A Cut-Cell Implementation of the Finite Element Method in deal.ii

The modeling of problems where the boundary changes significantly over time may become challenging as the mesh needs to be adapted constantly. In this context, computational methods where the mesh does not conform to the boundary are of great interest. This paper proposes a stabilized cut-cell approach to solve partial differential equations using unfitted meshes using the Finite Element Method. The open-source library deal.ii was used for implementation. In order to evaluate the method, three problems in two-dimensions were tested: the Poisson problem, a pure diffusion Laplace-Beltrami problem and a reaction diffusion case. Stabilization effects on the stiffness matrix were studied for the first two test cases, and the theoretical dependence of the condition number with mesh size was confirmed. In addition, an optimal stabilization parameter was defined. Optimal convergence rates were obtained for the first two test cases