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

**ON ERROR ESTIMATION FOR APPROXIMATION METHODS**

INVOLVING DOMAIN DISCRETIZATION II:

DETERMINISTIC PROBLEMS I. FULL DISCRETIZATION

OF SEMI-DISCRETE FINITE DIFFERENCE SCHEMES

FOR LINEAR EVOLUTIONARY PDES I:

THE PARABOLIC CASE

INVOLVING DOMAIN DISCRETIZATION II:

DETERMINISTIC PROBLEMS I. FULL DISCRETIZATION

OF SEMI-DISCRETE FINITE DIFFERENCE SCHEMES

FOR LINEAR EVOLUTIONARY PDES I:

THE PARABOLIC CASE

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 second of a sequence of 12 papers, preceded by
[#!ee-dd-1!#] and followed by [#!ee-dd-3!#,#!ee-dd-4!#,#!ee-dd-5!#,#!ee-dd-6!#,#!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.

This paper, together with [#!ee-dd-3!#], presents a new approach to error estimation
of the numerical solution of initial-boundary problems for
linear differential equations by *full discretization of semi-discrete approximating problems*. The material in the present communication
and [#!ee-dd-3!#] covers all previously unpublished results in Chapter 2 of
[#!ltd-phd!#], which extend, generalize and complement the results
of [#!45!#,#!24!#,#!25!#,#!26!#] and improve upon results of Bergh, Brenner,
Löfström, Peetre, Thomée, Wahlbin, Widlund and others
obtained in the case of semi-discrete approximation of a Cauchy
problem for a general class of linear evolutionary partial
differential equations. The present paper treats the parabolic case which features
higher rate of approximation due to the smoothing properties of the parabolic resolving operator, in contrast with the general, possibly non-parabolic, case considered in
[#!ee-dd-3!#] which exhibits lower rate of approximation.

**Received: **May 8, 2009

**AMS Subject Classification: **65M15, 65M22, 26A15, 26A16, 35G16, 35K05, 35K20, 39A06, 39A14, 39A70, 41A25, 41A55, 41A63, 46E35, 46E39, 46N20, 46N40, 47B38, 47B39, 47D03, 47D06, 47F05, 47N20, 47N40, 65D15, 65D25, 65D30, 65J10, 65M05, 65M06, 65M10, 65M12

**Key Words and Phrases: **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, differential equation, partial, non-stationary, linear, univariate, multivariate, multidimensional

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

**ISSN:** 1311-8080

**Year:** 2009

**Volume:** 57

**Issue:** 5