A computational three-field methodology for non-conforming finite elements over partitioned domains

Abstract

In this thesis, we describe a computational methodology to couple physical processes defined over independent sub-domains, that are partitions of a global domain in three-dimensions. The methodology presented helps to compute the numerical solution on the global domain by appropriately piecing the local solutions from each sub-domain. We discuss the mixed method formulation for the technique applied to a model problem and derive an error estimate for the finite element solution. We demonstrate through numerical experiments, that the method is robust and reliable in higher-dimensions.

Additionally, this thesis is concerned with the application of non-conforming finite element methods to stochastic partial differential equations. We present a mixed formulation of a three-field finite element method applied to an elliptic model problem involving stochastic loads. We then derive the exact form for the expected value and variance of the solution. Additionally the rate of convergence for the stochastic error is presented. Finally, we demonstrate the reliability of the method by comparing our exact results with numerical experiments.

The method is then extended for use in parabolic partial differential equations (e.g., time-dependent systems). After providing the derivation for the semi-discretization of the parabolic problem, we consider two classical full discretizations of a model problem: the backward Euler method and the Crank-Nicolson method. This method is implemented and used to model actual physical phenomena, namely, we consider the heat conduction/diffusion equation.

Finally, we report on the specifics of the implementation of the method in various distributed computing environments, including a computational grid and a shared memory multi-processor.