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

**ON ERROR ESTIMATION FOR APPROXIMATION METHODS**

INVOLVING DOMAIN DISCRETIZATION III:

DETERMINISTIC PROBLEMS II. FULL DISCRETIZATION

OF SEMI-DISCRETE FINITE DIFFERENCE SCHEMES

FOR LINEAR EVOLUTIONARY PDES II:

THE GENERAL, POSSIBLY NON-PARABOLIC, CASE

INVOLVING DOMAIN DISCRETIZATION III:

DETERMINISTIC PROBLEMS II. FULL DISCRETIZATION

OF SEMI-DISCRETE FINITE DIFFERENCE SCHEMES

FOR LINEAR EVOLUTIONARY PDES II:

THE GENERAL, POSSIBLY NON-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 third of a sequence of 12 papers, preceded by
[#!ee-dd-1!#,#!ee-dd-2!#] and followed by [#!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-2!#], 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 [#!ee-dd-2!#]
and the present communication covers all previously unpublished results in Chapter 2 of [#!ltd-phd!#], which extend, generalize and complement the results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. While [#!ee-dd-2!#] treated the parabolic case which features
higher rate of approximation due to the smoothing properties of the parabolic resolving operator, the present paper considers the general, possibly non-parabolic, case which exhibits lower rate of approximation.
The class of essentially non-parabolic partial differential equations (PDEs) included in the present consideration contains, e.g., the linear hyperbolic PDEs of first order *with variable coefficients*. This particular class of PDEs has been studied earlier in [#!22!#], for the case of semi-discrete approximation, and restricted only to the case of *constant coefficients*. The results of [#!22!#] have been upgraded in [#!25!#,#!26!#] for the case of fully discrete approximation, based on the estimates obtained in [#!22!#] for the semi-discrete case. For this particular narrow subclass of PDEs, the convergence rate provided by the general error estimate proved in Theorem of the present paper coincides with the respective convergence rates in the error estimates obtained in [#!22!#,#!25!#,#!26!#]; however, the present result extends also to the case of *variable coefficients*.

**Received: **May 13, 2009

**AMS Subject Classification: **65M15, 65M22, 26A15, 26A16, 35G16, 35K05, 35K20, 39A06, 39A14, 39A70, 41A25, 41A55, 41A63, 46E35, 46E39, 46N20, 46N40, 47B38, 47B39, 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, 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