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

**ON ERROR ESTIMATION FOR APPROXIMATION METHODS**

INVOLVING DOMAIN DISCRETIZATION

VI: DETERMINISTIC PROBLEMS V.

FULLY DISCRETE FINITE DIFFERENCE SCHEMES FOR

LINEAR ODES III: ESTIMATES IN TERMS OF PROPERTIES

OF THE PROBLEM'S DATA FUNCTIONS

II: INHOMOGENEOUS BOUNDARY CONDITIONS

INVOLVING DOMAIN DISCRETIZATION

VI: DETERMINISTIC PROBLEMS V.

FULLY DISCRETE FINITE DIFFERENCE SCHEMES FOR

LINEAR ODES III: ESTIMATES IN TERMS OF PROPERTIES

OF THE PROBLEM'S DATA FUNCTIONS

II: INHOMOGENEOUS 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 sixth of a sequence of 12 papers, preceded by
[#!ee-dd-1!#,#!ee-dd-2!#,#!ee-dd-3!#,#!ee-dd-4!#,#!ee-dd-5!#] and followed by
[#!ee-dd-7!#,#!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 third one, preceded by [#!ee-dd-4!#,#!ee-dd-5!#] and followed by [#!ee-dd-7!#,#!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!#] and here we apply 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!#,#!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 [#!ee-dd-4!#] 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 [#!1!#] using more advanced tools for error estimation, such as integral and averaged moduli of smoothness. In [#!ee-dd-4!#] we provided a systematic exposition of the results of [#!1!#], sharpened, generalized, and upgraded these results, 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 ordinary differential equations, in [#!ee-dd-5!#] and the present paper.

In [#!ee-dd-5!#] we addressed two major topics (corresponding to Stages 5 and 6 of the extended Lax principle), as follows.

- Stage 5: We developed
*a priori*estimates for the continual problem () which are*consistent*with the error estimates in terms of properties of the solution from [#!ee-dd-4!#]. - Stage 6: Based on the results obtained on Stage 4 in [#!ee-dd-4!#] and the results obtained on Stage 5 in [#!ee-dd-5!#], we derived sharp error estimates directly in terms of the data (variable coefficients and variable right-hand side) of the continual boundary-value problem () for the case of
*homogeneous boundary conditions*. These new error estimates implied a diversity of corollaries providing certain approximation rates under minimal assumptions about regularity of the data.

The considerations made in [#!ee-dd-5!#] were made only for the special case of homogeneous boundary conditions. The present paper is dedicated to extending the results obtained in [#!ee-dd-5!#] for homogeneous boundary conditions to *the general case of inhomogeneous boundary conditions*.

**Received: **May 19, 2009

**AMS Subject Classification: **65L12, 65L70, 26A15, 34A30, 34G05, 34G10, 39A06, 39A70, 41A25, 46N20, 46N40, 47B39, 47E05, 47N20, 47N40, 65D25, 65J10, 65L10, 65L20

**Key Words and Phrases: **error, estimate, approximation, step, convergence, rate, order, domain, mesh, discretization, discrete, continual, numerical, analysis, differentiation, finite difference scheme, modulus of smoothness, integral, averaged, Riemann sum, uniform, non-uniform, sequence space, Wiener amalgam, metric, norm, bound, boundary, differential equation, ordinary, stationary, template, functional, linear, non-linear, positive, univariate

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

**ISSN:** 1311-8080

**Year:** 2009

**Volume:** 57

**Issue:** 5