An “immersed” finite element method based on a locally anisotropic remeshing for the incompressible Stokes problem

In the present paper we study a finite element method for the incompressible Stokes problem with a boundary immersed in the domain on which essential constraints are imposed. Such type of methods may be useful to tackle problems with complex geometries, interfaces such as multiphase flow and fluid-structure interaction. The method we study herein consists in locally refining elements crossed by the immersed boundary such that newly added elements, called subelements, fit the immersed boundary. In this sense, this approach is of a fitted type, but with an original mesh given independently of the location of the immersed boundary. We use such a subdivision technique to build a new finite element basis, which enables us to represent accurately the immersed boundary and to impose strongly Dirichlet boundary conditions on it. However, the subdivision process may imply the generation of anisotropic elements, which, for the incompressible Stokes problem, may result in the loss of inf-sup stability even for well-known stable element schemes. We therefore use a finite element approximation, which appears stable also on anisotropic elements. We perform numerical tests to check stability of the chosen finite elements. Several numerical experiments are finally presented to illustrate the capabilities of the method. The method is presented for two-dimensional problems.