Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation
In this work we propose a hybridizable discontinuous Galerkin (hdG) discretization of the high-frequency Helmholtz equation in the presence of point sources and highly heterogeneous and discontinuous wave speed models. We show that it delivers solutions that are provably second-order accurate and do...
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| Format: | Article |
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Society of Exploration Geophysicists
2018
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| Online Access: | http://hdl.handle.net/1721.1/116247 https://orcid.org/0000-0001-7052-5097 |
| _version_ | 1811091636774502400 |
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| author | Núñez, Leonardo Zepeda Hewett, Russell Taus, Matthias F Demanet, Laurent |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Núñez, Leonardo Zepeda Hewett, Russell Taus, Matthias F Demanet, Laurent |
| author_sort | Núñez, Leonardo Zepeda |
| collection | MIT |
| description | In this work we propose a hybridizable discontinuous Galerkin (hdG) discretization of the high-frequency Helmholtz equation in the presence of point sources and highly heterogeneous and discontinuous wave speed models. We show that it delivers solutions that are provably second-order accurate and do not suffer from the pollution error, as long as a slightly higher order hdG method is used where the polynomial degree is chosen such that p = O(logw). These results hold even if the discontinuities in the wave speed are not resolved by the hdG mesh, as long as the integration procedure used in the assembly of the stiffness matrix respects the discontinuities. Further, we show that the associated linear systems can be solved using a modification of the method of polarized traces resulting in a method with linear complexity up to a poly-logarithmic factor, or sub-linear complexity in a parallel environment.
To our knowledge and surprise, this note contains the first instance of a numerical method that is at the same time fast (O(N) runtime) and accurate (second-order, pollution-free) in the context of models of geophysical interest. Keywords: finite element, frequency-domain, numerical, wave equation, acoustic |
| first_indexed | 2024-09-23T15:05:32Z |
| format | Article |
| id | mit-1721.1/116247 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T15:05:32Z |
| publishDate | 2018 |
| publisher | Society of Exploration Geophysicists |
| record_format | dspace |
| spelling | mit-1721.1/1162472022-10-02T00:32:05Z Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation Núñez, Leonardo Zepeda Hewett, Russell Taus, Matthias F Demanet, Laurent Massachusetts Institute of Technology. Department of Mathematics Taus, Matthias F Demanet, Laurent In this work we propose a hybridizable discontinuous Galerkin (hdG) discretization of the high-frequency Helmholtz equation in the presence of point sources and highly heterogeneous and discontinuous wave speed models. We show that it delivers solutions that are provably second-order accurate and do not suffer from the pollution error, as long as a slightly higher order hdG method is used where the polynomial degree is chosen such that p = O(logw). These results hold even if the discontinuities in the wave speed are not resolved by the hdG mesh, as long as the integration procedure used in the assembly of the stiffness matrix respects the discontinuities. Further, we show that the associated linear systems can be solved using a modification of the method of polarized traces resulting in a method with linear complexity up to a poly-logarithmic factor, or sub-linear complexity in a parallel environment. To our knowledge and surprise, this note contains the first instance of a numerical method that is at the same time fast (O(N) runtime) and accurate (second-order, pollution-free) in the context of models of geophysical interest. Keywords: finite element, frequency-domain, numerical, wave equation, acoustic TOTAL (Firm) United States. Air Force. Office of Scientific Research (Grant FA9550-12-1-0328) United States. Air Force. Office of Scientific Research (Grant FA9550-15-1-0078) United States. Office of Naval Research (Grant N00014-16-1-2122) National Science Foundation (U.S.) (Grant DMS-1255203) 2018-06-12T14:44:56Z 2018-06-12T14:44:56Z 2017-09 2018-05-17T17:07:27Z Article http://purl.org/eprint/type/JournalArticle 1949-4645 http://hdl.handle.net/1721.1/116247 Taus, Matthias, et al. "Pollution-Free and Fast Hybridizable Discontinuous Galerkin Solvers for the High-Frequency Helmholtz Equation." SEG Technical Program Expanded Abstracts 2017, pp. 4068–73. https://orcid.org/0000-0001-7052-5097 http://dx.doi.org/10.1190/SEGAM2017-17728116.1 SEG Technical Program Expanded Abstracts 2017 Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Society of Exploration Geophysicists MIT Web Domain |
| spellingShingle | Núñez, Leonardo Zepeda Hewett, Russell Taus, Matthias F Demanet, Laurent Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation |
| title | Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation |
| title_full | Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation |
| title_fullStr | Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation |
| title_full_unstemmed | Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation |
| title_short | Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation |
| title_sort | pollution free and fast hybridizable discontinuous galerkin solvers for the high frequency helmholtz equation |
| url | http://hdl.handle.net/1721.1/116247 https://orcid.org/0000-0001-7052-5097 |
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