Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

  author = {G. W. Zumbusch},
  title = {Symmetric Hierarchical Polynomials and the Adaptive
  journal = {Houston Journal of Mathematics},
  address = {Houston, Texas},
  year = {1996},
  pages = {529--540},
  editor = {A.V. Ilin and L. R. Scott},
  note = {Proceedings of the Third International Conference on
		  Spectral and High Order Methods, ICOSAHOM'95, also as
		  report SC-95-18 ZIB, Berlin},
  ps = { 1},
  pdf = { 1},
  annote = {refereed},
  abstract = {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.}