|
|
Prof. Jan Martin Nordbotten (University of Bergen)
| Speaker: |
Prof. Jan Martin Nordbotten
(University of Bergen)
|
| Date: |
Wednesday February 9, 2011
|
| Title: |
Multiscale
approximation
and preconditioning for linear elliptic problems
|
Abstract
|
Multiscale
numerical
methods have gained attention as a topic of research within
numerical methods for porous media flows. This invites several
different perspectives, each of which leads to different
interpretations of the applicability and value of multiscale methods:
A) Multiscale discretizations are enhancements of regular
discretizations on the coarser scale, and should be compared to
upscaling methods. B) Multiscale discretizations are approximations to
the fine scale solution in their own right. C) Multiscale methods are
variations of domain decomposition methods (or preconditioners) which
are designed with particular emphasis on certain features of the
problem.
Herein, we discuss multiscale methods in the setting of linear elliptic
problems arising from discretizing the pressure equation for porous
media. Our emphasis will be on the design of methods that allow for all
three perspectives outlined above. To achieve this, we pursue an
algebraic approach, which within certain approximations lead to both
classical domain decomposition preconditioners applicable within
iterative solvers as well as multiscale methods. An additional benefit
of the algebraic construction of multiscale methods comes in the form
of grid-independent implementation, allowing for fully unstructured
grids.
Through application to two- and three-dimensional problems, we discuss
the role of geometric information, and the construction of algebraic
approximations to fine-scale problems. Further, we highlight the
concept of auxiliary variables on the coarse scale, introduced as
supporting variables to improve approximation and convergence
properties. We illustrate how the auxiliary variables in multiscale
discretizations play a similar role to enhancing the nodal basis within
classical single-scale discretizations.
Our discussion is phrased in terms of numerical results of the proposed
methods, both as upscaling methods and as preconditioners. This duality
is emphasized in the construction, as it allows for a wide range of
options to provide adaptivity between approximation quality and error
control.
This is joint work with A. Sandvin and E. Keilegavlen (both Universtiy
of Bergen).
|
|
