Asymptotic expansions for the error of discretization algorithms for non-linear functional equations

Abstract

Assume that the solution r/(h) of a finite algorithm depending upon a parameter h> 0 converges for h-->0 to the solution y of a certain infinitesimal problem. We consider asymptotic expansions o/ the discretization error e(h) :=~(h)--y:where 0<pl< ..-<PN and the e~ do not depend on h. polation from several values ~(h~), h,>0. Except in the case of the EulerMaclaurin sum formula representing the expansion (t.1) for the approximation of definite integrals by trapezoidal sums, the existence of an asymptotic expansion and its sequence of exponents {p,} had only been conjectured in applications of Richardson-extrapolation to functional equations. Quite recently, GRAGG treated the case of initial value problems for first order differential equations (see [7]).In w 2 of this paper, we will --under suitable conditions --prove the existence of such expansions (usually with p~=p +v--1) for a very general class of diseretization algorithms for non-linear functional equations in Banach-spaces. In the proof, the sequence {e~} will be recursively constructed. If the expansion of the local discretization error (see (2.2)) contains only even powers of h, this fact is preserved ~ in the expansion of e(h). In these cases, Richardson-extrapolation is particularly effective in improving the numerical results.In w 3, we will apply our abstract theorem to several important functional equations and their discretizations : Initial and boundary value problems for both ordinary and partial differential equations, integral equations and integrodifferential equations. For all these infinitesimal functional equations our theorem will provide hypotheses under which the application of Richardson-extrapolation is justified for a given discretization algorithm.In w 4, we will actually compute the first terms of the expansion (I

The Defect Correction Approach

“…The deliberate consideration and use of asymptotic expansions in discretization algorithms began with Gragg's thesis [26]; the first general discussion was presented in Stetter [ 48], see also Stetter [ 49]. At first, the main objective was the justification of Richardson extrapolation where the mere existence of an asymptotic expansion (4.18) is used for the construction of a higher order approximation from several approximations z\ h;E Yf, i = 1 (l)r, obtained on grids Gh•with a non-empty intersection.…”

Approximation Methods for Nonlinear Problems with Application to Two-Point Boundary Value Problems

“…Asymptotic error expansions are not discussed in Section 3 as they are easily obtained from our results by employing the techniques indicated in [3] or [9].…”

Difference Methods and Deferred Corrections for Ordinary Boundary Value Problems

“…Here the e,(t) are solutions of linear two point boundary value problems similar to (1.1)--(1.2) or (3.2) but with inhomogeneous data determined by the truncation operators on y (t) and thee"' (t) for f.L < v. These derivations are by now standard, see for example [5], [14], [19].…”

“…Symbolic procedures for generating the weights w ;·• and complete tables of them for a wide range of derivatives are given in [10]. Of course, fast Vandermonde solvers [19] can also be used to generate these weights if storage space is at a premium.…”