Sparse Approximation of Singularity Functions

Abstract We are concerned with the sparse approximation of functions on the d-dimensional unit cube [0,1]d, which contain powers of distance functions to lower-dimensional k-faces (corners, edges, etc.). These functions arise, e.g., from corners, edges, etc., of domains in solutions to elliptic PDEs. Usually, they deteriorate the rate of convergence of numerical algorithms to approximate these solutions. We show that functions of this type can be approximated with respect to the H1 norm by sparse grid wavelet spaces VL, (VL) = NL, of biorthogonal spline wavelets of degree p essentially at the rate p: \[ \|u - P_Lu\|_{H^1([0,1]^d)} \leq CN_L^{-p}\,(\log_2 N_L)^s \|u\|, \qquad s = s(p,d), \] where || · || is a weighted Sobolev norm and PLu \in VL.

[1]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[2]  S. Nicaise,et al.  Approximation par préondelettes augmentée de l'équation de la plaque libre polygonale , 1999 .

[3]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[4]  Reinhold Schneider,et al.  Multiwavelets for Second-Kind Integral Equations , 1997 .

[5]  Wolfgang Dahmen,et al.  Adaptive Wavelet Schemes for Elliptic Problems - Implementation and Numerical Experiments , 2001, SIAM J. Sci. Comput..

[6]  Serge Nicaise,et al.  Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III: finite element methods on polygonal domains , 1992 .

[7]  J. Roßmann,et al.  On the Agmon-Miranda maximum principle for solutions of strongly elliptic equations in domains of ℝn with conical points , 1992 .

[8]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[9]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[10]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .

[11]  Olivier Pironneau,et al.  46. Best N-term capacitance approximation on sparse grids , 2001 .

[12]  J. Roßmann,et al.  Elliptic Boundary Value Problems in Domains with Point Singularities , 2002 .

[13]  Michael Griebel,et al.  Sparse grids for boundary integral equations , 1999, Numerische Mathematik.

[14]  W. Dahmen,et al.  Wavelets with Complementary Boundary Conditions — Function Spaces on the Cube , 1998 .

[15]  T. Apel.  Anisotropic Finite Elements: Local Estimates and Applications , 1999 .

[16]  R. DeVore,et al.  Nonlinear approximation , 1998, Acta Numerica.

[17]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

[18]  Ricardo H. Nochetto,et al.  Data Oscillation and Convergence of Adaptive FEM , 2000, SIAM J. Numer. Anal..

[19]  M. Griebel,et al.  Optimized Tensor-Product Approximation Spaces , 2000 .

[20]  Hans-Joachim Bungartz,et al.  A Note on the Complexity of Solving Poisson's Equation for Spaces of Bounded Mixed Derivatives , 1999, J. Complex..