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...at does not require multilevel meshing, has been successfully applied to scalar positive definite operators, but has not been extended to vector PDEs or indefinite operators like the Stokes equations =-=[55]-=-. Cartesian grid methods. State-of-the-art methods for problems in complex geometries, most often found in applications with dynamic interfaces, are based on regular grid or Cartesian grid discretizat...

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...ts ~ 4Dc. Alg. 2: v = 0, 1=1 (8 not used). Costs ~ 5Dc. Alg. 3: v = v(k), with v(k) = 0 if k ::j:. e -1, vee - 1) = 1 (smoothing on second finest level only), 1 = 1. Costs ~ 9Dc· W.r.t. Algorithm 3 we note that we do not use smoothing on level f because I - All Ac = 0, and numerical experiments have shown that smoothing on level k < f - 1 in general does not improve the efficiency of the method. In all the experiments we take a fixed arbitrary starting vector x(O). If the error after the m-th iteration is denoted by e(m), we use as a. measure for the error reduction: or EXPERIMENT 3.1. (as in [12,14]). We take: a(x, y) = (2y - 1)(1 - x 2 ), b(x, y) = 2xy(y - 1) ; g(x, y) = sin(7rx) + sin(137rx) + sin(7rY) + sin(137rY) . In Table 1 we show values of h D . 11 algorithm 1 algorithm 2 algorithm 3 € h£ = 1/64 he = 1/128 he = 1/64 h£ = 1/128 he = 1/64 10° 0.76 0.78 0.25 0.26 0.056 10-2 0.84 0.80 0.43 0.45 0.21 10-4 0.46 0.90 0.85 0.91 0.025 Table 1. EXPERIMENT 3.2. (strong convection in part of the domain). We take a( x, y) = as in Experiment 1 if y ::; 1.2.5x, zero otherwise; b( x, y) = as in Experiment 1 if y ::; 1.25x, zero otherwise. g as in Experiment 1, € = 10-4. In Table 2 we show values...

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...ongation matrices and convergent smoothing iterations, then through the use of the variational conditions applied to the symmetric positive definite system the convergence of the solver is guaranteed =-=[58]-=-. All the above preconditioners are mesh oriented in the sense that the prolongation operators are created using the geometry and connectivity information of the mesh. An alternative would be the use ...

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... research. Markov chains can possess one or more of the above characteristics. The branch of multigrid research which attempts to deal with general sparse systems is known as algebraic multigrid, see =-=[12]-=-. Given an appropriate choice of smoother, multigrid algorithms can be parallelized with a high degree of efficiency, see [9] for a recent survey. This is, of course, an added advantage in the context...

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...on of several algorithms on some representative problems in vibrational analysis. Several recent studies have demonstrated the viability of AMG preconditioners for problems in computational mechanics =-=[1, 38]-=- but to the best of our knowledge, there are no comparable studies for vibration analysis. The paper is organized as follows. We present a general derivation of the algorithms in Section 2 followed by...

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...rsening algorithms, algebraic multigrid, complexities, preconditioners AMS subject classifications. 65F10, 65N55, 65Y05 DOI. 10.1137/040615729 1. Introduction. The algebraic multigrid (AMG) algorithm =-=[1, 13, 14, 2]-=- is one of the most efficient algorithms for solving large unstructured sparse linear systems that arise in a wide range of science and engineering applications. One of AMG’s most desirable properties...

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...ver. Multigrid methods [41] are among the fastest algorithms for preconditioning and solving linear systems of equations arising from certain partial differential equations. Algebraic multigrid (AMG) =-=[39]-=- is an extension to cases where no problem-related information such as geometry is available. It constructs a multilevel hierarchy based on weighted interpolation with a carefully chosen set of nodes ...

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...cular, SA is often very efficient at solving the systems that arise from problems of three-dimensional (3D) thin-body elasticity, a task that can tax traditional AMG techniques. As with classical AMG =-=[4, 18, 19]-=-, the standard SA method bases its transfer operators on certain assumptions about the nature of algebraically smooth error, that is, error that simple relaxation schemes, such as Jacobi or Gauss–Seid...