Unfitted finite element method for fully coupled poroelasticity with stabilization

Abstract

The complex geometries of real-world problems pose a considerable challenge to the preprocessing of the standard finite element method (FEM), wherein the mesh conforms to the physical domain. In this regard, an unfitted FEM with the physical domain embedded in a relatively simple mesh can be a promising substitute for the fitted FEM. However, the use of the unfitted FEM for poroelasticity problems with continuous interpolation is seldom reported. Hence, in this study, we propose a symmetric weak formulation based on the modification of Nitsche’s method for the numerical computation of fully coupled two-field poroelasticity using the FEM with general unfitted meshes. Furthermore, three stabilization terms based on the ghost penalty method are introduced to address the ill-conditioning caused by the small cut ratio of the physical part of an element to the whole element. Benchmark problems and a geotechnical engineering problem are solved using the proposed methodology. It is numerically demonstrated that the condition number of the coefficient matrix is bounded even when there is a tiny cut ratio. Both the condition numbers and the accuracies remain stable irrespective of how the boundary cuts the mesh. More importantly, optimal convergence rates concerning both space and time are preserved. This study substantially facilitates the numerical computation of poroelasticity problems with complex geometries.