A meshfree method with domain decomposition for Helmholtz boundary value problems

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...

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Vydané v:2020 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency (SUMMA) s. 139 - 144
Hlavný autor: Valtchev, Svilen S.
Médium: Konferenčný príspevok..
Jazyk:English
Vydavateľské údaje: IEEE 11.11.2020
IEEE Canada
Predmet:
domain decomposition
Energy efficiency
Geometry
Iterative methods
L-shaped domain
Mathematical model
meshfree method
method of fundamental solutions
modified Helmholtz equation
non-homogeneous PDE
Numerical models
Partial differential equations
plane wave functions
Wave functions
ISBN:1728188407, 1728181135, 9781728188409, 9781728181134
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Shrnutí: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.
ISBN:1728188407
1728181135
9781728188409
9781728181134
DOI:10.1109/SUMMA50634.2020.9280636