Local Projection Stabilization for the Stokes System on Anisotropic Quadrilateral Meshes

The local projection stabilization (LPS) is suitable to stabilize the saddle point structure of the Stokes system when equal-order finite elements are used, as well as convective terms for Navier-Stokes. Hence, it has already been applied with large success to different fields of computational fluid dynamics, e.g., in 3D incompressible flows [7], compressible flows [12], reactive flows [8], parameter estimation [4, 5] and optimal control problems [11]. Although locally refined meshes have been used for this stabilization technique in all of these applications, the meshes have been isotropic so far. The solution of partial differential equations on anisotropic meshes are of substantial importance for efficient solutions of problems with interior layers or boundary layers, as for instance in fluid dynamics at higher Reynolds number. It is well known that stabilized finite element schemes, e.g. streamline upwind Petrov-Galerkin (SUPG), see [10], or pressure stabilized Petrov-Galerkin (PSPG), see [9], must be modified in the case of anisotropy. Becker has shown in [2] how the PSPG stabilization should be modified on anisotropic Cartesian grids. In this work, we make the first step of formulating LPS on anisotropic quadrilateral meshes by considering the Stokes system in the domain Ω ⊂ R for velocity v and pressure p: