| Title:
|
Finite element variational crimes in the case of semiregular elements (English) |
| Author:
|
Ženíšek, Alexander |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
41 |
| Issue:
|
5 |
| Year:
|
1996 |
| Pages:
|
367-398 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $\Omega $ whose boundary $\partial \Omega $ is formed by two circles $\Gamma _1$, $\Gamma _2$ with the same center $S_0$ and radii $R_1$, $R_2=R_1+\varrho $, where $\varrho \ll R_1$. On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u=0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus triangles obeying only the maximum angle condition and narrow quadrilaterals are used. The restrictions of test functions on triangles are linear functions while on quadrilaterals they are four-node isoparametric functions. Both the effect of numerical integration and that of approximation of the boundary are analyzed. The rate of convergence $O(h)$ in the norm of the Sobolev space $H^1$ is proved under the following conditions: 1. the (English) |
| Keyword:
|
finite element method |
| Keyword:
|
elliptic problems |
| Keyword:
|
semiregular elements |
| Keyword:
|
maximum angle condition |
| Keyword:
|
variational crimes |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 0870.65094 |
| idMR:
|
MR1404547 |
| DOI:
|
10.21136/AM.1996.134332 |
| . |
| Date available:
|
2009-09-22T17:52:18Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134332 |
| . |
| Reference:
|
[1] I. Babuška, and A.K. Aziz: On the angle condition in the finite element method.SIAM J. Numer. Anal. 13 (1976), 214–226. MR 0455462, 10.1137/0713021 |
| Reference:
|
[2] M. Feistauer, and A. Ženíšek: Finite element solution of nonlinear elliptic problems.Numer. Math. 50 (1987), 451–475. MR 0875168, 10.1007/BF01396664 |
| Reference:
|
[3] P. Jamet: Estimations d’erreur pour des éléments finis presque dégénérés.RAIRO Anal. Numér. 10 (1976), 43–61. MR 0455282 |
| Reference:
|
[4] M. Křížek: On semiregular families of triangulations and linear interpolation.Appl. Math. 36 (1991), 223–232. MR 1109126 |
| Reference:
|
[5] J. Nečas: Les Méthodes Directes en Théorie des Equations Elliptiques.Academia-Masson, Prague-Paris, 1967. MR 0227584 |
| Reference:
|
[6] L.A. Oganesian, and L.A. Rukhovec: Variational-Difference Methods for the Solution of Elliptic Problems.Izd. Akad. Nauk ArSSR, Jerevan, 1979. (Russian) |
| Reference:
|
[7] A. Ženíšek: Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations.Academic Press, London, 1990. MR 1086876 |
| Reference:
|
[8] A. Ženíšek, and M. Vanmaele: The interpolation theorem for narrow quadrilateral isoparametric finite elements.Numer. Math. 72 (1995), 123–141. MR 1359711, 10.1007/s002110050163 |
| . |
