Numerical studies on the stability of mixed finite elements over anisotropic meshes arising from immersed boundary Stokes problems

Motivated by a recently proposed local refinement strategy for immersed interface problems,in this work we aim at dealing with the behavior of mixed finite elements for the Stokes problem in (strongly) anisotropic mesh situations,leading to severely distorted elements. In fact,the majority of the theoretical results present in the finite element literature has been carried out under the assumption of well-shaped elements. In the case such a condition is not satisfied,the inf-sup constant may degenerate,thus leading to the instability of the system. To this aim,we herein use a generalized eigenvalue test problem that allows to conveniently investigate the behavior of different mixed finite elements over anisotropic mesh patterns,arising,e.g.,in immersed interface Stokes problems. We then test and study the numerical stability of two 2D finite element pairs,namely the Hood-Taylor (P2/P1) and the Hood-Taylor (P+ 2/P1) with a velocity field enhanced by a cubic bubble. On the contrary to the results presented in [3],we herein provide additional results on the potential presence of spurious modes and their locations. The present results corroborate those obtained in more complex cases described in [3] and [4].