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

  author = {M.~Griebel and S.~Knapek},
  title = {Optimized tensor-product approximation spaces},
  journal = {Constructive Approximation},
  year = {2000},
  optkey = {},
  volume = {16},
  number = {4},
  pages = {525--540},
  optmonth = {},
  abstract = {This paper is concerned with the construction of optimized
		  grids and approximation spaces for elliptic differential
		  and integral equations. The main result is the analysis of
		  the approximation of the embedding of the intersection of
		  classes of functions with bounded mixed derivatives in
		  standard Sobolev spaces. Based on the framework of
		  tensor-product biorthogonal wavelet bases and stable
		  subspace splittings, the problem is reduced to diagonal
		  mappings between Hilbert sequence spaces. We construct
		  operator adapted finite-element subspaces with a lower
		  dimension than the standard full-grid spaces. These new
		  approximation spaces preserve the approximation order of
		  the standard full-grid spaces, provided that certain
		  additional regularity assumptions are fulfilled. The form
		  of the approximation spaces is governed by the ratios of
		  the smoothness exponents of the considered classes of
		  functions. We show in which cases the so called curse of
		  dimensionality can be broken. The theory covers elliptic
		  boundary value problems as well as boundary integral
  optnote = {},
  annote = {article,245C},
  ps = { 1}