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An Adaptive Algorithm for PDE Problems with Random Data

Speaker

David J. Silvester, University of Manchester

Time

2017.12.14 15:00-16:00

Venue

Large Conference Room, Math Building

Abstract

We present a new adaptive algorithm for computing stochastic Galerkin finite element approximations for a class of elliptic PDE problems with random data. Specifically, we assume that the underlying differential operator has affine dependence on a large, possibly infinite, number of random parameters. Stochastic Galerkin approximations are then sought in the tensor product space $X \otimes {\cal P}$, where $X$ is a finite element space associated with a physical domain and ${\cal P}$ is a set of multivariate polynomials over a finite-dimensional manifold in the (stochastic) parameter space.

Our adaptive strategy is based on computing two error estimators (the spatial estimator and the stochastic one) that reflect the two distinct sources of discretisation error and, at the same time, provide effective estimates of the error reduction for the corresponding enhanced approximations. In particular, our algorithm adaptively `builds’ a polynomial space over a low-dimensional manifold in the infinitely-dimensional parameter space such that the discretisation error is reduced most efficiently (in the energy norm). Convergence of the adaptive algorithm is demonstrated numerically.

This is joint work with Alex Bespalov (University of Birmingham) and Catherine Powell (University of Manchester)