Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

/pdf/iterative-methods-for-sparse-linear-systems-34ozhpwq89.pdf

Iterative Methods for Sparse Linear Systems

Gmsh is an open-source 3-D finite element grid generator with a build-in CAD engine and post-processor. Its design goal is to provide a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright (C) 2009 John Wiley & Sons, Ltd.

Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

/pdf/the-local-discontinuous-galerkin-method-for-time-dependent-23o5v9uy8q.pdf

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

Introduction. A Simple Model Problem. Abstract Nonlinear Equations. Finite Element Discretizations of Elliptic PDEs. Practical Implementation. Bibliography. Subject Index.

A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques

Integrated space-time adaptive hp -refinement methods for parabolic systems

Adjerid, Babuska, and Flaherty [ Math Models Methods Appl Sci, 9 (1999), pp 261--286] and Yu [Math Numer Sinica, 13 (1991), pp 89--101] and [Math Numer Sinica, 13 (1991), pp 307--314] show that a posteriori estimates of spatial discretization errors of piecewise bi- p polynomial finite element solutions of elliptic and parabolic problems on meshes of square elements may be obtained from jumps in solution gradients at element vertices when p is odd and from local elliptic or parabolic problems when p is even We show that these simple error estimates are asymptotically correct for other finite element spaces The key requirement is that the trial space contain all monomial terms of degree p + 1 except for $ x_1^{p+1}$ and $ x_2^{p+1}$ in a Cartesian (x1,x2 ) frame Computational results show that the error estimates are accurate, robust, and efficient for a wide range of problems, including some that are not supported by the present theory These involve quadrilateral-element meshes, problems with singularities, and nonlinear problems

A Posteriori Finite Element Error Estimation for Diffusion Problems

A two-dimensional mesh moving technique for time-dependent partial differential equations

Foreword Preface hp-finite element procedures on non-uniform geometric meshes: Adaptivity and constrained approximation Tetrahedral bisection and adaptive finite elements Resolution of boundary layers on triangular meshes A general concept of adaptivity in finite element methods with applications to problems in fluid and structural mechanics A solution based H1 norm triangular mesh quality indicator Experiments with repartitioning and load balancing adaptive meshes Distributed octree data structures and local refinement method for the parallel solution of three-dimensional conservation laws Adaptive finite element methods for elastostatic contact problems The full domain partition approach to parallel adaptive refinement Adaptive solution of phase change problems over unstructured tetrahedral meshes

Grid Generation and Adaptive Algorithms

Cluster and grid computing has made hierarchical and heterogeneous computing systems increasingly common as target environments for large-scale scientific computation. A cluster may consist of a network of multiprocessors. A grid computation may involve communication across slow interfaces. Modern supercomputers are often large clusters with hierarchical network structures. For maximum efficiency, software must adapt to the computing environment. We focus on partitioning and dynamic load balancing, in particular on hierarchical procedures implemented within the Zoltan Toolkit, guided by DRUM, the Dynamic Resource Utilization Model. Here, different balancing procedures are used in different parts of the domain. Preliminary results show that hierarchical partitionings are competitive with the best traditional methods on a small hierarchical cluster.

Joseph E. Flaherty

Papers

Integrated space-time adaptive hp -refinement methods for parabolic systems

A Posteriori Finite Element Error Estimation for Diffusion Problems

A two-dimensional mesh moving technique for time-dependent partial differential equations

Grid Generation and Adaptive Algorithms

Hierarchical partitioning and dynamic load balancing for scientific computation