Doctoral Dissertations

Date of Award

5-2022

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Xiaobing Feng

Committee Members

OHANNES KARAKASHIAN, SANJAYA DEVINA, TUOC PHAN, CHEN XIA, CORY HAUCK

Abstract

This dissertation consists of three main parts with each part focusing on numerical approximations of the stochastic Stokes and Navier-Stokes equations.

Part One concerns the mixed finite element methods and Chorin projection methods for solving the stochastic Stokes equations with general multiplicative noise. We propose a modified mixed finite element method for solving the Stokes equations and show that the numerical solutions converge optimally to the PDE solutions. The convergence is under energy norms (strong convergence) for the velocity and in a time-averaged norm (weak convergence) for the pressure. In addition, after establishing the error estimates in second moment, high moment estimates are also established which serve as a bridge for deriving pathwise error estimates. The second portion of Part One focuses on Chorin projection methods for solving the stochastic Stokes equation with general multiplicative noise. Two types of the Chorin projection methods, the standard Chorin and modified Chorin projection methods, are considered. We establish sub-optimal order error estimates for the velocity and pressure approximations of the standard Chorin projection method, and optimal order estimates for the modified Chorin method Numerical experiments are also provided to verify the proved theoretical results.

Part Two studies the mixed finite element methods for solving the stochastic Navier-Stokes equations with general additive noise. We establish the strong convergence for the velocity and pressure approximations. In addition, high moment error estimates are also established which serve as a bridge for deriving pathwise error estimates for both the velocity and pressure approximations.

Part Three develops an iterative framework for parameter-dependent and random convection-diffusion problems which arise from many engineering and scientific applications. The main idea of the framework is to reformulate the underlying PDE problem into another problem with a parameter-independent convection-diffusion coefficient and a parameter-dependent (and solution-dependent) right-hand side, a fixed-point iteration is then employed to compute the solution of the reformulated problem. The computational saving is achieved at the solver level by using the LU direct method or block Krylov subspace methods as linear solvers so many computations can be reused. Numerical experiments are presented to validate the efficiency of the iterative method.

Download
Files over 3MB may be slow to open. For best results, right-click and select "save as..."

Share

COinS