Geodesic
Curved triangular finite $C^m$-elements
Applications of Mathematics, Tome 23 (1978) no. 5, pp. 346-377
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Curved triangular $C^m$-elements which can be pieced together with the generalized Bell's $C^m$-elements are constructed. They are applied to solving the Dirichlet problem of an elliptic equation of the order $2(m+1)$ in a domain with a smooth boundary by the finite element method. The effect of numerical integration is studied, sufficient conditions for the existence and uniqueness of the approximate solution are presented and the rate of convergence is estimated. The rate of convergence is the same as in the case of polygonal domains when the generalized Bell's $C^m$-elements are used.
Curved triangular $C^m$-elements which can be pieced together with the generalized Bell's $C^m$-elements are constructed. They are applied to solving the Dirichlet problem of an elliptic equation of the order $2(m+1)$ in a domain with a smooth boundary by the finite element method. The effect of numerical integration is studied, sufficient conditions for the existence and uniqueness of the approximate solution are presented and the rate of convergence is estimated. The rate of convergence is the same as in the case of polygonal domains when the generalized Bell's $C^m$-elements are used.
DOI : 10.21136/AM.1978.103761
Classification : 35A35, 35J40, 65M99, 65N30, 65N99
Keywords: generalized Bell’s $C^m$-elements; approximate solution; rate of convergence 
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Ženíšek, Alexander. Curved triangular finite $C^m$-elements. Applications of Mathematics, Tome 23 (1978) no. 5, pp. 346-377. doi: 10.21136/AM.1978.103761

[1] Bramble J. H., Zlámal M.: Triangular elements in the finite element method. Math. Соmр. 24 (1970), 809-820. | MR

[2] Ciarlet P. G., Raviart P. A.: The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Editor), pp. 409-474, Academic Press, New York 1972. | MR | Zbl

[3] Ciarlet P. G.: Numerical Analysis of the Finite Element Method. Université de Montréal, 1975. | MR

[4] Holuša L., Kratochvíl J., Zlámal M., Ženíšek A.: The Finite Element Method. Technical Report. Computing Center of the Technical University of Brno, 1970. (In Czech.)

[5] Kratochvíl J., Ženíšek A., Zlámal M.: A simple algorithm for the stiffness matrix of triangular plate bending finite elements. Int. J. numer. Meth. Engng. 3 (1971), 553 - 563. | DOI

[6] Mansfield L.: Approximation of the boundary in the finite element solution of fourth order problems. SIAM J. Numer. Anal. 15 (1978), the June issue. | DOI | MR | Zbl

[7] Nečas J.: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. | MR

[8] Stroud A. H.: Approximate Calculation of Multiple Integrals. Prentice-Hall., Englewood Cliffs, N. J., 1971. | MR | Zbl

[9] Zlámal M.: The finite element method in domains with curved boundaries. Int. J. numer. Meth. Engng. 5 (1973), 367-373. | DOI | MR

[10] Zlámal M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal. 10(1973), 229-240. | DOI | MR

[11] Zlámal M.: Curved elements in the finite element method. II. SlAM J. Numer. Anal. 1.1 (1974), 347-362. | DOI | MR

[12] Ženíšek A.: Interpolation polynomials on the triangle. Numer. Math. 15 (1970), 283 - 296. | DOI | MR

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