Numerical analysis of some multiscale and stochastic partial differential equations

Multiscale partial differential equations (PDEs) and stochastic PDEs arise from many technological and engineering situations, such as composite materials, ground water flow and oil recovery. For multiscale PDEs, as the scales differ from each other by several orders of magnitude, classical finite element (FE) methods are prohibitively expensive as the mesh width has to be of the order of the smallest scale for the approximating solution to represent correctly the exact solution of the multiscale equation. For stochastic PDEs, the cost of computing the statistical properties of the solution is high. The complexity of these problems may surpass the current available computing power. Studying new computational and approximation methods that can solve these problems within acceptable computational time, using reasonable computational resources without sacrificing accuracy is one of the central topics in applied mathematics today. This thesis aims to make novel contributions to this timely scientific challenge. We study novel computational methods for multiscale wave equations, multiscale elasticity equations and multiscale elastic wave equations. We also devote a part of this thesis to studying approximation for random and parametric elasticity equations.DOCTOR OF PHILOSOPHY (SPMS