# IJPAM: Volume 57, No. 5 (2009)

**ON ERROR ESTIMATION FOR APPROXIMATION METHODS**

INVOLVING DOMAIN DISCRETIZATION

VII: DETERMINISTIC PROBLEMS VI.

FULLY DISCRETE FINITE DIFFERENCE SCHEMES FOR

LINEAR PDES I: ESTIMATES IN TERMS OF PROPERTIES

OF THE SOLUTION

INVOLVING DOMAIN DISCRETIZATION

VII: DETERMINISTIC PROBLEMS VI.

FULLY DISCRETE FINITE DIFFERENCE SCHEMES FOR

LINEAR PDES I: ESTIMATES IN TERMS OF PROPERTIES

OF THE SOLUTION

Priority R&D Group for Mathematical Modelling

Numerical Simulation and Computer Visualization

Narvik University College

2, Lodve Lange's St., P.O. Box 385, N-8505 Narvik, NORWAY

e-mail: ltd@hin.no

url: https://ansatte.hin.no/ltd/

**Abstract.**This is the seventh of a sequence of 12 papers, preceded by
[#!ee-dd-1!#,#!ee-dd-2!#,#!ee-dd-3!#,#!ee-dd-4!#,#!ee-dd-5!#,#!ee-dd-6!#] and followed by [#!ee-dd-8!#,#!ee-dd-9!#,#!ee-dd-10!#,#!ee-dd-11!#,#!ee-dd-12!#]
(in this order), dedicated to the study of error estimates for
approximation problems based on discretization of the domain of the
approximated functions.
Within this sequence, in [#!ee-dd-2!#] and [#!ee-dd-3!#], for a model example of a Cauchy
problem for a linear differential equation with variable coefficients, we applied an indirect method for error estimation based
on results known in advance for the respective semi-discrete
approximating problems.

In a follow-up subsequence of six of these papers, of which this is the fourth one, preceded by [#!ee-dd-4!#,#!ee-dd-5!#,#!ee-dd-6!#] and followed by [#!ee-dd-8!#,#!ee-dd-9!#] (in this order) we develop a direct discrete method for error estimation based on an *extended Lax principle*, essentially proposed first in [#!ltd-phd!#] but explicitly formulated for the first time in [#!ee-dd-4!#, Section 2].
In [#!ee-dd-4!#,#!ee-dd-5!#,#!ee-dd-6!#] we applied the proposed method to obtain sharp error estimates for a model boundary
problem for a linear ordinary differential equation of second order with variable coefficients and right-hand side. This study is being continued in the present paper and [#!ee-dd-8!#,#!ee-dd-9!#] of the subsequence by an analogous application of the proposed method to obtain sharp error estimates for a model initial-boundary problem for a linear parabolic partial differential equation of second order.

In the present paper we discuss the first 4 stages of the extended Lax principle for the model problem in consideration. These stages essentially correspond to the classical Lax principle, but classical error estimates obtained via this principle are rather coarse [#!13!#]. They were essentially sharpened in
[#!44!#,#!ltd-phd!#] using more advanced tools for error estimation, such as integral and averaged moduli of smoothness. Here we provide a systematic exposition of the results of [#!ltd-phd!#], sharpening, generalizing, and upgrading the results of [#!44!#], and complementing them with some additional ones, in order to prepare the error estimates at Stage 4 for use in the derivation of appropriate *a priori* estimates at Stage 5 and the final results of the new method at Stage 6 of the extended Lax principle, related to partial differential equations, in the subsequent papers [#!ee-dd-8!#,#!ee-dd-9!#].

**Received: **May 25, 2009

**AMS Subject Classification: **65M15, 65M22, 26A15, 26A16, 35G16, 35K05, 35K20, 39A06,
39A14, 39A70, 41A25, 41A63, 46E35, 46E39, 46N20,
46N30, 46N40, 47B38, 47B39, 47D03, 47D06, 47E05, 47F05, 47N20,
47N40, 65D15, 65D25,
65D30, 65J10, 65M05, 65M06, 65M10,
65M12, 65N05, 65N06, 65N08, 65N10, 65N12, 65N15, 65N22

**Key Words and Phrases: **extended Lax principle, intermediate approximation,
Steklov-means,
error, estimate, approximation, step, convergence, rate,
order, domain, mesh, discretization, discrete, semi-discrete,
continual, numerical, analysis, differentiation, integration,
finite
difference scheme,
modulus of
smoothness, integral, averaged, Riemann sum, uniform, non-uniform,
-functional, Lebesgue space, sequence space, Sobolev space,
Triebel-Lizorkin space, Besov space, interpolation space, real,
complex, -space, Wiener amalgam, metric, norm, quasi-norm,
isomorphism, embedding, bound, equivalence constant, initial,
boundary, initial-boundary, differential equation,
partial,
non-stationary, template, functional, linear,
positive, univariate, multivariate, multidimensional

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2009

**Volume:** 57

**Issue:** 5