Numerical Approximation Methods for Partial Differential Equations

Author: A. Quarteroni et al.
Publisher:
Publish Date: 1998-03-01
Features:
Segment: 1. Introduction
Numerical approximation of partial differential equations is an important branch of Numerical Analysis. Often, it demands a knowledge of many aspects of the problem. First of all, the physical background of the problem is required in order to understand the behavior of expected solutions. This may often lead to the choice of convenient numerical methods. Secondly, modern formulation of the problem based on the variational (weak) form ought to be considered, as it allows the search for generalized solutions in Hilbert (or Banach) functionals spaces. Variational techniques yield a priori estimates for the solution, which in turn indicate in which kind of norms any virtual numerical solution can be prevented to be stable. Furthermore, results about smoothness of the mathematical solutions may suggest the numerical methodology to be used, and consequently, determine the kind of accuracy that can be achieved. The latter is pointed out from the error analysis. Clearly, specific attentions should be paid to the algorithmic aspects concerned with the choice of any numerical method. This book aims at providing general ideas on numerical approximation of partial differential equations, although (obviously) not all possible existing methods will be considered. In this respect, we mainly focus on variational numerical methods for the discretization of space derivatives, and on finite difference and fractional-step methods for advancing, in time, unsteady problems. Whenever possible, we present the unifying approach behind a priori different numerical strategies, provide general theory for analysis and illustrate a variety of algorithms that can be used to compute the effective numerical solution of the problem at hand, taking into consideration its algebraic structure. Consequently, we try to avoid using technicalities (or tricks, or algorithms) that work only in very specific situations, or that are not sustained from a sound theoretical background. Some problems (and methods) are discussed on a case-to-case basis, but very often they are included in a single logical unit (say Chapter, or Section).
1.1 The Conceptual Path Behind the Approximation
We consider a great number of mathematical problems, and numerical methods for their solution. For the approximation of any given boundary value problem, we schematically illustrate in Fig. 1.1.1 the decision path that needs to be followed. Level [1] is the boundary value problem at hand under its weak formulation accounting for the prescribed boundary conditions. Level [2] provides the kind of discretization (or numerical method) that can be pursued in order to reduce the given problem to one having finite dimension. Of course, the strategy adopted will determine the structure of the numerical problem. Throughout this book we mainly consider two kinds of discretization. The former is the Galerkin method, together with its remarkable variant, the Petrov-Galerkin method, which is based on an integral formulation of the differential problem. The second discretization we consider is the collocation method, which is, instead, based on the fulfillment of the differential equations at some selected points of the computational domain. We then reformulate the collocation method under a generalized Galerkin mode, precisely combining the Galerkin approach with numerical evaluation of integrals using Gaussian formulae. At a lower extent, we will address finite difference schemes for space discretization, especially for nonlinear convection-diffusion equations and for problems of wave propagation. For the latter we will also present the approach based on the finite volume method, which is very popular in computational fluid dynamics. Finally, we will illustrate shortly the elementary principles of the domain decomposition method, an approach which offers the best promise for the parallel solution of large problems in the field of scientific computing. Other approaches are often encountered in the literature as well, but they will only be addressed incidentally in this book. Level [3] specifies the nature of the subspaces used in the approximation. Typically, we have piecewise-polynomial functions of low degree when using finite elements, and global algebraic polynomials of high degree for spectral methods. These two remarkable cases will be discussed and analyzed in some of their variants (mixed finite elements, Legendre and Chebyshev spectral collocation methods). The choice operated at this level determines the functional structure of the numerical solution, the kind of accuracy that can be achieved, besides affecting the topological form of the resulting algebraic system.