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Literatur
Agmon, S., Lectures on Elliptic Boundary Value Problems. Princeton: Van Nostrand, 1965.
Birkhoff, G., M.H. Schultz, & R.S. Varga, Piecewise hermite interpolation in one and two variables with applications to partial differential equations. Numer. Math. 11, 232–256 (1968).
Bramble, J.H., & B.E. Hubbard, A priori bounds on the discretization error in the numerical solution of the Dirichlet problem. Contributions to Differential Equations 2, 229–252 (1963).
Bramble, J.H., & B.E. Hubbard, A theorem on error estimation for finite difference analogues of the Dirichlet problem for elliptic equations. Contributions to Differential Equations 2, 319–340 (1963).
Bramble, J. H., & B. E. Hubbard, Approximation of solutions of mixed boundary value problems for Poisson's equation by finite differences. J. Association for Computing Machinery 12, 114–123 (1965).
Bramble, J. H., & L. E. Payne, Bounds for solutions of second-order elliptic partial differential equations. Contributions to Differential Equations 1, 95–127 (1963).
Bramble, J. H., R. B. Kellogg, & V. Thomée, On the rate of convergence of some difference schemes for second order elliptic equations. Technical Note BN-534, Inst. for Fluid Dynamics and Appl. Math., University of Maryland, 1968.
Céa, J., Approximation variationnelle des problèmes aux limites. Ann. Inst. Fourier, 14, 345–444 (1964).
Courant, R., Variational methods for the solution of problems of equilibrium and vibrations. Bull. Amer. Math. Soc. 49, 1–23 (1943).
Demjanovic, Ju. K., The net method for some problems in mathematical physics. Dokl. Akad. Nauk SSSR 159 (1964); Soviet Math. 5, 1452–1456 (1964).
Demjanovic, Ju. K., Approximation and convergence of the net method in elliptic problems. Dokl. Akad. Nauk SSSR 170 (1966); Soviet Math. 7, 1129–1133 (1966).
Fix, G., Higher-order Rayleigh-Ritz approximations. J. Math. Mech. 18, 645–657 (1969).
Friedrichs, K. O., & H. B. Keller, A Finite Difference Scheme for Generalized Neumann Problems. In: Numerical Solution of Partial Differential Equations. New York: Academic Press 1966, S. 1–19.
Helfrich, H. P., Optimale lineare Approximation beschränkter Mengen in normierten Räumen. Erscheint demnächst.
Hubbard, B. E., Remarks on the Order of Convergence in the Discrete Dirichlet Problem. In: Numerical Solution of Partial Differential Equations. New York: Academic Press 1966, S. 21–34.
Katsanis, Th., A numerical method for the solution of certain Neumann problems. SIAM J. Appl. Math. 16, 723–731 (1968).
Kellogg, R. B., Difference equations on a mesh arising from a general triangulation. Math. Comp. 18, 203–210 (1964).
Kellogg, R. B., An error estimate for elliptic difference equations on a convex polygon. SIAM J. Numer. Anal. 3, 79–90 (1966).
Laasonen, P., On the solution of Poisson's difference equation. J. the Association for Computing Machinery 5, 370–382 (1958).
Lorentz, G. G., Approximation of Functions. New York: Holt, Rinehart and Winston 1966.
Mihlin, C. G., Some properties of polynomial approximations according to Ritz. Dokl. Akad. Nauk SSSR 180, 276–278 (1968); Soviet Math. 9, 614–616 (1968).
Necas, J., Les méthodes discrètes en théorie des équations élliptiques. Paris: Masson & Cie. Academia Prague 1967.
Nitsche, J., Zur Frage optimaler Fehlerschränken bei Differenzenverfahren. Rend. Circ. Mat. Palermo 16, 69–80 und 233–238 (1967).
Nitsche, J., Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens. Numer. Math. 11, 346–348 (1968).
Nitsche, J. A., & J. C. C. Nitsche, Error estimates for the numerical solution of elliptic differential equations. Arch. Rational Mech. Anal. 5, 293–306 (1960).
Oganesjan, L. A., Convergence of difference schemes in case of improved approximation of the boundary. Z. Vycisl. Mat. i Mat. Fiz. 6, 1029–1042 (1966).
Sobolev, S. L., Applications of Functional Analysis in Mathematical Physics. Am. Math. Soc., Providence, 1963.
Thomée, V., On the convergence of difference quotients in elliptic problems. Technical Note BN-537, Chalmers Institute of Technology, Göteborg, April 1968.
Veidinger, L., Über die Abschätzung des Fehlers bei finiten Differenzen. Stud. Sci. Mat. Hungarica 2, 185–191 (1967).
Zlamal, M., On the finite element method. Numer. Math. 12, 394–409 (1968).
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Nitsche, J. Lineare spline-funktionen und die methoden von ritz für elliptische randwertprobleme. Arch. Rational Mech. Anal. 36, 348–355 (1970). https://doi.org/10.1007/BF00282271
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DOI: https://doi.org/10.1007/BF00282271