An Isaac Newton Institute Workshop

Effective Computational Methods for Highly Oscillatory Problems: The Interplay between Mathematical Theory and Applications

Domain decomposition for multiscale elliptic PDEs

Authors: Ivan Graham (University of Bath), Rob Scheichl (University of Bath)

Abstract

In the Monte-Carlo simulation of fluid flow in stochastic media one typically has to solve elliptic PDEs with highly oscillatory coefficients. In practical applications in hydrogeology, these (random) oscillatory coefficients can have wavelength of order 10^{-3} or smaller and amplitude of the order of 10^{8}. Accurate finite element approximation in 2D requires of the order of 10^{8} degrees of freedom and the resulting linear systems can have condition number close to 10^{16}.

In this talk we will discuss domain decomposition preconditioning for such linear systems. Our overall aim is to solve such problems in a time close to the time required for solving a discretisation of a standard Poisson problem with constant coefficient with the same number of degrees of freedom.

The essential step in constructing two- or multi-level preconditioners is to replace the discretisation on the finest grid with a suitable discretisation on a coarser grid or grids. In the present application, standard (piecewise polynomial based) coarsening fails because, even if the fine mesh resolves the oscillations in the coefficient, coarser meshes typically fail to do so.

By extending the classical domain decomposition theory to this case, we show that a suitable coarsening strategy for heterogeneous media involves the construction of low energy coarse space basis functions. This naturally suggests that multiscale finite element methods can provide good coarse spaces, and leads to a new class of domain decomposition preconditioners. Recent results of Scheichl and Vainikko used the same theoretical technique to explain the robustness of certain algebraically defined preconditioners.

The theoretical results are illustrated by numerical examples on deterministic and random problems.

Reference:

I.G. Graham, P. Lechner and R. Scheichl, Domain Decomposition for Multiscale PDEs, Numer. Math. DOI 10.1007/s00211-007-0074-1 (2007) .