Domain Decomposition for Multiscale PDEs
joint work with I.G. Graham, P. Lechner, C. Pechstein, E. Vainikko, and
J. Van lent
Project Description
We consider additive Schwarz domain decomposition preconditioners for
piecewise linear finite element approximations of
elliptic PDEs with highly variable coefficients. In contrast to
standard analyses, we do not assume
that the coefficients
can be resolved by a coarse mesh.
This situation arises often in
practice, for example
in the computation of flows in
heterogeneous porous media, in both the deterministic and
(Monte-Carlo simulated) stochastic
cases. We
consider
preconditioners which combine local solves on general
overlapping subdomains together with a global solve on a
general coarse space of functions on a
coarse grid.
We perform a new analysis of the preconditioned matrix,
which shows rather explicitly how its condition number depends on
the variable coefficient in the
PDE as well as on the coarse mesh and
overlap parameters.
The classical estimates for this preconditioner with linear
coarsening
guarantee
good conditioning only when the coefficient varies mildly inside
the coarse grid elements. By contrast, our new results show that,
with a good choice of subdomains and coarse space basis functions,
the preconditioner can still be robust even for
large coefficient variation inside domains, when the classical method
fails to be robust. In particular our
estimates prove very
precisely the previously made empirical observation that
the use of low-energy coarse spaces can
lead to robust preconditioners. We go on to consider coarse
spaces constructed (a) via smoothed aggregation and (b) from
multiscale finite elements, and we prove that
preconditioners using these types of
coarsenings lead to robust
preconditioners for a variety of binary (i.e. two-scale)
media model problems. Moreover numerical experiments show that the
new preconditioners have greatly improved performance over
standard preconditioners/solvers even in the
random coefficient case.
The details of this work can be found in:
- R. Scheichl and E. Vainikko,
Robust Aggregation-Based Coarsening for Additive Schwarz in the
Case of Highly Variable Coefficients, in Proceedings of the European
Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006
(P. Wesseling, E. Onate, J. Periaux, Eds.), TU Delft, 2006.
[Preprint]
- I.G. Graham and R. Scheichl,
Robust Domain Decomposition Algorithms for Multiscale
PDEs, Numerical Methods for Partial Differential Equations
23(4):859-878, 2007.
[Preprint]
- I.G. Graham, P. Lechner and R. Scheichl,
Domain Decomposition for Multiscale PDEs, Numerische Mathematik
106:589-626, 2007.
[Preprint]
- R. Scheichl and E. Vainikko,
Additive Schwarz with Aggregation-Based Coarsening for Elliptic
Problems with Highly Variable Coefficients, Computing 80(4):319-343,
2007.
[Preprint]
- I.G. Graham and R. Scheichl,
Coefficient-explicit Condition Number Bounds for Overlapping
Additive Schwarz, in Proceedings of the 17th International
Conference on Domain Decomposition Methods, Strobl, Austria, 3-7 July
2006, Langer, U., Discacciati, M. Keyes, D. et al.
(Eds.), Domain Decomposition Methods in Science and
Engineering XVII, Lecture Notes in Computational Science and Engineering,
Vol. 60, Springer, New York, 2008.
[Preprint]
Our latest results on the extension to energy minimising coarse spaces and to
FETI methods can be found in
- J. Van lent, R. Scheichl and I.G. Graham,
Energy Minimizing Coarse Spaces for Two-level Schwarz Methods for
Multiscale PDEs, BICS Preprint 12/08, Bath Institute for Complex Systems,
University of Bath, 2008.
[Preprint]
- C. Pechstein and R. Scheichl,
Analysis of FETI Methods for Multiscale PDEs, RICAM Report 2008-20,
Johann Radon Institute for Computational and Applied Mathematics (RICAM),
Austrian Academy of Sciences, to appear in Numerische Mathematik, 2008.
[Preprint]
- C. Pechstein and R. Scheichl,
Robust FETI Solvers for Multiscale Elliptic PDEs, submitted to the
Proceedings of the 6th Int. Conf. on Scientific Computing in Electrical
Engineering, "Mathematics in Industry -- Scientific Computing in Electrical
Engineering (SCEE 2008)", Springer, 2008.
[Preprint]
- C. Pechstein and R. Scheichl,
Analysis of FETI Methods for Multiscale PDEs - Part II: Interface
Variation, in preparation.
Recently I have also started a collaboration with
Frederic Nataf (Paris VI)
on alternative (Robin or order-2) boundary conditions for two-level Schwarz
methods in the context of multiscale PDEs. Results to follow.
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Last updated 26/06/2006