Convergence Analysis of Domain Decomposition Methods for the Helmholtz Equation
B. Varkevisser (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Vandana Dwarka – Mentor (TU Delft - Numerical Analysis)
Henk M. Schuttelaars – Graduation committee member (TU Delft - Mathematical Physics)
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Abstract
The Helmholtz equation is a famous equation in the world of physics with which the behavior of waves can be described. The equation is frequently used when studying, for example, seismic waves from earthquakes or electromagnetic waves of an MRI. It is also infamous for the difficulties that arise when trying to solve this equation with a computer. A great deal of research has been conducted over the last decades in order to find and improve numerical solution methods. As of today, no method is known that is feasible for general Helmholtz problems. This thesis investigates the root cause of problems when trying to solve the Helmholtz equation with domain decomposition methods. To this end, we work with the most basic form of the Helmholtz equation, which we will first construct. We then perform mathematical analysis to gain some insights into the root cause of problems. Numerical experiments are also conducted in order to investigate the behavior of the
mathematical models.