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nonhomogeneous boundary conditions; Dirichlet; Neumann; finite element method; curved triangular elements; convergence
Approximation of nonhomogeneous boundary conditions of Dirichlet and Neumann types is suggested in solving boundary value problems of elliptic equations by the finite element method. Curved triangular elements are considered. In the first part of the paper the convergence of the finite element method is analyzed in the case of nonhomogeneous Dirichlet problem for elliptic equations of order $2m+2$, in the second part of the paper in the case of nonhomogeneous mixed boundary value problem for second order elliptic equations. In both parts of the paper of numerical integration is studied.
[1] J. J. Blair: Higher order approximations to the boundary conditions for the finite element method. Math. Соmр. 30 (1976), 250-262. MR 0398123 | Zbl 0342.65068
[2] J. H. Bramble S. R. Hilbert: Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970), 112-124. DOI 10.1137/0707006 | MR 0263214
[3] P. G. Ciarlet P. A. Raviart: 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 0421108
[4] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam 1978. MR 0520174 | Zbl 0383.65058
[5] S. Koukal: Piecewise polynomial interpolations in the finite element method. Apl. Mat. 18 (1973), 146-160. MR 0321318
[6] L. Mansfield: Approximation of the boundary in the finite element solution of fourth order problems. SIAM J. Numer. Anal. 15 (1978), 568-579. DOI 10.1137/0715037 | MR 0471373 | Zbl 0391.65047
[7] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague 1967. MR 0227584
[8] R. Scott: Interpolated boundary conditions in the finite element method. SIAM J. Numer. Anal. 12 (1975), 404-427. DOI 10.1137/0712032 | MR 0386304 | Zbl 0357.65082
[9] G. Strang: Approximation in the finite element method. Numer. Math. 19 (1972), 81-98. DOI 10.1007/BF01395933 | MR 0305547 | Zbl 0221.65174
[10] M. Zlámal: Curved elements in the finite element method. I. SIAM J. Numer. Anal. 10 (1973), 229-240. DOI 10.1137/0710022 | MR 0395263
[11] M. Zlámal: Curved elements in the finite element method. II. SIAM J. Numer. Anal. 11 (1974), 347-362. DOI 10.1137/0711031 | MR 0343660
[12] A. Ženíšek: Curved triangular finite $C^m$-elements. Apl. Mat. 23 (1978), 346-377. MR 0502072
[13] A. Ženíšek: Discrete forms of Friedrichs' inequalities in the finite element method. (To appear.) MR 0631681 | Zbl 0475.65072
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