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A S Silva - One of the best experts on this subject based on the ideXlab platform.

Roman Wienands - One of the best experts on this subject based on the ideXlab platform.

  • coarse grid approximation governed by local Fourier analysis
    2006
    Co-Authors: Roman Wienands
    Abstract:

    Solving discrete boundary value problems with the help of an appro- priate multigrid method (1, 4, 5, 6) necessitates the construction of a sequence of coarse grids with corresponding coarse grid approximations for the given flne grid discretization. Popular choices in this context are the Galerkin coarse grid approximation (GCA) and the use of the same discretization on the coarser grids as on the flne grid with properly adjusted mesh sizes (DCA). In this paper we propose an alternative strategy to select the required coarse grid discretizations within a multigrid solution method. It can be applied to flne grid operators that can be locally represented by a stencil. The coarse grid approx- imations are constructed by minimizing a certain low-frequency L 2 -norm. More precisely, a coarse grid discretization is chosen in such a way, that its Fourier Symbol is a best approximation (w.r.t. low frequencies) of the Fourier Symbol of the flne grid operator. This strategy is abbreviated by FCA since the design of coarse grid approximations is based on local Fourier analysis (5, 7). The entries of the coarse grid stencils are simply given by linear combinations of the flne grid entries. As a consequence, FCA can be considered as a black-box method to construct coarse grid operators. This method has been successfully applied to (anisotropic) difiusion equations, operators with mixed derivatives, problems with dominant convection, and operators involving jumping coe-cients. For nicely elliptic examples, FCA resembles the DCA approach, whereas for more di-cult applications (w.r.t. an e-cient multigrid treatment) it behaves similarly to GCA based on operator-dependent transfers.

L P Castro - One of the best experts on this subject based on the ideXlab platform.

Hamit A Serbest - One of the best experts on this subject based on the ideXlab platform.

  • diffraction of an obliquely incident plane wave by a two face impedance half plane wiener hopf approach
    Radio Science, 2000
    Co-Authors: Ernst Luneburg, Hamit A Serbest
    Abstract:

    The problem of diffraction of plane electromagnetic waves at an imperfectly conducting half plane with different impedances on upper and lower faces for oblique (skew) incidence either leads to a Wiener-Hopf equation with a 4×4 Fourier Symbol matrix for the tangential field components or to two formally decoupled Wiener-Hopf equations with 2×2 Symbol matrices of the Daniele-Khrapkov form for the electric and magnetic field components perpendicular to the diffracting edge. The higher-order edge singularity of the normal field components leads to undetermined constants in the classical Wiener-Hopf solution that are used to eliminate “unphysical” leaky wave poles that appear in the final solution by the residue calculus technique. The interrelation between both formulations involves an analytic family of polynomial transformation matrices. Consideration of the range restriction of this mapping is shown to be equivalent to the pole elimination procedure.

Rittich Hannah - One of the best experts on this subject based on the ideXlab platform.

  • Fourier Analysis of Periodic Stencils in Multigrid Methods
    'Society for Industrial & Applied Mathematics (SIAM)', 2018
    Co-Authors: Bolten M., Rittich Hannah
    Abstract:

    Many applications require the numerical solution of a partial differential equation (PDE), leading to large and sparse linear systems. Often a multigrid method can solve these systems efficiently. To adapt a multigrid method to a given problem, local Fourier analysis (LFA) can be used. It provides quantitative predictions about the behavior of the components of a multigrid method. In this paper we generalize LFA to handle what we call periodic stencils. An operator given by a periodic stencil has a block Fourier Symbol representation. It gives a way to compute the spectral radius and norm of the operator. Furthermore block Fourier Symbols can be used to find out how an operator acts on smooth/oscillatory input and whether its output will be smooth/oscillatory. This information can then be used to construct efficient smoothers and coarse grid corrections. We consider a particular PDE with jumping coefficients and show that it leads to a periodic stencil. LFA shows that the Jacobi method is a suitable smoother for this problem and an operator dependent interpolation is better than linear interpolation, as suggested by numerical experiments described in the literature. If an operator is given by an ordinary stencil, then block smoothers yield periodic stencils if the blocks correspond to rectangles in the domain. LFA shows that the block Jacobi and the red-black block Jacobi method efficiently reduce more frequencies than their pointwise versions. Further, it yields that a block smoother used in combination with aggressive coarsening can to some degree compensate for the reduced convergence rate caused by aggressive coarsening