A study of the Numerical Dispersion for the Continuous Galerkin discretization of the one-dimensional Helmholtz equation

Abstract : Although true solutions of Helmholtz equation are non-dispersive, their discretizations suffer from a phenomenon called numerical dispersion. While the true phase velocity is constant, the numerical one changes with the discretization scheme, order and mesh size. In our work, we study the dispersion associated with classical finite element. For arbitrary order of discretization, without using an ansatz, we construct the numerical solution on the whole real line, and obtain an asymptotic expansion for the phase difference between the exact wavenumber and the numerical one. We follow an approach analogous to that employed in the construction of true solutions at positive wavenumbers, which involves z-transform, contour deformation and limiting absorption principle. This perspective allows us to identify the numerical wavenumber with the angle of analytic poles. Such an identification is useful since the latter (analytic poles) can be numerically evaluated by an algorithm based on Theorem 2.4 in the 1999 paper by Philippe Guillaume, which then yields the value of numerical wavenumber.
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[Research Report] RR-9075, Inria Bordeaux Sud-Ouest; Magique 3D. 2017
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  • HAL Id : hal-01533177, version 1

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Hélène Barucq, Henri Calandra, Ha Pham, Sébastien Tordeux. A study of the Numerical Dispersion for the Continuous Galerkin discretization of the one-dimensional Helmholtz equation. [Research Report] RR-9075, Inria Bordeaux Sud-Ouest; Magique 3D. 2017. 〈hal-01533177〉

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