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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2012004234
pages 319-331

PSEUDO-MULTI-SCALE FUNCTIONS FOR THE STABILIZATION OF CONVECTION-DIFFUSION EQUATIONS ON RECTANGULAR GRIDS

Ali I. Nesliturk
Department of Mathematics, Izmir Institute of Technology, 35430, Izmir, Turkey
Onur Baysal
Department of Mathematics, Izmir Institute of Technology, 35430, Izmir, Turkey

ABSTRACT

We propose a finite element method of Petrov-Galerkin type for a singularly perturbed convection diffusion problem on a discretization consisting of rectangular elements. The method is based on enriching the finite-element space with a combination of multiscale and residual-free bubble functions. These functions require the solution of the original differential problem, which makes the method quite expensive, especially in two dimensions. Therefore, we instead employ their cheap, yet efficient approximations, using only a few nodes in each element. Several numerical tests confirm the good performance of the corresponding numerical method.

REFERENCES

  1. Brezzi, F. and Douglas, J., Stabilized mixed methods for the Stokes problem. DOI: 10.1007/BF01395886

  2. Brezzi, F., Franca, L. P., and Russo, A., Further considerations on residual-free bubbles for advective–diffusive equations. DOI: 10.1016/S0045-7825(98)00080-2

  3. Brezzi, F., Hauke, G., Marini, L. D., and Sangalli, G., Link-cutting bubbles for the stabilization of convection–diffusion–reaction problems. DOI: 10.1142/S0218202503002581

  4. Brezzi, F., Hughes, T. J. D., Marini, L. D., Russo, A., and Süli, E., A priori error analysis of residual-free bubbles for advection-diffusion problems. DOI: 10.1007/s002110050476

  5. Brezzi, F., Marini, D., and Russo, A., Applications of the pseudo residual-free bubbles to the stabilization of convection–diffusion problems. DOI: 10.1016/S0045-7825(98)00082-6

  6. Brezzi, F., Marini, D., and Russo, A., On the choice of a stabilizing subgrid for convection-diffusion problems. DOI: 10.1016/j.camwa.2010.04.002

  7. Brezzi, F. and Russo, A., Choosing bubbles for advection-diffusion problems. DOI: 10.1142/S0218202594000327

  8. Brooks, A. N. and Hughes, T. J. R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. DOI: 10.1016/0045-7825(82)90071-8

  9. Franca, L. P. and Frey, S. L., Stabilized finite element methods: II. The incompressible Navier-Stokes equations. DOI: 10.1016/0045-7825(92)90041-H

  10. Franca, L. P., Frey, S. L., and Hughes, T. J. R., Stabilized finite element methods: I. Application to the advective-diffusion model. DOI: 10.1016/0045-7825(92)90143-8

  11. Franca, L. P., Neslitürk, A. I., and Stynes, M., On the stability of residual-free bubbles for convection–diffusion problems and their approximation by two-level finite element method. DOI: 10.1016/S0045-7825(98)00081-4

  12. Franca, L. P., Madureira, A. L., Tobiska, L., and Valentin, F., Convergence analysis of a multiscale finite element method for singularly perturbed problems. DOI: 10.1137/040608490

  13. Franca, L. P., Madureira, A. L., and Valentin, F., Towards multiscale functions: Enriching finite element spaces with local but not bubble-like functions. DOI: 10.1016/j.cma.2004.07.029

  14. Franca, L. P., Ramalho, J. V. A., and Valentin, F., Multiscale and residual-free bubble functions for reaction–advection–diffusion problems. DOI: 10.1615/IntJMultCompEng.v3.i3.40

  15. Franca, L. P. and Valentin, F., On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation. DOI: 10.1016/S0045-7825(00)00190-0

  16. Harari, I. and Hughes, T. J. R., Stabilized finite element methods for steady advection-diffusion with production. DOI: 10.1016/0045-7825(94)90193-7

  17. Franca, L. P. and Balestra, M., A new finite element formulation for computational fluid dynamics. 5. Circumventing the Babuska–Brezzi condition—A stable Petrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations. DOI: 10.1016/0045-7825(86)90025-3

  18. Neslitürk, A. I., A stabilizing subgrid for convection-diffusion problem. DOI: 10.1142/S0218202506001121

  19. Neslitürk, A. I., On the choice of stabilizing subgrid for convection-diffusion problem on rectangular grids. DOI: 10.1016/j.camwa.2010.04.002

  20. Sangalli, G., Global and local error analysis for the residual-free bubbles method applied to advection-dominated problems. DOI: 10.1137/S0036142999365382