×

zbMATH — the first resource for mathematics

Stabilized finite element methods for singularly perturbed parabolic problems. (English) Zbl 0838.65095
The authors consider two stabilized numerical schemes for solving Dirichlet-type initial-boundary value problems of the parabolic equation \((\partial_t + L_\varepsilon) u : = \partial_t u - \varepsilon \Delta u + b \cdot \nabla u + cu = f\), modelling convection-reaction-diffusion. Weighted least squares forms of the underlying equation are added to the basic Galerkin finite element semidiscretization in order to accomodate the method to (possibly locally varying) convective, reactive or diffusive terms. Time integration is performed with standard stable one-step methods. Stability and error estimates are derived on unstructured grids and the design of the numerical damping parameters are considered. Numerical results are given for typical model problems, including “translating cone”.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Behr, M., Stabilized finite element methods for incompressible flows with emphasis on moving boundaries and interfaces, ()
[2] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[3] Codina, R., A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation, Comput. methods appl. mech. engrg., 110, 325-342, (1993) · Zbl 0844.76048
[4] Eriksson, K.; Johnson, C., Adaptive streamline diffusion finite element method for convection-diffusion problems, () · Zbl 0795.65074
[5] Franca, L.P.; Frey, S.L., Stabilized finite element methods, II: the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 99, 209-233, (1992) · Zbl 0765.76048
[6] Franca, L.P.; Frey, S.L.; Hughes, T.J.R., Stabilized finite element methods, I: application to the advective-diffusive model, Comput. methods appl. mech. engrg., 95, 253-276, (1992) · Zbl 0759.76040
[7] Gajewski, H.; Gröger, K.; Zacharias, K., Nichtlineare operatorgleichungen und operatordifferentialgleichungen, (1974), Akademie-Verlag Berlin · Zbl 0289.47029
[8] Goering, H.; Felgenhauer, A.; Lube, G.; Roos, H.G.; Tobiska, L., Singularly perturbed differential equations, (1983), Akademie-Verlag Berlin · Zbl 0522.35003
[9] Heywood, J.G.; Rannacher, R., Finite-element approximation of the nonstationary Navier-Stokes problem, part IV:error analysis for second-order time discretization, SIAM J. numer. anal., 27, 2, 353-384, (1990) · Zbl 0694.76014
[10] Hughes, T.J.R.; Franca, L.P.; 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, 173-189, (1989) · Zbl 0697.76100
[11] Johnson, C., Numerical solution of partial differential equations by the finite element method, (1987), Cambridge University Press Cambridge
[12] Johnson, C., Adaptive finite element methods for diffusive and convective problems, Comput. methods appl. mech. engrg., 82, 301-322, (1990) · Zbl 0717.76078
[13] Johnson, C., A new approach to algorithms for convection problems which are based on exact transport + projection, (1990), Chalmers University Göteborg Sweden, Preprint 1990-24
[14] Johnson, C.; Nävert, U.; Pitkäranta, J., J. finite element methods for linear hyperbolic problems, Comput. methods appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[15] Khalifa, A.; Robert, J.L.; Quelet, Y., A Douglas-Wang finite element approach for transient advection-diffusion problems, Comput. methods appl. mech. engrg., 110, 113-129, (1993) · Zbl 0845.76044
[16] Ladyzshenskaya, O.A.; Uralceva, N.N.; Solonnikov, V.A., Lineare und quasilineare gleichungen parabolischen typs, (1973), Nauka Moskow, (in Russian)
[17] Lube, G.; Weiss, D., Numerische simulation des stationären Wärme- und stofftransports mittels stromliniendiffusion-FEM, Techn. mech., 11, 4, 229-237, (1990)
[18] Luskin, M.; Rannacher, R., On the smoothing property of the Galerkin method for parabolic equations, SIAM J. numer. anal., 19, 1, 93-113, (1982) · Zbl 0483.65064
[19] Mittal, S.; Tezduyar, T.E., Notes on the stabilized space-time finite-element formulation of unsteady incompressible flows, Comput. phys. comm., 73, 93-112, (1992)
[20] Nävert, U., A finite element method for convection-diffusion problems, ()
[21] Pironneau, O., Finite element methods for fluids, (1989), Masson Paris · Zbl 0665.73059
[22] Pironneau, O.; Liou, J.; Tezduyar, T., Characteristic-Galerkin and Galerkin/least-squares formulations for the advection-diffusion equation with time-dependent domains, Comput. methods appl. mech. engrg., 100, 117-141, (1992) · Zbl 0761.76073
[23] Shakib, F., Finite element analysis of the compressible Euler and Navier-Stokes equations, ()
[24] Simo, J.C.; Armero, F., Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations, Comput. methods appl. mech. engrg., 111, 111-154, (1994) · Zbl 0846.76075
[25] Tezduyar, T.E.; Behr, M.; Liou, J.; Tezduyar, T.E.; Behr, M.; Liou, J., A new strategy for finite element computations involving moving boundaries and interfaces—the deforming spatial-domain/space-time procedure, Comput. methods appl. mech. engrg., Comput. methods appl. mech. engrg., 94, 353-371, (1992), Part II · Zbl 0745.76045
[26] Tezduyar, T.E.; Liou, J., Element-by-element and implicit-explicit finite element formulations for CFD, (), 281-300 · Zbl 0652.76021
[27] Tezduyar, T.E.; Mittal, S.; Ray, S.R.; Shin, R., Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Comput. methods appl. mech. engrg., 95, 221-242, (1992) · Zbl 0756.76048
[28] Tezduyar, T.E.; Park, Y.J.; Deans, H.A., Finite element procedures for time-dependent convection-diffusion-reaction systems, Internat. J. numer. methods fluids, 7, 1013-1033, (1987) · Zbl 0634.76088
[29] Walter, A.R., Ein finite-elemente-verfahren zur numerischen Lösung von erhaltungsgleichungen, () · Zbl 0692.65058
[30] Wloka, J., Partielle differentialgleichungen, (1982), Teubner Verlagsgesellschaft Leipzig · Zbl 0482.35001
[31] Zeidler, E., Nonlinear functional analysis, (1989), Springer-Verlag Berlin, IIa
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.