Multigrid methods

The First Copper Mountain Multigrid Conference on Multigrid Methods was organized in 1983 by Steve McCormick, who chaired the conference until 2003. Widely regarded as one of the premier international conferences on multigrid methods, it occurs every other year, alternating (as of 1990) with the equally successful Copper Mountain Conference on Iterative Methods. McCormick’s successor as chairman, Van Henson, chaired the 2011 multigrid meeting, assisted by program co-chairmen Ulrich Rüde and Irad Yavneh. The conference began with two tutorial sessions given by Irad Yavneh and Van Henson. The sessions covered multigrid fundamentals as well as algebraic multigrid (AMG) and other advanced topics. The remaining five days of the conference were organized around a sequence of twentyfive-minute talks, with ample time for individual research discussions between colleagues. The winners of the student paper competition were Yao Chen (Penn State University), Lei Tang (University of Colorado at Boulder), and Eran Treister (Technion). They presented their papers in the session for students. The Fifteenth Copper Mountain Multigrid Conference on Multigrid Methods was held in the Colorado Mountains on March 27 – April 1, 2011. This special issue contains 14 papers, 13 of which were presented at the conference. The papers address a variety of applications and topics. Brief summaries of each paper follow, listed in order of submission. In [1], Brezina et al. consider an extension of bootstrap algebraic multigrid, in which the interpolation weights used in AMG for the problem Ax D b are automatically determined by locally fitting a set of test vectors that have been relaxed as solutions to the homogeneous equation Ax D 0. The extension, rBAMG, involves adding scaled residuals of the test vectors to the leastsquares equations. In [2], Yaveneh et al. consider the classic Petrov–Galerkin for black box multigrid for nonsymmetric problems with factor-of-three coarsening. With symmetric line Gauss–Seidel as a smoother, the algorithm achieves fast and reliable convergence for both first-order and second-order discretizations of the convection operator for a wide range of diffusion coefficients. In [3], Emams reports on experiences with aggregation AMG relying on Krylov acceleration as preconditioners for linear systems in general purpose fluid flow simulation. In benchmarks that reflect the requirements for industrial situations, the performance of recently published algorithms for semi-definite problems is shown to be very attractive except for proposed variants that occasionally fail for nonsymmetric problems; modifications are suggested to lead to reliable solvers for these situations. In [4], Reps et al. analyze the Krylov convergence rate of a Helmholtz problem preconditioned with multigrid, which is applied to the Helmholtz problem formulated on a complex contour and uses polynomial smoothers at each level. For a one-dimensional model, it is shown that the Krylov