Graduate Thesis Or Dissertation

 

Dual-Norm Least-Squares Finite Element Methods for Hyperbolic Problems Public Deposited

https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/44558d306
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 L² norm, are discussed.

    In the linear case, the convergence in the L² norm is explicitly and naturally guaranteed by suitable formulations that are founded upon the original LL* method developed for elliptic PDEs. The approaches considered here are the LL*-based and LL*⁻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 L²-orthogonal projection of the exact solution. The formulations are analyzed and studied in detail.

     

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  • 2018
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  • 2020-01-13
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