×

zbMATH — the first resource for mathematics

A discontinuous enrichment method for variable-coefficient advection-diffusion at high Péclet number. (English) Zbl 1242.76125
Summary: A discontinuous Galerkin method with Lagrange multipliers is presented for the solution of variable-coefficient advection-diffusion problems at high Péclet number. In this method, the standard finite element polynomial approximation is enriched within each element with free-space solutions of a local, constant-coefficient, homogeneous counterpart of the governing partial differential equation. Hence in the two-dimensional case, the enrichment functions are exponentials, each exhibiting a sharp gradient in a carefully chosen flow direction. The continuity of the enriched approximation across the element interfaces is enforced weakly by the aforementioned Lagrange multipliers. Numerical results obtained for two benchmark problems demonstrate that elements based on the proposed discretization method are far more competitive for variable-coefficient advection-diffusion analysis in the high Péclet number regime than their standard Galerkin and stabilized finite element comparables.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76R50 Diffusion
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Brezzi, Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion problems, Computer Methods in Applied Mechanics and Engineering 166 pp 51– (1998) · Zbl 0932.65113
[2] Brooks, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 32 pp 199– (1982) · Zbl 0497.76041
[3] Harari, Streamline design of stability parameters for advection-diffusion problems, Journal of Computational Physics 171 pp 115– (2001) · Zbl 0985.65146
[4] Hughes, Finite Element Methods for Convection Dominated Flows 34 pp 19– (1979)
[5] Sheu, A monotone finite element method with test space of Legendre polynomials, Computer Methods in Applied Mechanics and Engineering 143 pp 349– (1997) · Zbl 0898.76067
[6] Corsini, A quadratic Petrov-Galerkin formulation for advection-diffusion-reaction problems in turbulence modelling, Journal of Computational and Applied Mechanics 5 pp 237– (2004) · Zbl 1150.76437
[7] Harari, Galerkin/least squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains, Computer Methods in Applied Mechanics and Engineering 98 pp 411– (1992) · Zbl 0762.76053
[8] Hughes, A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations, Computer Methods in Applied Mechanics and Engineering 73 (2) pp 173– (1989) · Zbl 0697.76100
[9] Franca, Stabilized finite element methods, I. Application to the advective-diffusive model, Journal of Computational and Applied Mechanics 95 pp 253– (1992) · Zbl 0759.76040
[10] Franca, Bubble functions prompt unusual stabilized finite element methods, Computer Methods in Applied Mechanics and Engineering 123 pp 299– (1995) · Zbl 1067.76567
[11] Delsaute, A conformal Petrov-Galerkin method for convection-dominated problems, International Journal for Numerical Methods in Fluids 56 pp 1077– (2008) · Zbl 1135.65397
[12] Delsaute B Dupret F A Petrov-Galerkin method for convection-dominated problems 11 15 · Zbl 1135.65397
[13] Farhat, The discontinuous enrichment method, Computer Methods in Applied Mechanics and Engineering 190 pp 6455– (2001) · Zbl 1002.76065
[14] Farhat, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime, Computer Methods in Applied Mechanics and Engineering 192 pp 1389– (2003) · Zbl 1027.76028
[15] Farhat, A higher-order discontinuous enrichment method for the solution of high Peclet advection-diffusion problems on unstructured meshes, International Journal for Numerical Methods in Engineering 81 pp 604– (2010) · Zbl 1183.76805
[16] Kalashnikova, A discontinuous enrichment method for the solution of advection-diffusion problems in high Peclet number regimes, Finite Elements in Analysis and Design 45 pp 238– (2009)
[17] Tezaur, Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems, International Journal for Numerical Methods in Engineering 66 pp 796– (2006) · Zbl 1110.76319
[18] Tezaur, A discontinuous enrichment method for capturing evanescent waves in multi-scale fluid and fluid/solid problems, Computer Methods in Applied Mechanics and Engineering 197 pp 1680– (2008) · Zbl 1194.74476
[19] Zhang, The discontinuous enrichment method for elastic wave propagation in the medium-frequency regime, International Journal for Numerical Methods in Engineering 66 pp 2086– (2006) · Zbl 1110.74860
[20] Massimi, A discontinuous enrichment method for three-dimensional multiscale harmonic wave propagation problems in multi-fluid and fluid-solid media, International Journal for Numerical Methods in Engineering 76 pp 400– (2008) · Zbl 1195.74292
[21] Brezzi, Choosing bubbles for advection-diffusion problems, Mathematical Models and Methods in Applied Sciences 4 (4) pp 571– (1994) · Zbl 0819.65128
[22] Franca, A two-level finite element method and its application to the Helmholtz equation, International Journal for Numerical Methods in Engineering 43 pp 23– (1998) · Zbl 0935.65117
[23] Babuška, A discontinuous hp finite element method for diffusion problems: 1-D analysis, Computers and Mathematics with Applications 37 pp 103– (1999) · Zbl 0940.65076
[24] Baumann, A discontinuous hp finite element method for the Euler and Navier-Stokes equations, International Journal for Numerical Methods in Fluids 31 pp 79– (1999) · Zbl 0985.76048
[25] El Alaoui, Nonconforming finite element methods with subgrid viscosity applied to advection-diffusion-reaction equations, Numerical Methods for Partial Differential Equations 22 (5) pp 1106– (2006) · Zbl 1104.65110
[26] Georgoulis, Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes, SIAM Journal on Scientific Computing 30 (1) pp 246– (2007) · Zbl 1159.65092
[27] El-Zein, Exponential finite elements for diffusion-advection problems, International Journal for Numerical Methods in Engineering 62 pp 2086– (2005) · Zbl 1118.76323
[28] Wang, A novel exponentially fitted triangular finite element method for an advection-diffusion problem with boundary layers, Journal of Computational Physics 134 pp 253– (1997) · Zbl 0897.76056
[29] Babuška, The partition of unity method, International Journal for Numerical Methods in Engineering 40 pp 727– (1997) · Zbl 0949.65117
[30] Melenk, The partition of unity method finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099
[31] Gracie, Concurrently coupled atomistic and XFEM models for dislocations and cracks, International Journal for Numerical Methods in Engineering 78 (3) pp 354– (2009) · Zbl 1183.74278
[32] Duarte, High-order generalized FEM for through-the-thickness branched cracks, International Journal for Numerical Methods in Engineering 72 pp 325– (2007) · Zbl 1194.74385
[33] Belytschko, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 (5) pp 601– (1999) · Zbl 0943.74061
[34] Farhat, Higher-order extensions for a discontinuous Galerkin methods for mid-frequency Helmholtz problems, International Journal for Numerical Methods in Engineering 61 pp 1938– (2004) · Zbl 1075.76572
[35] Oliveira SP Discontinuous enrichment methods for computational fluid dynamics 2002
[36] Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers, Revie Francaise d’Automatique Informatique Recherche Operationnelle 8-R2 pp 129– (1978)
[37] Babuška, The finite element method with Lagrange multipliers, Numerical Mathematics 20 pp 179– (1973) · Zbl 0258.65108
[38] Brezzi, Mixed and Hybrid Finite Element Methods (1991) · Zbl 0788.73002
[39] Franca, Refining the submesh strategy in the two-level finite element method: application to the advection-diffusion equation, International Journal for Numerical Methods in Engineering 39 pp 161– (2002) · Zbl 1016.76047
[40] Franca LP Oliveira SP Resolving boundary layers with the discontinuous enrichment method 2003
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.