Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
[1] G. W. Zumbusch. Symmetric hierarchical polynomials and the adaptive h-p-version. Houston Journal of Mathematics, pages 529-540, 1996. Proceedings of the Third International Conference on Spectral and High Order Methods, ICOSAHOM'95, also as report SC-95-18 ZIB, Berlin.
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The h-p-version of finite-elements delivers a sub-exponential convergence in the energy norm. A step towards a full adaptive implementation is taken in the context of unstructured meshes of simplices with variable order p in space. Both assumptions lead to desirable properties of shape functions like symmetry, p-hierarchy and simple coupling of elements.
In a first step it is demonstrated that for standard polynomial vector spaces on simplices not all of these features can be obtained simultaneously. However, this is possible if these spaces are slightly extended or reduced. Thus a new class of polynomial shape functions is derived, which are especially well suited for three dimensional tetrahedra.
The construction is completed by directly minimizing the condition numbers of the arising preconditioned local finite element matrices. The preconditioner is based on two-step domain decomposition techniques using a multigrid solver for the global linear problem p=1 and direct solvers for local higher order problems.
Some numerical results concerning an adaptive (feedback) version of h-p finite elements are presented.