticity problems with interfaces, using meshes that do not align with the interface. They prove optimal H1 and divergence norm error estimates and demonstrate through numerical experiments that the method is optimal for various Lamé parameters and locking-free as λ→∞.
We develop a new finite element method for solving planar elasticity problems involving of heterogeneous materials with a mesh not necessarily aligning with the interface of the materials. This method is based on the `broken' Crouzeix-Raviart $P_1$-nonconforming finite element method for elliptic interface problems \cite{Kwak-We-Ch}. To ensure the coercivity of the bilinear form arising from using the nonconforming finite elements, we add stabilizing terms as in the discontinuous Galerkin (DG) method \cite{Arnold-IP},\cite{Ar-B-Co-Ma},\cite{Wheeler}. The novelty of our method is that we use meshes independent of the interface, so that the interface may cut through the elements. Instead, we modify the basis functions so that they satisfy the Laplace-Young condition along the interface of each element. We prove optimal $H^1$ and divergence norm error estimates. Numerical experiments are carried out to demonstrate that the our method is optimal for various Lamè parameters $μ$ and $λ$ and locking free as $λ\to\infty$.