A high-resolution Petrov-Galerkin method for the 1D convection-diffusion-reaction problem

Computer Methods in Applied Mechanics and Engineering

Volumn 199, Issue 9-12, 2010, Pages 525-546

  Nadukandi, Prashanth ^a   Oñate, Eugenio ^a   Garcia, Julio ^a  

^a INTERNATIONAL CENTER FOR NUMERICAL METHODS IN ENGINEERING CIMNE   (Spain)

Author keywords

Convection diffusion reaction; Finite element; Petrov Galerkin; Stabilized high resolution methods

Indexed keywords

ADVECTION - DIFFUSION; ADVECTIVE-DIFFUSIVE TRANSPORT; CHARACTERISTIC LENGTH; CONVECTION-DIFFUSION-REACTION; CONVECTION-DIFFUSION-REACTION PROBLEMS; CONVECTION-DOMINATED; DESIGN PROCEDURE; FINITE ELEMENT; FLUID FLOW PROBLEMS; GIBBS PHENOMENA; HIGH RESOLUTION; HIGH-RESOLUTION METHODS; LINEAR FINITE ELEMENTS; NUMERICAL SOLUTION; PETROV-GALERKIN; PETROV-GALERKIN METHODS; SECOND-ORDER ACCURACY; SHOCK-CAPTURING; STABILIZATION PARAMETERS; STABILIZED FORMULATION;

CALCULATIONS; DIFFUSION IN LIQUIDS; FINITE ELEMENT METHOD; FLOW OF FLUIDS; HEAT CONVECTION; STABILIZATION;

GALERKIN METHODS;

EID: 73449148688     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2009.10.009     Document Type: Article

References (57)

