Summary
The distinction is described between qualitative error estimations (order of magnitude of the error) and quantitative error bounds (numerically computable strong mathematical pointwise error bounds). Progress by using approximation methods is made in the last years in singular nonlinear boundary value problems, in the method of finite elements, in free boundary value problems (exact inclusion for the free boundary in simple cases) and other areas.
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References
Bredendiek, E. [69] Simultant Approximation, Arch. Rat. Mech. Anal., 33 (1969), 307–330.
Bredendiek, E. and L, Collatz [76] Simultant Approximation bei Randwertaufgaben, to appear in Intern. Ser. Num. Math. 1976.
Cheney E.W. [66] Introduction to approximation theory N.W. McGraw-Hill, 1966.
Collatz, L. [56] Approximation von Funktionen bei einer und bei mehreren unabhängigen Veränderlichen, Z. Angew. Math. Mech. 36 (1956), 198–211.
Collatz, L. [69] Nichtlineare Approximation bei Randwertaufgaben V.JKM. Weimer 1969, 169–182.
Collatz, L. [75a] Methods for Solution of Partial Differential Equations, Symp. Numer. Solution Part. Diff. Equ. Kjeller, Norway 1975, 1–16.
Collatz, L. [74] Monotonicity with discontinuities in partial differential equations, Proc. Conf. Ord. Part. Diff. Equ. Dundee 1974, Lect. Notes Math. Bd. 415 (1974) 85–102.
Collatz, L. [75] Bemerkungen Zur verketteten Approximation, Intern. Ser. Num. Math. 26 (1975), 41–45.
Collatz, L. [77] The numerical treatment of some singular boundary value problems, Conference on Numer. Analysis, Dundee 1977, to appear.
Collatz, L. [78] Application of approximation to some singular boundary value problems, Proc. Conference Num. Anal. Dundee, Springer Lect. Not. Math. Bd. 630 (1978) 41–50.
Collatz, L. — W. Krabs [73] Approximationstheorie, Teubner, Stuttgart, 1973, 208 S.
Hoffmann, K.H. [78] Monotonie bei nichtlinearen Stefan-Problemen, Internat. Ser. Num. Math. 39 (1978), 162–190.
Meinardus, G. [67] Approximation of Functions, Theory and Numerical Methods, Springer Verlag, 1967, 198 S.
Meinardus, G. [76] Periodische Splines Functionen, Lect. Notes Math. Bd. 501 (1976) 177–199.
Meyer, A.G. [60] Schranken für die Lösungen von Randwertaufgaben mit elliptischer Differentialgleichung. Arch. rat. Mech. Anal. 6 (1960), 277–298.
Meyn, K.-H. -B. Werner [78] Randomaximum- und Monotonieprinzipien für elliptische Randwertaufgaben mit Gebietszerlegungen, erscheint demnächst, to appear.
Mitchell, A. R. — R. Wait [77] The finite Element Method in Partial Differential Equations, 1977, 192 p.
Natterer, F. [78] Berechnung oberer Schranken für die Norm der Ritz-Projektion auf finite Elemente Internat. Ser. Num. Math. 39 (1978) 236–245.
Osborne, M.R. — G.A. Watson [78] Nonlinear Approximation Problems in Vektornorms, Lect. Notes Math, 630 Springer 1978, 117–132.
Prenter, P. [71] Polynomial operators and equations, in Proc. Symp. Nonlinear functional analysis and applications, ed. L.B. Rail, Academic Press 1971.
Redheffer, R.M. [63] Die Collatzshce Monotonie bei Anfangswertproblemen, Arch. Rat. Mech. Anal. 14 (1963), 196–212.
Schröder, J. [56] Das Iterationsverfahren bei allgemeinerem Abstandbegriff, Math. Z. 1956, 66, 111–116.
Schröder, J. [59] Vom Defekt ausgehende Fehlerabschätzungen bei Differentialgleichungen. Arch. Rat. Mech. Anal. 3 (1959) 219–228.
Sprekels, J. [78] Iterationsverfahren zur Einschliebung positiver Lösungen superlinearer Integralgleichungen. Intern. Ser. Num. Math. 39, (1978) 261–279.
Taylor, G.D. [72] On minimal H-sets, J. Approximation Theory 5 (1972), 113–117.
Voss, H. [77] Existenz und Einschliebung positiver Losungen superlinearer Integralgleichungen und Randowertaufgaben, Habil. Schrift, Hamburg, 1977.
Walter, W. [70] Differential and Integral Inequalities, Springer 1970, 352 S.
Weinitschke, H. [77] Verzweigungsprobleme bei kreisförmigen elastischen Platten, Intern. Ser. Num. Math. 38 (1977) 195–212.
Werner, B. [75] Monotonie und finite Elemente bei elliptischen Differentialgleichungen, Intern. Ser. Num. Math. 27 (1975) 309–329.
Wetterling, W. [68] Lösungsschranken bei elliptischen Differen-tialgleichungen, Intern. Ser. Num. Math. 9 (1968) 393–401.
Wetterling, W. [77] Quotienteneinschliebung bei Eignewertaufgaben mit partieller Differentialgleichung, Int. Ser. Num. Math. 38 (1977) 213–218.
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Collatz, L. (1979). Some Applications of Polynomial and Spline Approximation. In: Sahney, B.N. (eds) Polynomial and Spline Approximation. NATO Advanced Study Institutes Series, vol 49. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9443-0_1
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