Fully discrete approximation of parametric and stochastic elliptic PDEs
For researchers in computational PDEs and uncertainty quantification, this provides rigorous convergence guarantees for practical approximations, though the results are incremental extensions of existing theory.
The paper establishes convergence rates for fully discrete approximations of parametric and stochastic elliptic PDEs, combining spatial and parametric discretizations, and quantifies the total number of degrees of freedom needed to achieve a given accuracy.
It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent solutions. These results by themselves do not yield practically realizable approximations, since they do not cover the approximation of the arising expansion coefficients, which are functions of the spatial variable. In this work, we study the combined spatial and parametric approximability for elliptic PDEs with affine or lognormal parametrizations of the diffusion coefficients and corresponding Taylor, Jacobi, and Hermite expansions, to obtain fully discrete approximations. Our analysis yields convergence rates of the fully discrete approximation in terms of the total number of degrees of freedom. The main vehicle consists in $\ell^p$ summability results for the coefficient sequences measured in higher-order Hilbertian Sobolev norms. We also discuss similar results for non-Hilbertian Sobolev norms which arise naturally when using adaptive spatial discretizations.