Mathematikgebäude, Seminarraum M614/616 (6. Etage)

We propose a continuous interior penalty (CIP) method for the pure transport problem and for the viscosity dependent 'Stokes-Brinkman' problem where the gradient jump penalty is localized to faces in the interior of subdomains. Special focus is given to the case, where the subdomains are so-called composite finite elements, i.e., for instance, quadrilateral, hexahedral or prismatic elements which are composed by simplices such that the arising global simplicial mesh is regular.
The big advantage of this local CIP is that it allows for static condensation in contrast to the classical CIP method. If the degrees of freedom in the interior of the composite finite elements are eliminated using static condensation then the resulting couplings of the skeleton degrees of freedom are comparable to those for classical conforming finite element methods which leads to a substantially smaller matrix stencil than for the standard global CIP-method. Optimal stability and error estimates are proved and numerical tests are presented.
For the Stokes-Brinkman model, our error bound does not increase if the viscosity parameter tends to zero which is mainly caused by adding a penalty term for the divergence of the velocity in the discretization. Moreover, the reduction effect of the static condensation is much stronger for this model since, beside of the elimination of all velocity degrees of freedom in the interior of each composite cell, all pressure degrees of freedom except for the cell-wise constants can be eliminated.

Prof. Dr. Friedhelm Schieweck

Otto-von-Guericke Universität Magdeburg