511 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
An efficient and accurate MPI based parallel simulator for streamer discharges in three dimensions
We propose an efficient and accurate message passing interface (MPI) based
parallel simulator for streamer discharges in three dimensions using the fluid
model. First, we propose a new second-order semi-implicit scheme for the
temporal discretization of the model, which relaxes the dielectric relaxation
time restriction. Moreover, it solves the Poisson equation only once at each
time step, while classical second-order semi-implicit and explicit schemes
typically need twice. Second, we introduce a geometric multigrid preconditioned
FGMRES solver, which dramatically improves the efficiency of solving the
Poisson equations with either constant or variable coefficients. It is
numerically shown that the solver is faster than other Krylov subspace solvers,
and it takes no more than 4 iterations for the Poisson solver to converge to a
relative residual of during streamer simulations. Last but not the
least, all the methods are implemented using MPI, and the good parallel
efficiency of the code and great performance of the numerical algorithms are
demonstrated by a series of numerical experiments, using up to 2560 cores on
the Tianhe2-JK clusters. A double-headed streamer discharge as well as the
interaction of two streamers is studied, using up to 10.7 billion mesh cells
Algebraic multigrid for stabilized finite element discretizations of the Navier Stokes equation
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2002.Includes bibliographical references (p. 141-152).A multilevel method for the solution of systems of equations generated by stabilized Finite Element discretizations of the Euler and Navier Stokes equations on generalized unstructured grids is described. The method is based on an elemental agglomeration multigrid which produces a hierarchical sequence of coarse subspaces. Linear combinations of the basis functions from a given space form the next subspace and the use of the Galerkin Coarse Grid Approximation (GCA) within an Algebraic Multigrid (AMG) context properly defines the hierarchical sequence. The multigrid coarse spaces constructed by the elemental agglomeration algorithm are based on a semi-coarsening scheme designed to reduce grid anisotropy. The multigrid transfer operators are induced by the graph of the coarse space mesh and proper consideration is given to the boundary conditions for an accurate representation of the coarse space operators. A generalized line implicit relaxation scheme is also described where the lines are constructed to follow the direction of strongest coupling. The solution algorithm is motivated by the decomposition of the system characteristics into acoustic and convective modes. Analysis of the application of elemental agglomeration AMG (AMGe) to stabilized numerical schemes shows that a characteristic length based rescaling of the numerical stabilization is necessary for a consistent multigrid representation.by Tolulope Olawale Okusanya.Ph.D
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