WIAS Preprint No. 1420, (2009)

Shifted linear systems in electromagnetics. Part I: Systems with identical right-hand sides


  • Schlundt, Rainer
    ORCID: 0000-0002-4424-4301
  • Schmückle, Franz-Josef
  • Heinrich, Wolfgang

2010 Mathematics Subject Classification

  • 35Q60 65F10 65F15 65N22 78M25


  • Microwave device, Maxwell's equations, Scattering matrix, Boundary value problem, PML boundary condition, Eigenvalue problem, Linear algebraic equations, Multiple shifts, Krylov subspace method, Polynomial preconditioning




We consider the solution of multiply shifted linear systems for a single right-hand side. The coefficient matrix is symmetric, complex, and indefinite. The matrix is shifted by different multiples of the identity. Such problems arise in a number of applications, including the electromagnetic simulation in the development of microwave and mm-wave circuits and modules. The properties of microwave circuits can be described in terms of their scattering matrix which is extracted from the orthogonal decomposition of the electric field. We discretize the Maxwell's equations with orthogonal grids using the Finite Integration Technique (FIT). Some Krylov subspace methods have been used to solve multiply shifted systems for about the cost of solving just one system. We use the QMR method based on coupled two-term recurrences with polynomial preconditioning.

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