Stabilization techniques and adaptive conjugate gradient solver tolerances for the finite cell method

Immersed boundary methods such as the finite cell method provide a versatile tool for the analysis of structures, which are difficult to discretize in a boundary conforming manner due to their complex geometries. Using Cartesian grids and a fictitious domain approach, the effort is shifted from meshing toward the quadrature, which needs to be adapted to discontinuous integrands for cut cells, i.e., elements cut by the immersed boundary. Further, the condition number of the stiffness matrix of such discretizations is typically much larger compared to boundary fitted discretizations, making the use of iterative solvers challenging. In order to restore the performance of iterative solvers, i.e., lower the condition number of the stiffness matrix, several stabilization methods are available. In this work, we compare the performance of two stabilization methods. The classical $ \alpha $-stabilization method uses a material with low (but non-zero) stiffness in the fictitious domain. The so called eigenvalue- or $ \epsilon $-stabilization is based on an eigendecomposition of the stiffness matrix and selectively stabilizes modes, which are associated with very low eigenvalues. Generally, both stabilization methods introduce an error, which, however, can be corrected. This work includes an investigation of such correction mechanisms for linear and nonlinear problems.