16 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
Towards Accuracy and Scalability: Combining Isogeometric Analysis with Deflation to Obtain Scalable Convergence for the Helmholtz Equation
Finding fast yet accurate numerical solutions to the Helmholtz equation
remains a challenging task. The pollution error (i.e. the discrepancy between
the numerical and analytical wave number k) requires the mesh resolution to be
kept fine enough to obtain accurate solutions. A recent study showed that the
use of Isogeometric Analysis (IgA) for the spatial discretization significantly
reduces the pollution error.
However, solving the resulting linear systems by means of a direct solver
remains computationally expensive when large wave numbers or multiple
dimensions are considered. An alternative lies in the use of (preconditioned)
Krylov subspace methods. Recently, the use of the exact Complex Shifted
Laplacian Preconditioner (CSLP) with a small complex shift has shown to lead to
wave number independent convergence while obtaining more accurate numerical
solutions using IgA.
In this paper, we propose the use of deflation techniques combined with an
approximated inverse of the CSLP using a geometric multigrid method. Numerical
results obtained for both one- and two-dimensional model problems, including
constant and non-constant wave numbers, show scalable convergence with respect
to the wave number and approximation order p of the spatial discretization.
Furthermore, when kh is kept constant, the proposed approach leads to a
significant reduction of the computational time compared to the use of the
exact inverse of the CSLP with a small shift
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