Summary
In the paper we consider a singularly perturbed linear parabolic initialboundary value problem in one space variable. Two exponential fitted schemes are derived for the problem using Petrov-Galerkin finite element methods with various choices of trial and test spaces. On rectangular meshes which are either arbitrary or slightly restricted, we derive global energy norm andL 2 norm and localL ∞ error bounds which are uniform in the diffusion parameter. Numerical results are also persented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bobisud, L. (1967): Second-order linear parabolic equations with a small parameter. Arch. Rational Mech. Anal.27, 385–397
Celia, M.A., Russell, T.F., Herrera, I., Ewing, R.E. (1990): An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation. Adv. Water Resour.13, 187–206
Dawson, C.N., Russell, T.F., Wheeler, M.F. (1989): Some improved error estimates for the modified method of characteristics. SIAM J. Numer. Anal.26, 1487–1512
Farrell, P.A., Hegarty, A. (1992): On the determination of the order of uniform convergence. In: R. Vichnevetsky, J.J.H. Miller eds., Proceedings of 13th IMACS World Congress on Computation and Applied Mathematics IMACS, pp. 501–502, North-Holland, Amsterdam
Guo, W. Ph.D. Dissertation. Mathematics Department, University College, Cork, Ireland (in preparation)
Johnson, C. (1987): Numerical solution of partial differential equations by the finite element method. Cambridge University Press, Cambridge
Morton, K.W., Sobey, I.J. (1991): Discretisation of a convection-diffusion equation. Technical Report 91/4. Oxford University Computing Laboratory, Numerical Analysis Group Oxford
Nävert, U. (1982): A finite element method for convection-diffusion problems. Thesis. Chalmers University of Technology and University of Gothenburg, Gothenburg
Ng-Stynes, M.J., O'Riordan, E., Stynes, M. (1988): Numerical methods for time-dependent convection-diffusion equations. J. Comput. Appl. Math.21, 289–310
Niijima, K. (1990): Pointwise error estimates for a streamline diffusion finite element scheme. Numer. Math.56, 707–719
Stynes, M. (1989): An adaptive uniformly convergent numerical method for a semilinear singular perturbation problem. SIAM J. Numer. Anal.26, 442–455
Stynes, M., Guo, W. (1991): Finite element analysis of an advection-diffusion equation, In: Numerical methods in singularly perturbed problems. H.G. Roos, A. Felgenhauer, L. Angermann, eds., Proceedings International Seminar on Appl. Mathmatic. 91, pp. 123–128. Technische Universität Dresden, Dresden, Germany
Stynes, M., O'Riordan, E. (1989): Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points. Numer. Math.55, 521–544
Stynes, M., O'Riordan, E. (1991): An analysis of a two-point boundary value problem with a boundary layer, using only finite element techniques. Math. Comput.56, 663–675
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Guo, W., Stynes, M. Finite element analysis of exponentially fitted Lumped schemes for time-dependent convection-diffusion problems. Numer. Math. 66, 347–371 (1993). https://doi.org/10.1007/BF01385702
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01385702