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).