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.
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.
results concerning an adaptive (feedback) version of
h-p finite elements are presented.