Abstract
Discontinuous Galerkin methods are widely used in many practical fields. In this thesis, we focus on a new class of discontinuous Galerkin methods for second-order wave equations. This thesis is constructed by three main parts. In the first part, we study the convergence properties of the energy-based discontinuous Galerkin proposed in [3] for wave equations. We improve the existing suboptimal error estimates to an optimal convergence rate in the energy norm. In the second part, we generalize the energy-based discontinuous Galerkin method proposed in [3] to the advective wave equation and semilinear wave equation in second-order form. Energy-conserving or energy-dissipating methods follow from simple, mesh-independent choices of the interelement fluxes. Error estimates in the energy norm are established. In the third part, we focus on establishing methods to overcome the computa- tional stiffness from the high-order piecewise polynomial approximations in the energy-based discontinuous Galerkin methods and reduce the computational cost of the inversion of the stiffness matrix.
Degree Date
Spring 2020
Document Type
Dissertation
Degree Name
Ph.D.
Department
Mathematics
Advisor
Thomas Hagstrom
Number of Pages
158
Format
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License
Recommended Citation
Zhang, Lu, "A New Class of Discontinuous Galerkin Methods for Wave Equations in Second-Order Form" (2020). Mathematics Theses and Dissertations. 5.
https://scholar.smu.edu/hum_sci_mathematics_etds/5
