Multiscale Finite Element approach for "weakly" random problems and related issues
For researchers in computational science and engineering, this offers an efficient method for multiscale problems with small random perturbations, though it is an incremental extension of existing deterministic methods.
The paper develops a multiscale finite element method for elliptic problems with weakly random coefficients, modifying the deterministic basis to handle randomness efficiently. Numerical experiments show computational speed-up compared to standard approaches.
We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale Finite Element Method. Using numerical experiments, we demonstrate the efficiency of our approach and the computational speed-up with respect to a more standard approach. We provide a complete analysis of the approach, extending that available for the deterministic setting.