A higher-order discontinuous enrichment method for the solution of high Péclet advection-diffusion problems on unstructured meshes.

*(English)*Zbl 1183.76805Summary: A higher-order discontinuous enrichment method (DEM) with Lagrange multipliers is proposed for the efficient finite element solution on unstructured meshes of the advection-diffusion equation in the high Péclet number regime. Following the basic DEM methodology, the usual Galerkin polynomial approximation is enriched with free-space solutions of the governing homogeneous partial differential equation (PDE). In this case, these are exponential functions that exhibit a steep gradient in a specific flow direction. Exponential Lagrange multipliers are introduced at the element interfaces to weakly enforce the continuity of the solution. The construction of several higher-order DEM elements fitting this paradigm is discussed in detail. Numerical tests performed for several two-dimensional benchmark problems demonstrate their computational superiority over stabilized Galerkin counterparts, especially for high Péclet numbers.

##### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

74R99 | Fracture and damage |

80A20 | Heat and mass transfer, heat flow (MSC2010) |

##### Keywords:

advection-diffusion; discontinuous Galerkin method; discontinuous enrichment method; high Péclet number; Lagrange multipliers; high-order
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\textit{C. Farhat} et al., Int. J. Numer. Methods Eng. 81, No. 5, 604--636 (2010; Zbl 1183.76805)

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##### References:

[1] | Farhat, The discontinuous enrichment method, Computer Methods in Applied Mechanics and Engineering 190 pp 6455– (2001) · Zbl 1002.76065 |

[2] | 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) |

[3] | 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 |

[4] | 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 |

[5] | 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 |

[6] | 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 |

[7] | 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) |

[8] | Oliveira SP. Discontinuous enrichment methods for computational fluid dynamics. Ph.D. Thesis, University of Colorado, Denver, CO, 2002. |

[9] | 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 |

[10] | Hughes, Finite Element Methods for Convection Dominated Flows 34 pp 19– (1979) |

[11] | 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 |

[12] | 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 |

[13] | Araya, An adaptive stabilized finite element scheme for the advection-reaction-diffusion equation, Applied Numerical Mathematics 54 pp 491– (2005) · Zbl 1078.65101 |

[14] | 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 |

[15] | 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 |

[16] | Franca, Stabilized finite element methods, I. Application to the advective-diffusive model, Computer Methods in Applied Mechanics and Engineering 95 pp 253– (1992) · Zbl 0759.76040 |

[17] | Franca, Bubble functions prompt unusual stabilized finite element methods, Computer Methods in Applied Mechanics and Engineering 123 pp 299– (1995) · Zbl 1067.76567 |

[18] | Babuška, The partition of unity method, International Journal for Numerical Methods in Engineering 40 pp 727– (1997) |

[19] | 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 |

[20] | 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 |

[21] | 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 |

[22] | 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 |

[23] | Brezzi, Choosing bubbles for advection-diffusion problems, Mathematical Models and Methods in Applied Sciences 4 (4) pp 571– (1994) · Zbl 0819.65128 |

[24] | 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 |

[25] | 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 |

[26] | Baumann, A discontinuous hp finite element method for the Euler and Navier-Stokes equations, Tenth International Conference on Finite Elements in Fluids (Tucson, AZ, 1998), International Journal for Numerical Methods in Fluids 31 pp 79– (1999) |

[27] | Babuška, A discontinuous hp finite element method for diffusion problems: 1-D analysis, Computers and Mathematics with Applications 37 pp 103– (1999) |

[28] | 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 |

[29] | 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 |

[30] | 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 |

[31] | El-Zein, Exponential finite elements for diffusion-advection problems, International Journal for Numerical Methods in Engineering 62 pp 2086– (2005) · Zbl 1118.76323 |

[32] | 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 |

[33] | Babuška, The finite element method with Lagrange multipliers, Numerical Mathematics 20 pp 179– (1973) |

[34] | 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) |

[35] | Brezzi, Mixed and Hybrid Finite Element Methods (1991) · Zbl 0788.73002 · doi:10.1007/978-1-4612-3172-1 |

[36] | Harari, Streamline design of stability parameters for advection-diffusion problems, Journal of Computational Physics 171 pp 115– (2001) · Zbl 0985.65146 |

[37] | 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 |

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