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Welcome to the MultiScaleFEM.jl wiki!

This wiki contains details about the implementation of the multiscale method for different problems. This work is a part of my time as a postdoc at Umeå University, Sweden. I am attempting to make these notes as I learn the method and implement them from scratch. Multiscale methods are a class of numerical methods used to solve problems arising in Heterogeneous media. What does that mean? Consider the standard steady state equation

$$ \begin{align} -\nabla\cdot(A_\epsilon(x,y), \nabla u(x,y)) &= f(x,y), \quad (x,y) \in \Omega,\\ u &= 0, \quad x \in \partial \Omega \end{align} $$

Here the function $A_\epsilon(x,y)$ varies rapidly in space, and makes the problem difficult to solve using traditional methods. Multiscale methods aim to incorporate the oscillations in the material properties into the bases functions which then results in a smaller system which is usually easier to solve. If you are from the scientific computing community, its a fairly standard study with the method applied to different problems to study the convergence rates etc. I will simply add the 1d and 2d material you see on the repository's README.md here and then attempt to slowly populate the wiki. Right now, it contains all that I have on 1D and 2D Higher Order Multiscale Methods along with a small section on Localized Orthognal Decomposition Method. Follow the Table of Contents on the Wiki to check out the pages!

References

• Målqvist, A. and Peterseim, D., 2020. Numerical homogenization by localized orthogonal decomposition. Society for Industrial and Applied Mathematics.
• Maier, R., 2021. A high-order approach to elliptic multiscale problems with general unstructured coefficients. SIAM Journal on Numerical Analysis, 59(2), pp.1067-1089.
• Abdulle, A. and Henning, P., 2017. Localized orthogonal decomposition method for the wave equation with a continuum of scales. Mathematics of Computation, 86(304), pp.549-587.