1. Hughes T.J.R., and Brooks A.N. A theoretical framework for Petrov-Galerkin, methods with discontinuous weighting functions: application to the streamline upwind procedure. In: Gallagher R.H., Norrie D.M., Oden J.T., and Zienkiewicz O.C. (Eds). Finite Elements in Fluids vol. IV (1982), Wiley, Chichester
2. Brooks A.N., and Hughes T.J.R. Streamline Upwind/Petrov-Galerkin formulations for the convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 199-259
3. Douglas J., and Russell T.F. Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with the finite element or finite difference procedures. SIAM J. Numer. Anal. 19 (1982) 871-885
4. Donea J. A Taylor-Galerkin method for convective transport problems. Int. J. Numer. Methods Engrg. 20 (1984) 101-119
5. Löhner R., Morgan K., and Zienkiewicz O.C. The solution of non-linear hyperbolic equation systems by the finite element method. Int. J. Numer. Methods Fluids 4 (1984) 1043-1063
6. Hughes T.J.R., Franca L.P., and Hulbert G.M. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Engrg. 73 (1989) 173-189
7. Brezzi F., Bristeau M.O., Franca L.P., Mallet M., and Rogé G. A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Methods Appl. Mech. Engrg. 96 (1992) 117-129
8. Baiocchi C., Brezzi F., and Franca L.P. Virtual bubbles and the GaLS. Comput. Methods Appl. Mech. Engrg. 105 (1993) 25-141
9. Brezzi F., and Russo A. Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4 (1994) 571-587
10. Hughes T.J.R. Multiscale phenomena: Green's function, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized formulations. Comput. Methods Appl. Mech. Engrg. 127 (1995) 387-401
11. Hughes T.J.R., Feijoo G.R., Mazzei L., and Quincy J.B. The variational multiscale method: a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24
12. Zienkiewicz O.C., and Codina R. A general algorithm for compressible and incompressible flows. Part I: the split, characteristic based scheme. Int. J. Numer. Methods Fluids 20 (1995) 869-885
13. Oñate E. Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems. Comput. Methods Appl. Mech. Engrg. 151 (1998) 233-265
14. Oñate E., and Manzan M. Stabilization techniques for finite element analysis of convection-diffusion problems. In: Sundén B., and Comini G. (Eds). Computational Analysis of Convection Heat Transfer (2000), WIT Press, Southampton, UK 71-117
15. Brezzi F., Franca L.P., Hughes T.J.R., and Russo A. b = ∫ g. Comput. Methods Appl. Mech. Engrg. 142 (1997) 353-360
16. Codina R. Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comput. Methods Appl. Mech. Engrg. 156 (1998) 185-210
17. Mizukami A., and Hughes T.J.R. A Petrov-Galerkin finite element method for convection-dominated flows: an accurate upwinding technique for satisfying the maximum principle. Comput. Methods Appl. Mech. Engrg. 50 (1985) 181-193
18. Hughes T.J.R., Mallet M., and Mizukami A. A new finite element formulation for computational fluid dynamics: II. Beyond SUPG. Comput. Methods Appl. Mech. Engrg. 54 (1986) 341-355
19. Galeão A.C., and Dutra do Carmo E.G. A consistent approximate upwind Petrov-Galerkin method for the convection-dominated problems. Comput. Methods Appl. Mech. Engrg. 68 (1988) 83-95
20. Johnson C., Szepessy A., and Hansbo P. On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws. Math. Comput. 54 (1990) 107-129
21. Dutra do Carmo E.G., and Galeão A.C. Feedback Petrov-Galerkin methods for convection-dominated problems. Comput. Methods Appl. Mech. Engrg. 88 (1991) 1-16
22. Codina R. A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation. Comput. Methods Appl. Mech. Engrg. 110 (1993) 325-342
23. de Sampaio P.A.B., and Coutinho A.L.G.A. A natural derivation of discontinuity capturing operator for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6291-6308
24. Burman E., and Ern A. Nonlinear diffusion and discrete maximum principle for stabilized Galerkin approximations of the convection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg. 191 (2002) 3833-3855
25. Dutra do Carmo E.G., and Alvarez G.B. A new stabilized finite element formulation for the scalar convection-diffusion problems: the streamline and approximate upwind/Petrov-Galerkin method. Comput. Methods Appl. Mech. Engrg. 192 (2003) 3379-3396
26. Dutra do Carmo E.G., and Alvarez G.B. A new upwind function in stabilized finite element formulations, using linear and quadratic elements for the scalar convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 2383-2402
27. Knobloch P. Improvements of the Mizukami-Hughes method for convection-diffusion equations. Comput. Methods Appl. Mech. Engrg. 196 (2006) 579-594
28. Oñate E., Zarate F., and Idelsohn S.R. Finite element formulation for the convective-diffusive problems with sharp gradients using finite calculus. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1793-1825
29. John V., and Knobloch P. On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part I - a review. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2197-2215
30. Tezduyar T.E., and Park Y.J. Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations. Comput. Methods Appl. Mech. Engrg. 59 (1986) 307-325
31. Idelsohn S.R., Nigro N., Storti M., and Buscaglia G. A Petrov-Galerkin formulation for advective-reaction-diffusion problems. Comput. Methods Appl. Mech. Engrg. 136 (1996) 27-46
32. Franca L.P., and Dutra do Carmo E.G. The Galerkin gradient least-squares method. Comput. Methods Appl. Mech. Engrg. 74 (1989) 41-54
33. Harari I., and Hughes T.J.R. Stabilized finite element methods for steady advection-diffusion with production. Comput. Methods Appl. Mech. Engrg. 115 (1994) 165-191
34. Franca L.P., and Farhat C. Bubble functions prompt unusual stabilized finite element methods. Comput. Methods Appl. Mech. Engrg. 123 (1995) 299-308
35. Franca L.P., and Valentin F. On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation. Comput. Methods Appl. Mech. Engrg. 190 (2001) 1785-1800
36. Brezzi F., Hauke G., Marini L.D., and Sangalli G. Link-cutting bubbles for the stabilization of convection-diffusion-reaction problems. Math. Models Methods Appl. Sci. 13 3 (2003) 45-461
37. Codina R. On stabilized finite element methods for linear systems of convection-diffusion-reaction equations. Comput. Methods Appl. Mech. Engrg. 190 (2000) 2681-2706
38. Hauke G., and García-Olivares A. Variational subgrid scale formulations for the advection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6847-6865
39. Hauke G. A simple subgrid scale stabilized method for the advection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg. 191 (2002) 2925-2947
40. Hauke G., Sangalli G., and Doweidar M.H. Combining adjoint stabilized methods for the advection-diffusion-reaction problem. Math. Models Methods Appl. Sci. 17 2 (2007) 305-326
41. Oñate E., Miquel J., and Hauke G. Stabilized formulation for the advection-diffusion-absorption equation using finite calculus and linear finite elements. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3926-3946
42. Oñate E., Miquel J., and Zarate F. Stabilized solution of the multidimensional advection-diffusion-absorption equation using linear finite elements. Comput. Fluids 36 (2007) 92-112
43. Felippa C., and Oñate E. Nodally exact Ritz discretizations of the 1D diffusion-absorption and Helmholtz equations by variational FIC and modified equation methods. Comput. Mech. 39 2 (2007) 91-111
44. Idelsohn S.R., Heinrich J.C., and Oñate E. Petrov-Galerkin methods for the transient advective-diffusive equation with sharp gradients. Int. J. Numer. Methods Engrg. 39 (1996) 1455-1473
45. Yu C.C., and Heinrich J.C. Petrov-Galerkin methods for the time-dependent convective transport equations. Int. J. Numer. Methods Engrg. 23 (1986) 883-901
46. Hirsch C. Numerical Computation of Internal and External Flows Vols. 1 and 2 (1990), Wiley
47. LeVeque R.J. Numerical Methods for Conservation Laws (1990), Birkhauser-Verlag, Basel
48. Laney C.B. Computational Gas Dynamics (1998), Cambridge University Press
49. Toro E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics (1999), Springer-Verlag
50. Boris J.P., and Book D.L. Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11 (1973) 38-69
51. Zalesak S.T. Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31 (1979) 335-362
52. van Leer B. Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101-136
53. Harten A. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357-393
54. Harten A., Engquist B., Osher S., and Chakravarthy S. Uniformly higher order essentially non-oscillatory schemes, III. J. Comput. Phys. 71 (1987) 231-303
55. Kuzmin D., Löhner R., and Turek S. Flux-corrected Transport: Principles, Algorithms and Applications (2005), Springer
56. J.T. Oden, Historical comments on finite elements, in: Proceedings of the ACM Conference on History of Scientific and Numeric Computation Princeton, New Jersey, USA, 1987, pp. 125-130.
57. Zienkiewicz O.C., Taylor R.L., and Nithiarasu P. The Finite Element Method for Fluid Dynamics (2005), Elsevier, Butterworth-Heinemann