An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems

Mikhail A. Bochev, G.L.G. Sleijpen, A. Sopaheluwakan

Abstract

Numerical solution of the Helmholtz equation in an infinite domain often involves restriction of the domain to a bounded computational window where a numerical solution method is applied. On the boundary of the computational window artificial transparent boundary conditions are posed, for example, widely used perfectly matched layers (PMLs) or absorbing boundary conditions (ABCs). Recently proposed transparent-influx boundary conditions (TIBCs) resolve a number of drawbacks typically attributed to PMLs and ABCs, such as introduction of spurious solutions and the inability to have a tight computational window. Unlike the PMLs or ABCs, the TIBCs lead to a nonlinear dependence of the boundary integral operator on the frequency. Thus, a nonlinear Helmholtz eigenvalue problem arises. This paper presents an approach for solving such nonlinear eigenproblems which is based on a truncated singular value decomposition (SVD) polynomial approximation of the nonlinearity and subsequent solution of the obtained approximate polynomial eigenproblem with the Jacobi-Davidson method.
Original languageUndefined
Place of PublicationEnschede
PublisherDepartment of Applied Mathematics, University of Twente
Number of pages14
StatePublished - Sep 2008

Publication series

NameMemorandum / Department of Applied Mathematics
PublisherDepartment of Applied Mathematics, University of Twente
No.Supplement/1883
ISSN (Print)1874-4850
ISSN (Electronic)1874-4850

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Transparent boundary conditions
Absorbing boundary conditions
Eigenproblem
Numerical solution
Jacobi-Davidson method
Spurious solutions
Artificial boundary conditions
Infinite domain
Boundary integral
Hermann von Helmholtz
Polynomial approximation
Helmholtz equation
Singular value decomposition
Integral operator
Eigenvalue problem
Resolve
Nonlinearity
Restriction
Polynomial

Keywords

  • IR-65013
  • MSC-65F15
  • METIS-251205
  • EWI-13530
  • MSC-35J05
  • MSC-65F30

Cite this

Bochev, M. A., Sleijpen, G. L. G., & Sopaheluwakan, A. (2008). An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems. (Memorandum / Department of Applied Mathematics; No. Supplement/1883). Enschede: Department of Applied Mathematics, University of Twente.

Bochev, Mikhail A.; Sleijpen, G.L.G.; Sopaheluwakan, A. / An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems.

Enschede : Department of Applied Mathematics, University of Twente, 2008. 14 p. (Memorandum / Department of Applied Mathematics; No. Supplement/1883).

Research output: ProfessionalReport

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Bochev, MA, Sleijpen, GLG & Sopaheluwakan, A 2008, An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems. Memorandum / Department of Applied Mathematics, no. Supplement/1883, Department of Applied Mathematics, University of Twente, Enschede.

An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems. / Bochev, Mikhail A.; Sleijpen, G.L.G.; Sopaheluwakan, A.

Enschede : Department of Applied Mathematics, University of Twente, 2008. 14 p. (Memorandum / Department of Applied Mathematics; No. Supplement/1883).

Research output: ProfessionalReport

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N2 - Numerical solution of the Helmholtz equation in an infinite domain often involves restriction of the domain to a bounded computational window where a numerical solution method is applied. On the boundary of the computational window artificial transparent boundary conditions are posed, for example, widely used perfectly matched layers (PMLs) or absorbing boundary conditions (ABCs). Recently proposed transparent-influx boundary conditions (TIBCs) resolve a number of drawbacks typically attributed to PMLs and ABCs, such as introduction of spurious solutions and the inability to have a tight computational window. Unlike the PMLs or ABCs, the TIBCs lead to a nonlinear dependence of the boundary integral operator on the frequency. Thus, a nonlinear Helmholtz eigenvalue problem arises. This paper presents an approach for solving such nonlinear eigenproblems which is based on a truncated singular value decomposition (SVD) polynomial approximation of the nonlinearity and subsequent solution of the obtained approximate polynomial eigenproblem with the Jacobi-Davidson method.

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Bochev MA, Sleijpen GLG, Sopaheluwakan A. An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems. Enschede: Department of Applied Mathematics, University of Twente, 2008. 14 p. (Memorandum / Department of Applied Mathematics; Supplement/1883).