Print Email Facebook Twitter Domain Decomposition Helmholtz Solvers Title Domain Decomposition Helmholtz Solvers: Obtaining Wave Number Independence Author Sieburgh, Erik (TU Delft Electrical Engineering, Mathematics and Computer Science) Contributor Heinlein, A. (mentor) Dwarka, V.N.S.R. (mentor) Vuik, Cornelis (graduation committee) Schuttelaars, H.M. (graduation committee) Degree granting institution Delft University of Technology Programme Applied Mathematics Date 2022-10-31 Abstract Wave phenomena play an important role in many different applications such as MRI scans, seismology and acoustics [41, 49, 47]. At the core of such applications lies the Helmholtz equation, which represents the time-independent version of the wave equation. Simulating a Helmholtz problem numerically with accurate numerical solutions for large wave numbers is challenging. Numerical solvers for the Helmholtz problem have to balance having accurate numerical solutions, requiring a number of iterations to reach convergence that is independent of the wave number and solving with linear time complexity with respect to the grid nodes. Currently, there is no numerical Helmholtz solver that can satisfy these requirements at once. We developed Schwarz domain decomposition preconditioners which leads to wave number independent convergence for wave numbers in 2D and 3D, while remaining to have accurate numerical solutions. The preconditioners use two-level Schwarz preconditioners, with the coarse problem being constructed using higher-order interpolation with quadratic rational Bézier curves. The developed domain decomposition preconditioners are designed to leverage parallel computing in the future in an attempt for the preconditioners to acquire the ability to solve with linear time complexity.In this research, the preconditioner resulting in wave number independent convergence and the lowest iteration count is the two-level scaled hybrid Schwarz preconditioner with a coarse problem constructed using higher-order Bézier interpolation. This preconditioner uses a deflation method to remove unwanted eigenvalues. Removing these unwanted eigenvalues results in a clustering of the eigenvalues which is more favourable for GMRES. Currently, all the developed preconditioners suffer from high computational cost for large wave numbers, due to the coarse problem becoming large. Decreasing the coarse problem size of the preconditioners, while remaining to have wave number independent convergence, has shown to been unsuccessful. To better understand the required conditions for wave number independent convergence of the preconditioners, we investigated the relationship between the number of coarse grid nodes and the wave length, too see if there is anything generalizable about this relationship and wave number independent convergence of the solvers.In conclusion, the balancing for a Helmholtz solver to have accurate numerical solutions, requiring a number of iterations to reach convergence that is independent of the wave number and solving with linear time complexityis again shown to be difficult. This work provides the initial development and testing of promising wave number independent Helmholtz solvers, from which more research should follow that tackle its biggest computational problems. Subject Domain DecompositionHelmholtz equationDeflation To reference this document use: http://resolver.tudelft.nl/uuid:32f4b716-6f83-4a95-b8b5-c3d35a32349f Part of collection Student theses Document type master thesis Rights © 2022 Erik Sieburgh Files PDF Master_Thesis_Erik_Sieburgh.pdf 3.73 MB Close viewer /islandora/object/uuid:32f4b716-6f83-4a95-b8b5-c3d35a32349f/datastream/OBJ/view