HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II. Optimization of the Runge-Kutta smoother

Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge-Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge-Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space-time discontinuous Galerkin finite element discretization of the advection-diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained

Coarsening Strategies for Unstructured Multigrid Techniques with Application to Anisotropic Problems

Over the years, multigrid has been demonstrated as an efficient technique for solving inviscid flow problems. However, for viscous flows, convergence rates often degrade. This is generally due to the required use of stretched meshes (i.e., the aspect ratio AR = Δy/Δx < < 1) in order to capture the boundary layer near the body. Usual techniques for generating a sequence of grids that produce proper convergence rates on isotropic meshes are not adequate for stretched meshes. This work focuses on the solution of Laplace's equation, discretized through a Galerkin finite-element formulation on unstructured stretched triangular meshes. A coarsening strategy is proposed and results are discussed

The Sixth Copper Mountain Conference on Multigrid Methods, part 2

The Sixth Copper Mountain Conference on Multigrid Methods was held on April 4-9, 1993, at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Immersed finite element method for interface problems with algebraic multigrid solver

This thesis is to discuss the bilinear and 2D linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. In contrast to the body-fitting mesh restriction of the traditional finite element methods or finite difference methods for interface problems, a number of numerical methods based on structured meshes independent of the interface have been developed. When these methods are applied to the real world applications, we often need to solve the corresponding large scale linear systems many times, which demands efficient solvers. The algebraic multigrid (AMG) method is a natural choice since it is independent of the geometry, which may be very complicated in interface problems. However, for those methods based on finite difference formulation and a structured mesh independent of the interface, the stiffness matrix of the linear system is usually not symmetric positive-definite, which demands extra efforts to design efficient multigrid methods. On the other hand, the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite. Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and 2D linear IFE methods for both stationary and moving interface problems after the IFE and multi-grid methods are reviewed respectively. The numerical examples demonstrate the features of the proposed algorithm, including the optimal convergence in both Ł² and semi-H¹ norms of the IFE-AMG solutions, the high efficiency with proper choice of the components and parameters of AMG, the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems, and the relationship between the cost and the moving interface location --Abstract, page iii

A matrix-free ILU realization based on surrogates

Matrix-free techniques play an increasingly important role in large-scale simulations. Schur complement techniques and massively parallel multigrid solvers for second-order elliptic partial differential equations can significantly benefit from reduced memory traffic and consumption. The matrix-free approach often restricts solver components to purely local operations, for instance, the Jacobi- or Gauss--Seidel-Smoothers in multigrid methods. An incomplete LU (ILU) decomposition cannot be calculated from local information and is therefore not amenable to an on-the-fly computation which is typically needed for matrix-free calculations. It generally requires the storage and factorization of a sparse matrix which contradicts the low memory requirements in large scale scenarios. In this work, we propose a matrix-free ILU realization. More precisely, we introduce a memory-efficient, matrix-free ILU(0)-Smoother component for low-order conforming finite elements on tetrahedral hybrid grids. Hybrid grids consist of an unstructured macro-mesh which is subdivided into a structured micro-mesh. The ILU(0) is used for degrees-of-freedom assigned to the interior of macro-tetrahedra. This ILU(0)-Smoother can be used for the efficient matrix-free evaluation of the Steklov-Poincare operator from domain-decomposition methods. After introducing and formally defining our smoother, we investigate its performance on refined macro-tetrahedra. Secondly, the ILU(0)-Smoother on the macro-tetrahedrons is implemented via surrogate matrix polynomials in conjunction with a fast on-the-fly evaluation scheme resulting in an efficient matrix-free algorithm. The polynomial coefficients are obtained by solving a least-squares problem on a small part of the factorized ILU(0) matrices to stay memory efficient. The convergence rates of this smoother with respect to the polynomial order are thoroughly studied