Previous |  Up |  Next

# Article

 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 .

## Files

Files Size Format View
AplMat_36-1991-5_1.pdf 3.150Mb application/pdf View/Open

Partner of