# IJPAM: Volume 57, No. 6 (2009)

**ON ERROR ESTIMATION FOR APPROXIMATION METHODS**

INVOLVING DOMAIN DISCRETIZATION

VIII: DETERMINISTIC PROBLEMS VII.

FULLY DISCRETE FINITE DIFFERENCE SCHEMES

FOR LINEAR PDES II. ESTIMATES IN TERMS OF

PROPERTIES OF THE PROBLEM'S DATA FUNCTIONS

I: HOMOGENEOUS BOUNDARY CONDITIONS

INVOLVING DOMAIN DISCRETIZATION

VIII: DETERMINISTIC PROBLEMS VII.

FULLY DISCRETE FINITE DIFFERENCE SCHEMES

FOR LINEAR PDES II. ESTIMATES IN TERMS OF

PROPERTIES OF THE PROBLEM'S DATA FUNCTIONS

I: HOMOGENEOUS BOUNDARY CONDITIONS

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 eighth 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!#,#!ee-dd-7!#] and followed by
[#!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 fifth one, preceded by [#!ee-dd-4!#,#!ee-dd-5!#,#!ee-dd-6!#,#!ee-dd-7!#] and followed by [#!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 remaining three papers, [#!ee-dd-7!#], the present paper paper and [#!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 [#!ee-dd-7!#] we discussed 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 the rather coarse classical error estimates obtained via this principle have been essentially sharpened in [#!ltd-phd!#,#!44!#] using more advanced tools for error estimation, such as integral and averaged moduli of smoothness. In [#!ee-dd-7!#] we provided a systematic exposition of the results of [#!ltd-phd!#], sharpened, generalized, and upgraded the results of [#!44!#], and complemented 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 present paper and the subsequent paper [#!ee-dd-9!#].

In the present paper we address two major topics (corresponding to Stages 5 and 6 of the extended Lax principle), as follows.

- Stage 5: We develop
*a priori*estimates for the continual problem () with*homogeneous initial and boundary conditions*which are*consistent*with the error estimates in terms of properties of the solution from [#!ee-dd-7!#]. - Stage 6: Based on the results obtained on Stage 4 in [#!ee-dd-7!#] and the results obtained here on Stage 5, we derive sharp error estimates directly in terms of the data of the continual initial-boundary problem () for the case of
*homogeneous boundary conditions*. (In view of the homogeneity of the initial and boundary conditions and the constance of the linear PDE's coefficients, the data-function here is only the right-hand side.) These new error estimates imply a diversity of corollaries providing certain approximation rates under minimal assumptions about regularity of the data.

In [#!ee-dd-9!#] we shall extend the results obtained here for homogeneous initial and boundary conditions to the general case of inhomogeneous initial and/or boundary conditions.

**Received: **May 28, 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, Sobolev-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,
continuity, density, extension,
template, functional, linear,
positive, univariate, multivariate, multidimensional

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

**ISSN:** 1311-8080

**Year:** 2009

**Volume:** 57

**Issue:** 6