Numerical solution of the Helmholtz Equation using high-order continuous Galerkin methods
We show a continuous Galerkin formulation with high-order polynomials to solve the Helmholtz equation. High-order formulations obtain solutions with less numerical error, and can use curved high-order meshes to approximate the domain. To reduce the computational requirements of the high-order formul...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2017 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/106636 |
| Acceso en línea: | https://hdl.handle.net/2117/106636 |
| Access Level: | acceso abierto |
| Palabra clave: | Difference equations, Partial--Numerical solutions Numerical modeling Finite element method Helmholtz equation High-order Static condensation Equacions diferencials parcials--solucions numèriques Classificació AMS::65 Numerical analysis::65N Partial differential equations, boundary value problems Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi numèrica |
| Sumario: | We show a continuous Galerkin formulation with high-order polynomials to solve the Helmholtz equation. High-order formulations obtain solutions with less numerical error, and can use curved high-order meshes to approximate the domain. To reduce the computational requirements of the high-order formulation, we apply a static condensation technique. Using this technique, we eliminate a set of unknowns from the global linear system and therefore, we solve a smaller system of equations. Then, we recover the full solution by solving several systems that only involve the unknowns of a single element. In the examples we show that the proposed implementation converges optimally to the analytical solution both for two and three dimensional examples. We also show an application where the mesh is curved in order to better capture the geometry. |
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