Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading

Authors:
Alexey Chernov
Affiliations:
Hausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Venue:
Applied Numerical Mathematics
Year:
2012

Citing 9
Cited 0

The optimal convergence rate of the p-version of the finite element method

SIAM Journal on Numerical Analysis
Stochastic finite elements: a spectral approach

Stochastic finite elements: a spectral approach
A posteriori error estimates for boundary element methods

Mathematics of Computation
Sparse finite elements for stochastic elliptic problems: higher order moments

Computing
Karhunen-Loève approximation of random fields by generalized fast multipole methods

Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
Sparse second moment analysis for elliptic problems in stochastic domains

Numerische Mathematik
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

SIAM Journal on Numerical Analysis
Sparse p-version BEM for first kind boundary integral equations with random loading

Applied Numerical Mathematics
A finite element method for elliptic problems with stochastic input data

Applied Numerical Mathematics

Quantified Score

Hi-index	0.00

Visualization

Abstract

In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L^2 and tensorized H"0^1 simultaneously on a standard k-dimensional cube. In the special case k=2 the suggested approximation operator is also optimal in L^2 and tensorized H^1 (without essential boundary conditions). This allows to construct an optimal sparse p-version FEM with sparse piecewise continuous polynomial splines, reducing the number of unknowns from O(p^2), needed for the full tensor product computation, to O(plogp), required for the suggested sparse technique, preserving the same optimal convergence rate in terms of p. We apply this result to an elliptic differential equation and an elliptic integral equation with random loading and compute the covariances of the solutions with O(plogp) unknowns. Several numerical examples support the theoretical estimates.