Sponsor
Portland State University. Department of Mathematics and Statistics
Advisor
Jay Gopalakrishnan
Date of Award
3-21-2018
Document Type
Dissertation
Degree Name
Doctor of Philosophy (Ph.D.) in Mathematical Sciences
Department
Mathematics
Physical Description
1 online resource (viii, 137 pages)
DOI
10.15760/etd.6094
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
Recommended Citation
Sepùlveda Salas, Paulina Ester, "Spacetime Numerical Techniques for the Wave and Schrödinger Equations" (2018). Dissertations and Theses. Paper 4206.
https://pdxscholar.library.pdx.edu/open_access_etds/4206
10.15760/etd.6094