Hanson, P. M.; Walsh, J. E. Asymptotic theory of the global error and some techniques of error estimation. (English) Zbl 0535.65075 Numer. Math. 45, 51-74 (1984). The error of the approximate solution obtained by discretising a functional equation can be shown under certain conditions to possess an asymptotic expansion in terms of some parameter which is usually a representative step-length. We consider the case of two-parameter expansions, which is particularly relevant to parabolic equations. We derive results for the existence of the expansion and for the application of the classical difference correction and of defect correction. The theory is illustrated by the discussion of a simple parabolic problem. Cited in 1 Document MSC: 65N15 Error bounds for boundary value problems involving PDEs 35K20 Initial-boundary value problems for second-order parabolic equations Keywords:deferred correction; error estimation; asymptotic expansion; two- parameter expansions PDFBibTeX XMLCite \textit{P. M. Hanson} and \textit{J. E. Walsh}, Numer. Math. 45, 51--74 (1984; Zbl 0535.65075) Full Text: DOI EuDML References: [1] Frank, R.: The method of iterated defect correction and its application to two-point boundary value problems. Part I. Numer. Math.25, 409-419 (1976). Part II. Numer. Math.27, 407-420 (1977) · Zbl 0346.65034 · doi:10.1007/BF01396337 [2] Frank, R., Ueberhuber, C.W.: Iterated defect correction for differential equations. Report No. 29/77. Inst. f. Numer. Math., Technical University of Vienna 1977 · Zbl 0364.65053 [3] Hanson, P.M., Walsh, J.E.: Asymptotic theory of the global error and some techniques of error estimation for parabolic equations. Report No. 43. Department of Mathematics, University of Manchester 1979 [4] Hildebrand, F.B.: Introduction to Numerical Analysis. New York: McGraw-Hill, 1956 · Zbl 0070.12401 [5] Lindberg, B.: Compact deferred correction formulas. In: Numerical integration of differential equations and large linear systems (J. Hinze, ed). Lecture Notes in Mathematics, No. 968. Berlin, Heidelberg, New York: Springer 1982 · Zbl 0496.65039 [6] Pereyra, V.: On improving an approximate solution of a functional equation by deferred corrections. Numer. Math.8, 376-391 (1966) · Zbl 0173.18103 · doi:10.1007/BF02162981 [7] Stetter, H.J.: Analysis of discretization methods for ordinary differential equations. Berlin, Heidelberg, New York: Springer 1973 · Zbl 0276.65001 [8] Stetter, H.J.: Global error estimation in O.D.E. solvers. Proceedings of Dundee Conference on Numerical Analysis (G.A. Watson, ed.) No. 630. Berlin, Heidelberg, New York: Springer 1977 [9] Stetter, H.J.: The defect correction principle and discretization methods. Numer. Math.29, 425-443 (1978) · Zbl 0362.65052 · doi:10.1007/BF01432879 [10] Zadunaisky, P.E.: On the estimation of errors propagated in the numerical integration of ordinary differential equations. Numer. Math.27, 21-39 (1976) · Zbl 0324.65035 · doi:10.1007/BF01399082 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.