Algebraic multigrid for higher-order finite elements

Heys, Jeffrey J.; Manteuffel, Thomas A.; McCormick, Stephen F.; Olson, Luke N.

doi:10.1016/j.jcp.2004.10.021

Cited by 68 publications

(71 citation statements)

References 23 publications

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“…Also shown in the inset images are velocity vectors for specific branches. Because the triquartic finite element basis was based on Chebyshev-Lobatto node positions, which have been previously shown to work well with AMG solvers [20], the nodes are not evenly distributed but are somewhat clustered near the corners of elements. The global conservation of mass is approximately exact because C 3 was set to enforce this condition, but it is useful to examine the pressure gradient to determine the accuracy of the solution.…”

Section: Resultsmentioning

confidence: 99%

On Mass‐Conserving Least‐Squares Methods

Heys¹,

Lee²,

Manteuffel³

et al. 2006

SIAM J. Sci. Comput.

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Abstract. Least-squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations, including symmetric positive definite discrete operators and a sharp error measure. One of the potential drawbacks, especially in three dimensions, is that mass conservation is achieved only in a least-squares sense, and underresolved solutions are especially vulnerable to poor conservation. For the stationary Navier-Stokes equations, which are typically rewritten as a larger system of first-order equations, the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used. A new first-order system reformulation of the Navier-Stokes equations is presented that admits a wider range of mass-conserving boundary conditions. This new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems, a square channel, backwardfacing step, and branching tubes with two generations of bifurcations.

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Section: Resultsmentioning

confidence: 99%

On Mass‐Conserving Least‐Squares Methods

Heys¹,

Lee²,

Manteuffel³

et al. 2006

SIAM J. Sci. Comput.

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“…The use of the AMG methods for spectral elements has recently been studied in [34]. The number of connections between unknowns of the problem increases when higher-order elements are used.…”

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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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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“…One way to obtain this matrix is to rediscretize the PDE with first-order finite elements while preserving the number and the location of the degrees of freedom [12,24]. Assuming that the element matrices are available as well as the corresponding assembly routine, another option is to compute a sparse approximation of each element matrix, and then assemble them to form the intermediate matrix [1].…”

Section: Introductionmentioning

confidence: 99%

Algebraic Multigrid for Moderate Order Finite Elements

Napov

Notay

2014

SIAM J. Sci. Comput.

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We investigate the use of algebraic multigrid (AMG) methods for the solution of large sparse linear systems arising from the discretization of scalar elliptic partial differential equations with Lagrangian finite elements of order at most 4. The resulting system matrices do not have the M-matrix property that is required by standard analyzes of classical AMG and aggregation-based AMG methods. A unified approach is presented that allows to extend these analysis. It uses an intermediate M-matrix and highlights the role of the spectral equivalence constant that relates this matrix to the original system matrix. This constant is shown to be bounded independently of the problem size and jumps in the coefficients of the partial differential equations, provided that jumps are located at elements' boundaries. For two dimensional problems, it is further shown to be uniformly bounded if the angles in the triangulation also satisfy a uniform bound. This analysis validates the application of the AMG methods to the considered problems. On the other hand, because the intermediate M-matrix can be computed automatically, an alternative strategy is to define the AMG preconditioner from this matrix, instead of defining them from the original matrix. Numerical experiments are presented that asses both strategies using publicly available state-of-the-art implementations of classical AMG and aggregation-based AMG methods.

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Algebraic multigrid for higher-order finite elements

Cited by 68 publications

References 23 publications

On Mass‐Conserving Least‐Squares Methods

On Mass‐Conserving Least‐Squares Methods

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

Algebraic Multigrid for Moderate Order Finite Elements

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