Multiscale phenomena are integral to most processes which can be modeled as wave propagation problems. In this talk I will explore the use of variational multiscale (VMS) solution strategies for deterministic and stochastic time-harmonic wave propagation problems. Accurate solutions to time-harmonic Maxwell’s equations are essential in understanding several electromagnetic phenomena. Often, restrictions on discretization size can render the performance of the standard finite element method unsatisfactory. This primarily stems from fine-scale features not being resolved, and can lead to an accumulation of error with long wave propagation distances. I will prove the efficacy of the Galerkin least squares (GLS) technique in addressing this issue. It will be achieved by designing a numerical scheme with better dispersion characteristics, thus effectively capturing the phase of the solution. I will substantiate the claims using numerical examples on practical problems.
Complimenting accurate discretization methods with fast solvers is imperative. In the context of time-harmonic Maxwell’s equations, the poor conditioning of the discretized system renders standard iterative algorithms ineffective. I will describe a non-overlapping domain decomposition method developed from a variational multiscale paradigm as an alternative for a large class of problems. At each iteration, the solution within each sub-domain is sought while best accounting the effect of rest of the domain. Its accomplished by utilizing perfectly matched layers to enforce pseudo-differential interface conditions. I will demonstrate numerically how this turns out to be a computationally inexpensive method with fast convergence. Often parameters defining the mathematical model of a physical phenomena are stipulated by specifying their uncertainty. I will extend and analyze the VMS method to partial differential equations with stochastic coefficients that provide a framework for studying such systems. I will elaborate upon the fine-scale stochastic Green’s function, an operator which defines the VMS method completely. Using the theoretical insights, I will argue how approximations to enable a practical implementation of the VMS method is possible, and which result in improved statistics at a much lower computational cost. The VMS method also provide as a by-product an estimate of the error in the coarse approximation. I will showcase how this can potentially act as an error indicator to aid adaptive refinement in the joint physical and stochastic space.