| Title:
|
A Petrov-Galerkin approximation of convection-diffusion and reaction-diffusion problems (English) |
| Author:
|
Dalík, Josef |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
36 |
| Issue:
|
5 |
| Year:
|
1991 |
| Pages:
|
329-354 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation $-\epsilon u^n + pu' + qu=f$ are presented and analyzed theoretically. The positive number $\epsilon$ is supposed to be much less than the discretization step and the values of $\left|p\right|,q$. An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced. (English) |
| Keyword:
|
convection-diffusion problem with dominated convection |
| Keyword:
|
Petrov-Galerkin method |
| Keyword:
|
reaction-diffusion equation |
| Keyword:
|
test functions |
| Keyword:
|
Petrov-Galerkin method |
| Keyword:
|
Dirichlet problem |
| Keyword:
|
algorithm |
| Keyword:
|
numerical examples |
| MSC:
|
34B05 |
| MSC:
|
34E15 |
| MSC:
|
35J25 |
| MSC:
|
65L10 |
| MSC:
|
65L60 |
| MSC:
|
65L99 |
| MSC:
|
65N30 |
| MSC:
|
65N99 |
| MSC:
|
76M10 |
| MSC:
|
76R50 |
| idZBL:
|
Zbl 0748.65061 |
| idMR:
|
MR1125636 |
| DOI:
|
10.21136/AM.1991.104471 |
| . |
| Date available:
|
2008-05-20T18:42:10Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104471 |
| . |
| Reference:
|
[1] J. E. Akin: Application and implementation of finite element methods.Academic Press, London, New York, 1982. Zbl 0535.73063, MR 0693291 |
| Reference:
|
[2] J. W. Barret K. W. Morton: The mathematics of finite elements and applications IV.Academic Press, London, New York (1982), 403-411. |
| Reference:
|
[3] P. Bar-Yoseph M. Israeli: An asymptotic finite element method for improvement of solutions of boundary layer problems.Numer. Math. Vol. 49, 4 (1986), 425-438. MR 0853664, 10.1007/BF01389540 |
| Reference:
|
[4] J. H. Bramble B. E. Hubbard: New monotone type approximations for elliptic problems.Math. Соmр. 18 (1964), 349-367. MR 0165702 |
| Reference:
|
[5] A. N. Brooks T. J. R. Hughes: Streamline upwind Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations.Computer Math. in Appl. Mech. and Eng. 32 (1982), 199-259. MR 0679322, 10.1016/0045-7825(82)90071-8 |
| Reference:
|
[6] P. Ciarlet: The finite element method for elliptic problems.North-Holland, Amsterdam, 1978. Zbl 0383.65058, MR 0520174 |
| Reference:
|
[7] J. Dalík: An apriori error estimate of an approximation of a two-point boundary value problem by the Petrov-Galerkin method.(Czech). Knižnice obd. a věd. spisů VUT Brno, Sv. A-35 (1988), 19-28. MR 0960239 |
| Reference:
|
[8] E. P. Doolan J. J. H. Miller W. H. A. Schilders: Uniform numerical methods for problems with initial and boundary layers.Boole Press, Dublin, 1980. MR 0610605 |
| Reference:
|
[9] R. Ghwinski: Numerical nethods for nonlinear variational problems. Appendix II.Springer- -Verlag, New York, Berlin, 1984. MR 0737005 |
| Reference:
|
[10] P. W. Hemker P. M. De Zeeuw: Defect correction for the solution of a singular perturbation problem.(preprint). Math. centrum, 1982. MR 0685785 |
| Reference:
|
[11] P. W. Hemker: Numerical aspects of singular perturbation problems.(preprint). Math. centrum, Amsterdam, 1982. MR 0708292 |
| Reference:
|
[12] T. lkeda: Maximum principle in finite element models for convection-diffusion phenomena.North-Holland, Amsterdam, New York, Oxford, 1983. |
| Reference:
|
[13] C. Johnson U. Nävert: Analysis of some finite element methods for advection-diffusion problems.(research report). Chalmers Univ. of Techn., Goteborg, 1980. MR 0605502 |
| Reference:
|
[14] C. Johnson U. Nävert J. Pitkäranta: Finite elements method for linear hyperbolic problems.(research report). Chalmers Univ. of Techn., Göteborg, 1982. |
| Reference:
|
[15] U. Nävert: A finite element method for convection-diffusion problems.(thesis). Chalmers Univ. of Techn., Göteborg, 1982. |
| Reference:
|
[16] U. Nävert: The streamline diffusion method for timedependent convection-diffusion problems with small diffusion.(research report). Chalmers Univ. of Techn., Göteborg, 1981. |
| Reference:
|
[17] E. O'Riordan: Singularly perturbed finite element methods.Numer. Math. Vol. 44, 3 (1984), 425-434. Zbl 0569.65065, MR 0757497, 10.1007/BF01405573 |
| Reference:
|
[18] G. D. Raithby: Skew upstream differencing schemes for problems involving fluid flow.Comp. Meth. Appl. Mech. Eng. Vol 9 (1976), 153-164. Zbl 0347.76066, MR 0443576, 10.1016/0045-7825(76)90058-X |
| Reference:
|
[19] P. A. Raviart: Les méthodes d'élements finis en mécanique des fluides II.3. Edditions Eyrolles, Paris, 1981. MR 0631851 |
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