138 research outputs found
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
LFA-tuned matrix-free multigrid method for the elastic Helmholtz equation
We present an efficient matrix-free geometric multigrid method for the
elastic Helmholtz equation, and a suitable discretization. Many discretization
methods had been considered in the literature for the Helmholtz equations, as
well as many solvers and preconditioners, some of which are adapted for the
elastic version of the equation. However, there is very little work considering
the reciprocity of discretization and a solver. In this work, we aim to bridge
this gap. By choosing an appropriate stencil for re-discretization of the
equation on the coarse grid, we develop a multigrid method that can be easily
implemented as matrix-free, relying on stencils rather than sparse matrices.
This is crucial for efficient implementation on modern hardware. Using two-grid
local Fourier analysis, we validate the compatibility of our discretization
with our solver, and tune a choice of weights for the stencil for which the
convergence rate of the multigrid cycle is optimal. It results in a scalable
multigrid preconditioner that can tackle large real-world 3D scenarios.Comment: 20 page
A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory
We develop a new dispersion minimizing compact finite difference scheme for
the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly
developed ray theory for difference equations. A discrete Helmholtz operator
and a discrete operator to be applied to the source and the wavefields are
constructed. Their coefficients are piecewise polynomial functions of ,
chosen such that phase and amplitude errors are minimal. The phase errors of
the scheme are very small, approximately as small as those of the 2-D
quasi-stabilized FEM method and substantially smaller than those of
alternatives in 3-D, assuming the same number of gridpoints per wavelength is
used. In numerical experiments, accurate solutions are obtained in constant and
smoothly varying media using meshes with only five to six points per wavelength
and wave propagation over hundreds of wavelengths. When used as a coarse level
discretization in a multigrid method the scheme can even be used with downto
three points per wavelength. Tests on 3-D examples with up to degrees of
freedom show that with a recently developed hybrid solver, the use of coarser
meshes can lead to corresponding savings in computation time, resulting in good
simulation times compared to the literature.Comment: 33 pages, 12 figures, 6 table
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