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boundary value elliptic problems; finite element method; generalized splines; elastic plate
We prove that the finite element method for one-dimensional problems yields no discretization error at nodal points provided the shape functions are appropriately chosen. Then we consider a biharmonic problem with mixed boundary conditions and the weak solution $u$. We show that the Galerkin approximation of $u$ based on the so-called biharmonic finite elements is independent of the values of $u$ in the interior of any subelement.
[1] J.  H.  Ahlberg, E.  N.  Nilson and J. L.  Walsh: The Theory of Splines and Their Applications. Academic Press, London, 1967. MR 0239327
[2] I.  Babuška, M.  Práger and E. Vitásek: Numerical Processes in Differential Equations. John Wiley & Sons, London, 1966. MR 0223101
[3] C.  Baiocchi, F.  Brezzi and L. P.  Franca: Virtual bubbles and Galerkin-least-squares type methods. Comput. Methods Appl. Mech. Engrg. 105 (1993), 125–141. DOI 10.1016/0045-7825(93)90119-I | MR 1222297
[4] G.  H.  Behforooz, N.  Papamichael: Improved orders of approximation derived from interpolatory cubic splines. BIT 19 (1979), 19–26. DOI 10.1007/BF01931217 | MR 0530111
[5] M.  Bernadou: Convergence of conforming finite element methods for general shell problems. Internat. J. Engrg. Sci. 18 (1980), 249–276. DOI 10.1016/0020-7225(80)90049-X | MR 0661274 | Zbl 0429.73084
[6] J.  Brandts: Superconvergence phenomena in finite element methods. PhD thesis, Utrecht Univ. (1995).
[7] F. Brezzi, A. Russo: Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4 (1994), 571–587. DOI 10.1142/S0218202594000327 | MR 1291139
[8] P. G.  Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, New York, Oxford, 1978. MR 0520174 | Zbl 0383.65058
[9] G.  Heindl: Interpolation and approximation by piecewise quadratic $C^1$-functions of two variables. In: Multivariate Approximation Theory (W. Schempp and K. Zeller, eds.), ISNM vol. 51, Birkhäuser, Basel, 1979, pp. 146–161. MR 0560670
[10] I.  Hlaváček, J.  Nečas: On inequalities of Korn’s type. Arch. Rational Mech. Anal. 36 (1970), 305–334. DOI 10.1007/BF00249518
[11] V.  Hoppe: Finite elements with harmonic interpolation functions. In: Proc. Conf. MAFELAP (J. R.  Whiteman, ed.), Academic Press, London, 1973, pp. 131–142. Zbl 0278.73051
[12] D.  Huang, D. Wu: The superconvergence of the spline finite element solution and its second order derivative for the two-point boundary problem of a fourth order differential equation. J.  Zhejiang Univ. 3 (1982), 92–99.
[13] M.  Křížek, L.  Liu, P. Neittaanmäki: On harmonic and biharmonic finite elements. In: Finite Element Methods III: Three-dimensional problems, Vol. 15, M. Křížek, P. Neittaanmäki (eds.), GAKUTO Internat. Ser. Math. Sci. Appl., Tokyo, 2001, pp. 146–154.
[14] M.  Křížek, P.  Neittaanmäki: On time-harmonic Maxwell equations with nonhomogeneous conductivities: solvability and FE-approximation. Apl.  Mat. 34 (1989), 480–499. MR 1026513
[15] M.  Křížek, P.  Neittaanmäki: Finite Element Approximation of Variational Problems and Applications. Longman, Harlow, 1990. MR 1066462
[16] J. E.  Lagnese, J.-L.  Lions: Modelling Analysis and Control of Thin Plates. Masson, Paris and Springer-Verlag, Berlin, 1989.
[17] Q.  Lin: Full convergence order for hyperbolic finite elements. In: Proc. Conf. Discrete Galerkin Methods (B.  Cockburn, ed.), Newport 1999, pp. 167–177. MR 1842172
[18] J.  Nečas, I.  Hlaváček: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Elsevier, Amsterdam, Oxford, New York, 1981. MR 0600655
[19] K.  Rektorys: Variational Methods in Mathematics, Science and Engineering, chap. 23. Riedel, Dodrecht, 1980. MR 0596582
[20] G.  Strang, G.  Fix: An Analysis of the Finite Element Method. Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1973. MR 0443377
[21] P.  Tong: Exact solutions of certain problems by finite-element method. AIAA J. 7 (1969), 178–180. DOI 10.2514/3.5067
[22] C.  Wielgosz: Exact results given by finite element methods in mechanics. J.  Méch. Théor. Appl. 1 (1982), 323–329. MR 0700053 | Zbl 0503.73046
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