×

zbMATH — the first resource for mathematics

A discontinuous Galerkin method with Lagrange multipliers for spatially-dependent advection-diffusion problems. (English) Zbl 1439.76052
Summary: A higher-accuracy discontinuous Galerkin method with Lagrange multipliers (DGLM) is presented for the solution of the advection-diffusion equation with a spatially varying advection field in the high Péclet number regime, where the classical polynomial finite element method (FEM) produces spurious oscillations in the solution at practical mesh resolutions. The proposed DGLM method is based on discontinuous polynomial shape functions that are attached to an element rather than its nodes. It overcomes the aforementioned spurious oscillation issue by enriching these functions with approximate free-space solutions of homogeneous equations derived from an asymptotic analysis of the governing partial differential equation inspired by Prandtl’s boundary layer theory. These enrichment functions are capable of resolving exponential, parabolic, and corner boundary layers at relatively coarse mesh resolutions. The proposed method enforces a weak continuity of the solution approximation across inter-element boundaries using polynomial Lagrange multipliers, which makes it a hybrid method. However, unlike other hybrid methods, it operates directly on the second-order form of the advection-diffusion equation and does not require any stabilization. Its intrinsic performance and its superiority over the higher-order polynomial FEM are demonstrated for several test problems at Péclet numbers ranging from one thousand to one billion.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Brooks, A. N.; Hughes, T. J., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 13, 199-259 (1982) · Zbl 0497.76041
[2] Hughes, T. J.; 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, 2, 173-189 (1989) · Zbl 0697.76100
[3] Franca, L. P.; Farhat, C., Bubble functions prompt unusual stabilized finite element methods, Comput. Methods Appl. Mech. Engrg., 123, 1, 299-308 (1995) · Zbl 1067.76567
[4] Brezzi, F.; Russo, A., Choosing bubbles for advection-diffusion problems, Math. Models Methods Appl. Sci., 04, 04, 571-587 (1994) · Zbl 0819.65128
[5] Franca, L. P.; Farhat, C.; Macedo, A. P.; Lesoinne, M., Residual-free bubbles for the Helmholtz equation, Internat. J. Numer. Methods Engrg., 40, 21, 4003-4009 (1997) · Zbl 0897.73062
[6] Hughes, T. J., Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127, 1, 387-401 (1995) · Zbl 0866.76044
[7] Koobus, B.; Farhat, C., A variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshesapplication to vortex shedding, Comput. Methods Appl. Mech. Engrg., 193, 1516, 1367-1383 (2004), Recent Advances in Stabilized and Multiscale Finite Element Methods · Zbl 1079.76567
[8] Babuška, I.; Melenk, J. M., The partition of unity method, Internat. J. Numer. Methods Engrg., 40, 4, 727-758 (1997) · Zbl 0949.65117
[9] Farhat, C.; Harari, I.; Franca, L. P., The discontinuous enrichment method, Comput. Methods Appl. Mech. Engrg., 190, 48, 6455-6479 (2001) · Zbl 1002.76065
[10] Farhat, C.; Kalashnikova, I.; Tezaur, R., A higher-order discontinuous enrichment method for the solution of high pclet advectiondiffusion problems on unstructured meshes, Internat. J. Numer. Methods Engrg., 81, 5, 604-636 (2010) · Zbl 1183.76805
[11] Kalashnikova, I.; Tezaur, R.; Farhat, C., A discontinuous enrichment method for variable-coefficient advection-diffusion at high Péclet number, Internat. J. Numer. Methods Engrg., 87, 1-5, 309-335 (2011) · Zbl 1242.76125
[12] Brogniez, S.; Farhat, C.; Hachem, E., A high-order discontinuous Galerkin method with Lagrange multipliers for advection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 264, 49-66 (2013) · Zbl 1286.65150
[13] Farhat, C.; Harari, I.; Hetmaniuk, U., A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime, Comput. Methods Appl. Mech. Engrg., 192, 1112, 1389-1419 (2003) · Zbl 1027.76028
[14] Farhat, C.; Harari, I.; Hetmaniuk, U., The discontinuous enrichment method for multiscale analysis, Comput. Methods Appl. Mech. Engrg., 192, 2830, 3195-3209 (2003), Multiscale Computational Mechanics for Materials and Structures · Zbl 1054.76048
[15] Tezaur, R.; Farhat, C., Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems, Internat. J. Numer. Methods Engrg., 66, 5, 796-815 (2006) · Zbl 1110.76319
[16] Nguyen, N. C.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for Linear Convection-diffusion Equations, J. Comput. Phys., 228, 9, 3232-3254 (2009) · Zbl 1187.65110
[17] Di Pietro, D. A.; Droniou, J.; Ern, A., A discontinuous-skeletal method for advection-diffusion-reaction on General Meshes, SIAM J. Numer. Anal., 53, 5, 2135-2157 (2015) · Zbl 06481904
[18] Demkowicz, L.; Gopalakrishnan, J., A class of discontinuous PetrovGalerkin methods. Part I: The transport equation, Comput. Methods Appl. Mech. Engrg., 199, 2324, 1558-1572 (2010) · Zbl 1231.76142
[20] Tezaur, R.; Kalashnikova, I.; Farhat, C., The discontinuous enrichment method for medium-frequency Helmholtz problems with a spatially variable wavenumber, Comput. Methods Appl. Mech. Engrg., 268, 126-140 (2014) · Zbl 1295.76030
[21] Bar-Yoseph, P.; Israeli, M., An asymptotic finite element method for improvement of solutions of boundary layer problems, Numer. Math., 49, 4, 425-438 (1986) · Zbl 0616.65105
[22] Borker, R.; Farhat, C.; Tezaur, R., A high-order discontinuous Galerkin method for unsteady advection-diffusion problems, J. Comput. Phys., 332, 520-537 (2017) · Zbl 1380.65250
[23] Hunter, J. K., Asymptotic analysis and singular perturbation theory (2004), Department of Mathematics, University of California at Davis
[24] Kevorkian, J.; Cole, J. D., Multiple scale and singular perturbation methods, Vol. 114 (2012), Springer Science & Business Media · Zbl 0846.34001
[25] Raviart, P. A.; Thomas, J. M., Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp., 31, 138, 391-413 (1977) · Zbl 0364.65082
[27] Franca, L. P.; Frey, S. L.; Hughes, T. J., Stabilized finite element methods: I. Application to the advective-diffusive model, Comput. Methods Appl. Mech. Engrg., 95, 2, 253-276 (1992) · Zbl 0759.76040
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.