Algebraic Multi-grid for Discrete Elliptic Second-Order Problems

The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions. The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situated at the Gauss-Lobatto quadrature points of the elements, and the Gauss-Lobatto quadrature rule is used so that the mass matrix becomes diagonal. This method is combined with the second-order central finite difference or the fourth-order Runge-Kutta time discretization. As a consequence of these choices, only matrix-vector products are needed in time dependent simulation. This makes the controllability method computationally efficient.

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“…At this stage, we use a modification of Kickinger's [33] algebraic multigrid (AMG) introduced in [26].…”

Section: Preconditioned Conjugate Gradient Methodsmentioning

confidence: 99%

Time-harmonic elasticity with controllability and higher-order discretization methods

Mönkölä

Heikkola

Pennanen

et al. 2008

Journal of Computational Physics

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“…In [34], the geometric multigrid method was replaced by a more generic and more flexible algebraic multigrid method (AMG). In this paper, an AMG based on [39] is utilized, using the implementation that is described in [34], with modifications that make it suitable for vector valued problems, like the Navier equation.…”

Section: Algebraic Multigrid Based Damped Preconditionersmentioning

confidence: 99%

A damping preconditioner for time-harmonic wave equations in fluid and elastic material

Airaksinen

Pennanen

Toivanen

2009

Journal of Computational Physics

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“…Based on reduction techniques from algebraic multigrid methods in which strong couplings and weak couplings are distinguished [40,35,24], we separate strong edges from weak edges. For every vertex i ∈ V we neglect all inflow edges (i, j) ∈ E with a weight smaller than τ -times the average of the weights of all inflow edges of vertex i.…”

Section: Construction Of Reduced Matrix Graphĝmentioning

confidence: 99%

Numbering techniques for preconditioners in iterative solvers for compressible flows

Pollul

Reusken

2007

Numerical Methods in Fluids

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We consider Newton-Krylov methods for solving discretized compressible Euler equations. A good preconditioner in the Krylov subspace method is crucial for the efficiency of the solver. In this paper we consider a point-block Gauss-Seidel method as preconditioner. We describe and compare renumbering strategies that aim at improving the quality of this preconditioner. A variant of reordering methods known from multigrid for convection-dominated elliptic problems is introduced. This reordering algorithm is essentially black-box and significantly improves the robustness and efficiency of the point-block Gauss-Seidel preconditioner. Results of numerical experiments using the QUADFLOW solver and the PETSc library are given.

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Algebraic Multi-grid for Discrete Elliptic Second-Order Problems

Cited by 43 publications

References 8 publications

Time-harmonic elasticity with controllability and higher-order discretization methods

Time-harmonic elasticity with controllability and higher-order discretization methods

A damping preconditioner for time-harmonic wave equations in fluid and elastic material

Numbering techniques for preconditioners in iterative solvers for compressible flows

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