SEMINARS
An adaptive algorithm for PDE problems with random data

2017-12-14　15:00 — 16:00

Large Conference Room

David J. Silvester

University of Manchester

Jinglai Li

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)