On finite element solutions for bounded time-harmonic scattering problems with an exact nonlocal boundary condition.

Time-harmonic scattering problems in partial differential equations have been thoroughly analyzed for many years. These problems are originally posed in an unbounded domain, but to easily discretize the domain, we will instead restrict to a finite, bounded domain and utilize the finite element method to solve the resulting problem. This restriction demands an additional boundary condition that introduces an error component. An important goal in selecting an appropriate boundary condition is to minimize this added error. Many well-known boundary conditions have been shown to produce accurate numerical solutions as the mesh is refined, including PML and the DtN map. We add to this literature by expanding on a recently-introduced exact nonlocal boundary condition utilizing layer potentials and Green's formulas, and we will explore both the Helmholtz and Maxwell equations with this setting. For the latter problem, we obtain a best-approximation result for the convergence rate as the mesh is refined. We will use Firedrake to solve both problems and generate numerical results to confirm our theory. We also utilize Firedrake's external operator functionality to implement a fast multipole method (FMM) to efficiently evaluate our layer potentials.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Doctoral

Grantor

Baylor University.

Year dc:date.issued

2026

Author dc:creator

• Anderson, Drew, 1999-

Advisor dc:contributor.advisor

• Kirby, Robert C.

• Boundary conditions.
• Maxwell equation.
• Helmholtz equation.
• Fast multipole method (FMM)
• Layer potentials.

Language dc:language.iso

en

Handle dc:identifier.uri

https://hdl.handle.net/2104/15095

OAI identifier oai:identifier

oai:baylor-ir.tdl.org:2104/15095