Lottes - Towards Robust Algebraic Multigrid Methods for Nonsymmetric Problems

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Springer International Publishing AG 2017

James Lottes Towards Robust Algebraic Multigrid Methods for Nonsymmetric Problems Springer Theses Recognizing Outstanding Ph.D. Research 10.1007/978-3-319-56306-0_1

1. Introduction

James Lottes 1

(1)

Google Inc., Mountain View, CA, USA

James Lottes

Email:

The solution of large systems of linear equations,

(1.1)

is a ubiquitous task in scientific computing, forming the core computational kernel of scientific computations arising from a broad range of applications. One remarkable application, and the one for which multigrid (MG) was originally devised, is the numerical solution of some partial differential equation (PDE),

(1.2)

specified with suitable boundary conditions on some domain . After discretizing the infinite-dimensional problem () to be solved (at each time step, if time is involved).

Algebraic multigrid (AMG) is a class of iterative methods for solving such linear systems, specifically those that are sparse, as is the case, for example, for typical discretizations of PDEs by grid-based methods. When applicable, AMG is often able to produce a numerical solution using a number of operations scaling linearly or nearly linearly in the number of unknowns. Our specific interest is in AMG applied to inherently nonsymmetric problems, for example, discretized convection-diffusion equations with high Pclet number (convection-dominated), as opposed to, say, a nonsymmetric discretization of Poissons equation. Most of the AMG analysis that has been done, with Notays work being a notable exception, has been for the symmetric case. Analysis is essential for the development of improved, more robust, and more broadly applicable AMG methods.

We present in Chap..

In this chapter we first look at the simplest of symmetric model problems, Poissons equation, and briefly cover its discretization by the finite element method (FEM) and its solution by geometric multigrid. The purpose is to introduce in the simplest concrete setting all of the concepts that we will make use of later, including the variational setting, linear iterative methods, and multigrid in its general conception. We then look at a nonsymmetric model problem, advection-diffusion, as well as its discretization, which we will use to illustrate and test the abstract theory of later chapters. All of this material is background. Also, for the most part, later chapters can be read independently of the material here.

1.1 A Symmetric Model Problem

This section focuses on Poissons equation. We begin with a brief presentation of the finite element method (FEM) for this equation, before presenting geometric multigrid applied to the resulting problem. We end with the main ideas of the multigrid convergence theory due to Hackbusch. This is background material, dealing only with a specific, symmetric model problem, and only with geometric multigrid. The intent is to introduce and motivate, in a concrete setting, the ideas we will need in later chapters.

Algebraic multigrid (AMG) methods make as few assumptions about the input matrix as possible, and are certainly not limited to matrices resulting from finite element discretizations. However, one of the key elements of AMG, the Galerkin (or Petrov-Galerkin) coarse operator construction, is directly analogous to the finite element discretization procedure, which comes about from the variational setting in which the FEM is based. The analysis of later chapters will be rooted in just such a variational setting. The presentation of FEM here is meant to provide a concrete introduction to these concepts, not to rule out applicability of the material of later chapters to other discretizations.

1.1.1 Abstract Formulation of Poissons Equation

Let us consider the problem of finding a solution to Poissons equation on a bounded open subset of with Lipschitz continuous boundary :

(1.3)

A modern approach is to look for solutions u in a suitable function space. As usual, we will use the Sobolev space the subset of consisting of those functions with weak first derivatives also in The Dirichlet boundary conditions narrow the solution space to the subspace of of functions whose trace on vanishes. The variational form of () reads: given , find such that

(1.4)

where denotes the inner product.

In this and following chapters, the concept of dual space will play a fundamental role. Recall that the continuous dual of a Banach space , denoted by , consists of all continuous linear functionals on . Each element maps each vector

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