A meshfree method with domain decomposition for Helmholtz boundary value problems

Valtchev, Svilen

http://hdl.handle.net/10400.8/13613

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A meshfree method with domain decomposition for Helmholtz boundary value problems.pdf	In the framework of meshfree methods, we address the numerical solution of boundary value problems (BVP) for the non-homogeneous modified Helmholtz partial differential equation (PDE). In particular, the unknown solution of the BVP is calculated in two steps. First, a particular solution of the PDE is approximated by superposition of plane wave functions with different wavenumbers and directions of propagation. Then, the corresponding homogeneous BVP is solved, for the homogeneous part of the solution, using the classical method of fundamental solutions (MFS). The combination of these two meshfree techniques shows excellent numerical results for non-homogeneous BVPs posed in simple geometries and when the source term of the PDE is sufficiently regular. However, for more complex domains or when the source term is piecewise defined, the MFS fails to converge. We overcome this problem by coupling the MFS with Lions non-overlapping domain decomposition method. The proposed technique is tested for the modified Helmholtz PDE with a discontinuous source term, posed in an L-shaped domain.	3.83 MB	Adobe PDF	Download

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Authors

Valtchev, Svilen

Abstract(s)

In the framework of meshfree methods, we address the numerical solution of boundary value problems (BVP) for the non-homogeneous modified Helmholtz partial differential equation (PDE). In particular, the unknown solution of the BVP is calculated in two steps. First, a particular solution of the PDE is approximated by superposition of plane wave functions with different wavenumbers and directions of propagation. Then, the corresponding homogeneous BVP is solved, for the homogeneous part of the solution, using the classical method of fundamental solutions (MFS). The combination of these two meshfree techniques shows excellent numerical results for non-homogeneous BVPs posed in simple geometries and when the source term of the PDE is sufficiently regular. However, for more complex domains or when the source term is piecewise defined, the MFS fails to converge. We overcome this problem by coupling the MFS with Lions non-overlapping domain decomposition method. The proposed technique is tested for the modified Helmholtz PDE with a discontinuous source term, posed in an L-shaped domain.

Description

EISBN - 978-1-7281-8840-9
Article number - 9280636; Conference name - 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency, SUMMA 2020; Conference city - Lipetsk; Conference date - 10 November 2020 - 13 November 2020; Conference code - 165770

Keywords

meshfree method plane wave functions method of fundamental solutions domain decomposition modified Helmholtz equation non-homogeneous PDE L-shaped domain

URI

http://hdl.handle.net/10400.8/13613

Citation

S. S. Valtchev, "A meshfree method with domain decomposition for Helmholtz boundary value problems," 2020 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency (SUMMA), Lipetsk, Russia, 2020, pp. 139-144, doi: https://doi.org/10.1109/SUMMA50634.2020.9280636.