THE COMPOSITE MINI ELEMENT—COARSE MESHCOMPUTATION OF STOKES FLOWS ON COMPLICATEDDOMAINS

摘要

We introduce a new finite element method, the composite mini element, for the mixeddiscretization of the Stokes equations on two- and three-dimensional domains that may contain ahuge 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 meshsize of the underlying triangulation, while the computational effort is determined by its number ofelements. If the physical domain is very complicated, then the minimal number of simplices, which arenecessary to resolve the domain, can be very large and distributed in a nonoptimal way with respectto the approximation quality. In contrast to that, the minimal dimension of the composite minielement space is independent of the number of geometric details. Instead of a geometric resolutionof the domain and the boundary condition by the finite element mesh the shape of the finite elementfunctions is adapted to the geometric details. This approach allows low-dimensional approximationseven for problems with complicated geometric details such as holes or rough boundaries. We proveits linear (optimal order) approximability and its inf-sup stability. Further, we will be able to controlthe nonconformity in the space without increasing the space dimension in such a way that the a ,ΩuCME||1 priori error estimate ||u-IIP-PCME||0,Ω ≤ h||f||0,Ωholds. Thereby, in contrast to theclassical methods, the choice of the mesh size parameter h is not constrained by the size of geometricdetails. In addition, it turns out that the method can be viewed as a coarse-scale generalization ofthe classical mini element approach; i.e., it reduces the computational effort, while the approximationquality depends on the (coarse) mesh size in the usual way.