A general equivalence theorem in the theory of discretization methods

Authors:
J. M. Sanz-Serna and C. Palencia

Journal:
Math. Comp. **45** (1985), 143-152

MSC:
Primary 65J10; Secondary 65M10, 65N10

DOI:
https://doi.org/10.1090/S0025-5718-1985-0790648-7

MathSciNet review:
790648

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Lax-Richtmyer theorem is extended to work in the framework of Stetter's theory of discretizations. The new result applies to both initial and boundary value problems discretized by finite elements, finite differences, etc. Several examples are given, together with a comparison with other available equivalence theorems. The proof relies on a generalized Banach-Steinhaus theorem.

**[1]**R. Ansorge, "Survey of equivalence theorems in the theory of difference approximations for partial differential equations",*Topics in Numerical Analysis*III (J. J. M. Miller, ed.), Academic Press, London, 1977, pp. 1-16. MR**513198 (81d:65052a)****[2]**R. Ansorge,*Differenzenapproximationen partieller Anfangswertaufgaben*, Teubner, Stuttgart, 1978. MR**513022 (80f:65001)****[3]**P. G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam, 1978. MR**0520174 (58:25001)****[4]**G. Dahlquist, "Convergence and stability in the numerical integration of ordinary differential equations,"*Math. Scand.*, v. 4, 1956, pp. 33-53. MR**0080998 (18:338d)****[5]**P. Henrici,*Discrete Variable Methods in Ordinary Differential Equations*, Wiley, New York, 1962. MR**0135729 (24:B1772)****[6]**P. D. Lax & R. D. Richtmyer, "Survey of stability of linear finite difference equations,"*Comm. Pure Appl. Math.*, v. 9, 1956, pp. 267-293. MR**0079204 (18:48c)****[7]**J. K. Mountain, "The Lax equivalence theorem for linear inhomogeneous equations in spaces,"*J. Approx. Theory*, v. 33, 1981, pp. 126-130. MR**643908 (83b:39003)****[8]**C. Palencia & J. M. Sanz-Serna, "Equivalence theorems for incomplete spaces: an appraisal,"*IMA J. Numer. Anal.*, v. 4, 1984, pp. 109-115. MR**740788 (86a:65049)****[9]**C. Palencia & J. M. Sanz-Serna, "An extension of the Lax-Richtmyer theory,"*Numer. Math.*, v. 44, 1984, pp. 279-283. MR**753959 (86c:65096)****[10]**R. D. Richtmyer & K. W. Morton,*Difference Methods for Initial Value Problems*, Interscience, New York, 1967. MR**0220455 (36:3515)****[11]**M. N. Spijker,*Stability and Convergence of Finite-Difference Methods*, Thesis, Leiden, Rijksuniversiteit, 1968. MR**0239761 (39:1118)****[12]**M. N. Spijker, "Equivalence theorems for nonlinear finite difference methods,"*Numerische Behandlung Nichtlinearer Integrodifferential und Differentialgleichungen*(R. Ansorge and W. Tornig, eds.), Lecture Notes in Math., Vol. 395, Springer-Verlag, Berlin, 1974, pp. 109-122. MR**0339442 (49:4201)****[13]**H. J. Stetter,*Analysis of Discretization Methods for Ordinary Differential Equations*, Springer, Berlin, 1973. MR**0426438 (54:14381)****[14]**F. Stummel, "Diskrete Konvergenz linearer Operatoren. I,"*Math. Ann.*, v. 190, 1970, pp. 45-92. MR**0291870 (45:959)****[15]**F. Stummel, "Weak stability and weak discrete convergence of continuous mappings,"*Numer. Math.*, v. 26, 1976, pp. 301-315. MR**0436578 (55:9521)****[16]**R. Temam,*Navier-Stokes Equations, Theory and Numerical Analysis*, North-Holland, Amsterdam, 1977.

Retrieve articles in *Mathematics of Computation*
with MSC:
65J10,
65M10,
65N10

Retrieve articles in all journals with MSC: 65J10, 65M10, 65N10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0790648-7

Article copyright:
© Copyright 1985
American Mathematical Society