Summary
The paper presents two methods for a piecewise Hermite interpolation of a sufficiently smooth function. The interpolation function is on each elementary rectangle, into which the given region is divided, determined by all the derivatives of the function under consideration up to a certain predetermined order. The results obtained are utilized in the solution of a general quasi-linear equation and in the solution of a non-linear integral equation.
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Birkhoff, G., Schultz, M. H., Varga, R. S.: Piecewise Hermite interpolation in one and two variables with applications to partial differential equations. Numer. Math.11, 232–256 (1968).
Bramble, J. H., Hilbert, S. R.: Estimation of linear functional on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal.7, 112–124 (1970).
Bramble, J. H., Zlámal, M.: Triangular elements in the finite element method. Math. Comp.24, 12, 809–820 (1970).
Browder, F. E.: Existence and uniqueness theorems for solutions of nonlinear boundary value problems. Proceedings Symposia on Applied Math. Vol. 17, Amer. Math. Soc. 24–49 (1965).
Ciarlet, P. G., Schultz, M. H., Varga, R. S.: Numerical methods of high-order accuracy for nonlinear boundary value problems. I. One dimensional problem. Numer. Math.9, 396–430 (1967).
Ciarlet, P. G., Schultz, M. H., Varga, R. S.: Numerical methods of high-order accuracy for nonlinear boundary value problems, V. Monotone operator theory. Numer. Math.13, 51–77 (1969).
Katshurovski, R. I.: Nonlinear monotone operators in Banach spaces [in Russian]. Uspechi Mat. Nauk XXIII,2, 140, 121–168 (1968).
Melkes, F.: The finite element method for nonlinear problems. Aplikace matematiky, No.3, 15, 177–189 (1970).
Vainberg, M. M.: Variational methods for the study of nonlinear operators [in Russian]. Moscow 1956.
Yosida, K.: Functional analysis [in Russian]. Moscow 1967.
Zienkiewicz, O. C.: The finite element method in structural and continuum mechanics. London: McGraw Hill 1967.
Zlámal, M.: On the finite element method. Numer. Math.12, 394–409 (1968).
Ženíšek, A.: Interpolation polynomials on the triangle. Numer. Math.15, 283–296 (1970).
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Melkes, F. Reduced piecewise bivariate Hermite interpolations. Numer. Math. 19, 326–340 (1972). https://doi.org/10.1007/BF01404879
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DOI: https://doi.org/10.1007/BF01404879