First Advisor

Jay Gopalakrishnan

Date of Publication

Winter 3-20-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences

Department

Mathematics and Statistics

Subjects

Space and time, Wave equation, Boundary value problems, Schrödinger equation

DOI

10.15760/etd.6094

Physical Description

1 online resource (viii, 137 pages)

Abstract

The most common tool for solving spacetime problems using finite elements is based on semidiscretization: discretizing in space by a finite element method and then advancing in time by a numerical scheme. Contrary to this standard procedure, in this dissertation we consider formulations where time is another coordinate of the domain. Therefore, spacetime problems can be studied as boundary value problems, where initial conditions are considered as part of the spacetime boundary conditions.

When seeking solutions to these problems, it is natural to ask what are the correct spaces of functions to choose, to obtain wellposedness. This motivates the study of an abstract theory for unbounded partial differential operators associated with a general boundary value problem on a bounded domain. A framework for choosing the spaces is introduced, and conditions for the solvability of weak formulations are provided. We apply this framework to study wave problems on tents and to study wellposed discontinuous Petrov-Galerkin (DPG) formulations for the Schrödinger and wave equations. Several numerical issues are also discussed.

Persistent Identifier

http://archives.pdx.edu/ds/psu/24321

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