Abstract
We introduce a new finite element method, the composite mini element, for the mixed discretization of the Stokes equations on two- and three-dimensional domains that may contain a huge number of geometric details. In standard finite element discretizations of the Stokes problem, such as the classical mini element, the approximation quality is determined by the maximal mesh size of the underlying triangulation, while the computational effort is determined by its number of elements. If the physical domain is very complicated, then the minimal number of simplices, which are necessary to resolve the domain, can be very large and distributed in a nonoptimal way with respect to the approximation quality. In contrast to that, the minimal dimension of the composite mini element space is independent of the number of geometric details. Instead of a geometric resolution of the domain and the boundary condition by the finite element mesh the shape of the finite element functions is adapted to the geometric details. This approach allows low-dimensional approximations even for problems with complicated geometric details such as holes or rough boundaries. We prove its linear (optimal order) approximability and its inf-sup stability. Further, we will be able to control the nonconformity in the space without increasing the space dimension in such a way that the a priori error estimate $\|{\mathbf{u}-\mathbf{u}^{\mathrm{CME}}}\|_{1,\Omega}+\|{p-p^{\mathrm{CME}}} \|_{0,\Omega}\lesssim h\|{\mathbf{f}}\|_{0,\Omega}$ holds. Thereby, in contrast to the classical methods, the choice of the mesh size parameter $h$ is not constrained by the size of geometric details. In addition, it turns out that the method can be viewed as a coarse-scale generalization of the classical mini element approach; i.e., it reduces the computational effort, while the approximation quality depends on the (coarse) mesh size in the usual way.
Petersheim, Daniel