Multiscale Finite Elements (MsFEM)

Basic ideas

Numerical resolution of physical and engineering problems involves very often multiple scale phenomena. Transport in perforated media, air flow in urban or PDEs with highly oscilating coefficients are good examples. Direct simulations require enormous amount of computational resources. Multiscale finite element method (MsFEM) allows to deal with this difficulty. The problem is solved on coarse mesh with suitably taylored basis functions, which reflect small scale behaviour of the solution. The basis functions are precalculated on an underlying fine mesh. That is to say, they are numerical solutions of PDEs defined only on a single coarse mesh element with suitably choosen boundary condtitions. The basis functions need to be computed only once for each coarse mesh element and are used throughout entire resolution of the time dependent problem. As opposed to the methods based on the homogenization theory, MsFEM basis functions does not depend on any arbitrary analytical model of the microstructure.

The figure below shows an example of a precalculated basis functions for a Crouzeix-Raviart MsFEM variant (on the left) compared to the Q1 MsFEM variant (on the right).

Advection - diffusion problem in perforated domain

A good example of possible applications of the MsFEM is the advection-diffusion equation in perforated domains. One can imagine solving a poluant dispertion problem in an urban area, where the complex geometry of the domain leads to extremely time consuming direct simulations. MsFEM approach allows to obtain reliable results with much smaler meshes. In figures below a MsFEM solutions on grids ranging from 8x8 (figure a) to 128x128 (figure e) are compared to a standard FEM on 1024x1024 grid (figure f) on a computational domain with 400 random perforations with a circular advection field and a source term equal to 1 in the upper and lower parts. The relative L2 error is of the order of 10% for the 64x64 MsFEM comared to the FEM fine scale solution.

Stokes problem

Another example of the application of the MsFEM is the flow in porous media. The figures below show the x component of the velocity as well as the streamlines for the open chanel flow through a domain with 144 randomly placed perforation in the middle part. Boundary conditions for velocity field correspond to the Poisseuille flow through a clear chanel. The grid size is ranging from 2x4 (figure a) to 64x128 (figure f) for the MsFEM and 640x1280 for the standard FEM (figure g). The relative L2 error is of the order of 10% for the 32x64 MsFEM grid compared to the FEM fine scale solution.

Publications

• B. P. Muljadi, J. Narski, A. Lozinski, P. Degond, Non-Conforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media. Part I: Methodologies and Numerical Experiments, Multiscale Modelling and Simulation, 13 (2015) pp. 1146–1172, preprint: arXiv:1404.283
• P. Degond, A. Lozinski, B. P. Muljadi, J. Narski, Crouzeix-Raviart MsFEM with Bubble Functions for Diffusion and Advection-Diffusion in Perforated Media, Communications in Computational Physics, 17 (2015), pp. 887-907, preprint: arXiv:1310.8639