Date of Award

Spring 5-10-2018

Document Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Thomas A. Manteuffel

Abstract

Least-squares finite element discretizations of first-order hyperbolic partial differential equations (PDEs) are proposed and studied. Hyperbolic problems are notorious for possessing solutions with jump discontinuities, like contact discontinuities and shocks, and steep exponential layers. Furthermore, nonlinear equations can have rarefaction waves as solutions. All these contribute to the challenges in the numerical treatment of hyperbolic PDEs.

The approach here is to obtain appropriate least-squares formulations based on suitable minimization principles. Typically, such formulations can be reduced to one or more (e.g., by employing a Newton-type linearization procedure) quadratic minimization problems. Both theory and numerical results are presented.

A method for nonlinear hyperbolic balance and conservation laws is proposed. The formulation is based on a Helmholtz decomposition and closely related to the notion of a weak solution and a $H^{-1}$-type least-squares principle. Accordingly, the respective important conservation properties are studied in detail and the theoretically challenging convergence properties, with respect to the $L^2$ norm, are discussed.

In the linear case, the convergence in the $L^2$ norm is explicitly and naturally guaranteed by suitable formulations that are founded upon the original $\cL\cL^\star$ method developed for elliptic PDEs. The approaches considered here are the $\cL\cL^\star$-based and $(\cL\cL^\star)^{-1}$ methods, where the latter utilizes a special negative-norm least-squares minimization principle. These methods can be viewed as specific approximations of the generally infeasible quadratic minimization that determines the $L^2$-orthogonal projection of the exact solution. The formulations are analyzed and studied in detail.

Key words: first-order hyperbolic problems; hyperbolic balance laws; Burgers equation; weak solutions; Helmholtz decomposition; conservation properties; space-time discretization; least-squares methods; dual methods; adjoint methods; negative-norm methods; finite element methods; discontinuous coefficients; exponential layers

AMS subject classifications: 65N30, 65N15

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