The third day: numerical methods
Many problems in science and engineering have an inherent
multiscale character which poses severe challenges for both
theoretical and numerical analysis. Consequently, many types
of numerical methods have been constructbed for handling such
problems, e.g. homogenization, subgrid modelling, residual-free or
multiscale finite elements, upscaling (cf. the literature
below). Hierarchical ideas are also the basis for an
efficient solution of linear or non-linear systems of equations as
can be seen from multigrid or wavelet methods.
In my presentation, I will give an introduction into the subject
of the numerical solution of multiscale problems. For
achieving this goal, I will examine several of the above-mentioned
approaches and discuss their commonalities and differences.
The guideline will be how to compute the numerical solutions of
practically important problems faster and/or more accurate.
Recommended reading:
M. Braack and A. Ern: A posteriori control of modeling errors and
discretization errors, SIAMMMS 1, pp. 221-238 (2003).
B. Engquist and O. Runborg. Wavelet-based numerical homogenization
with applications. In: Springer Lecture Notes Comput. Sci. and
Eng. 20, pages
97-148 (2001).
M. Ohlberger: A posteriori error estimates for the heterogeneous
multiscale finite element method for elliptic homogenization
problems,
SIAMMMS 4, 88-114 (2005)
Y. Efendiev, T. Y. Hou: Multiscale Finite Element Methods: Theory
and Applications, Springer (2009).
B. Engquist and P. Lötstedt and O. Runborg: Multiscale
Methods in Science and Engineering, Springer (2008).
W. J"ager and A. Mikelic and N. Neuss: Asymptotic analysis of the
laminar viscous flow over a porous bed, SIAMSSC 22, 2006-2028
(2001),
A. M. Matache and I. Babu\v{s}ka and C. Schwab: Generalized
p-{FEM} in homogenization, Numerische Mathematik 86, 319--275
(2000).
N. Neuss and W. J"ager and G. Wittum: Homogenization and
Multigrid, Computing 66, 1-26 (2001).
J. Nolen, G. Papanicolaou, O. Pironneau, A Framework for Adaptive
Multiscale Methods for Elliptic Problems, SIAM Multiscale Modeling
and Simulation, vol. 7, 171-196 (2008).
R. Scheichl and E. Vainikko: Additive {Schwarz} with
aggregation-based coarsening for elliptic problems with highly
variable coefficients, Computing 80, 319-343 (2007).