| Title:
|
On exact results in the finite element method (English) |
| Author:
|
Hlaváček, Ivan |
| Author:
|
Křížek, Michal |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
46 |
| Issue:
|
6 |
| Year:
|
2001 |
| Pages:
|
467-478 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
boundary value elliptic problems |
| Keyword:
|
finite element method |
| Keyword:
|
generalized splines |
| Keyword:
|
elastic plate |
| MSC:
|
35J40 |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 1066.65126 |
| idMR:
|
MR1865517 |
| DOI:
|
10.1023/A:1013716729409 |
| . |
| Date available:
|
2009-09-22T18:08:06Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134478 |
| . |
| Reference:
|
[1] J. H. Ahlberg, E. N. Nilson and J. L. Walsh: The Theory of Splines and Their Applications.Academic Press, London, 1967. MR 0239327 |
| Reference:
|
[2] I. Babuška, M. Práger and E. Vitásek: Numerical Processes in Differential Equations.John Wiley & Sons, London, 1966. MR 0223101 |
| Reference:
|
[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. MR 1222297, 10.1016/0045-7825(93)90119-I |
| Reference:
|
[4] G. H. Behforooz, N. Papamichael: Improved orders of approximation derived from interpolatory cubic splines.BIT 19 (1979), 19–26. MR 0530111, 10.1007/BF01931217 |
| Reference:
|
[5] M. Bernadou: Convergence of conforming finite element methods for general shell problems.Internat. J. Engrg. Sci. 18 (1980), 249–276. Zbl 0429.73084, MR 0661274, 10.1016/0020-7225(80)90049-X |
| Reference:
|
[6] J. Brandts: Superconvergence phenomena in finite element methods.PhD thesis, Utrecht Univ. (1995). |
| Reference:
|
[7] F. Brezzi, A. Russo: Choosing bubbles for advection-diffusion problems.Math. Models Methods Appl. Sci. 4 (1994), 571–587. MR 1291139, 10.1142/S0218202594000327 |
| Reference:
|
[8] P. G. Ciarlet: The Finite Element Method for Elliptic Problems.North-Holland, Amsterdam, New York, Oxford, 1978. Zbl 0383.65058, MR 0520174 |
| Reference:
|
[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 |
| Reference:
|
[10] I. Hlaváček, J. Nečas: On inequalities of Korn’s type.Arch. Rational Mech. Anal. 36 (1970), 305–334. 10.1007/BF00249518 |
| Reference:
|
[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 |
| Reference:
|
[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. |
| Reference:
|
[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. MR 1896273 |
| Reference:
|
[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 |
| Reference:
|
[15] M. Křížek, P. Neittaanmäki: Finite Element Approximation of Variational Problems and Applications.Longman, Harlow, 1990. MR 1066462 |
| Reference:
|
[16] J. E. Lagnese, J.-L. Lions: Modelling Analysis and Control of Thin Plates.Masson, Paris and Springer-Verlag, Berlin, 1989. |
| Reference:
|
[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 |
| Reference:
|
[18] J. Nečas, I. Hlaváček: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction.Elsevier, Amsterdam, Oxford, New York, 1981. MR 0600655 |
| Reference:
|
[19] K. Rektorys: Variational Methods in Mathematics, Science and Engineering, chap. 23.Riedel, Dodrecht, 1980. MR 0596582 |
| Reference:
|
[20] G. Strang, G. Fix: An Analysis of the Finite Element Method.Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1973. MR 0443377 |
| Reference:
|
[21] P. Tong: Exact solutions of certain problems by finite-element method.AIAA J. 7 (1969), 178–180. 10.2514/3.5067 |
| Reference:
|
[22] C. Wielgosz: Exact results given by finite element methods in mechanics.J. Méch. Théor. Appl. 1 (1982), 323–329. Zbl 0503.73046, MR 0700053 |
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