A discontinuous enrichment method for variable‐coefficient advection–diffusion at high Péclet number

A discontinuous Galerkin method with Lagrange multipliers is presented for the solution of variable-coefficient advection–diffusion problems at high Peclet 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 Peclet number regime than their standard Galerkin and stabilized finite element comparables. Copyright © 2010 John Wiley & Sons, Ltd.

[1]  C. Farhat,et al.  The Discontinuous Enrichment Method , 2000 .

[2]  Charbel Farhat,et al.  A higher‐order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes , 2010 .

[3]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[4]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[5]  A. Corsini,et al.  A QUADRATIC PETROV-GALERKIN FORMULATION FOR ADVECTION-DIFFUSION-REACTION PROBLEMS IN TURBULENCE MODELLING , 2004 .

[6]  A. El-Zein.  Exponential finite elements for diffusion–advection problems , 2005 .

[7]  Linda El Alaoui,et al.  Nonconforming finite element methods with subgrid viscosity applied to advection‐diffusion‐reaction equations , 2006 .

[8]  A monotone finite element method with test space of Legendre polynomials , 1997 .

[9]  Charbel Farhat,et al.  Higher‐order extensions of a discontinuous Galerkin method for mid‐frequency Helmholtz problems , 2004 .

[10]  C. Farhat,et al.  Bubble Functions Prompt Unusual Stabilized Finite Element Methods , 1994 .

[11]  Song Wang,et al.  A Novel Exponentially Fitted Triangular Finite Element Method for an Advection-Diffusion Problem with Boundary Layers , 1997 .

[12]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[13]  J. Tinsley Oden,et al.  A discontinuous hp finite element method for the Euler and Navier–Stokes equations , 1999 .

[14]  Charbel Farhat,et al.  A discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid/solid problems , 2008 .

[15]  Thomas J. R. Hughes,et al.  Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains , 1992 .

[16]  L. Franca,et al.  Refining the submesh strategy in the two‐level finite element method: application to the advection–diffusion equation , 2002 .

[17]  François Dupret,et al.  A conformal Petrov–Galerkin method for convection‐dominated problems , 2008 .

[18]  C. Farhat,et al.  The discontinuous enrichment method for elastic wave propagation in the medium‐frequency regime , 2006 .

[19]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[20]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[21]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[22]  S. P. Oliveira,et al.  Discontinuous enrichment methods for computational fluid dynamics , 2003 .

[23]  Carlos Armando Duarte,et al.  A high‐order generalized FEM for through‐the‐thickness branched cracks , 2007 .

[24]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[25]  Charbel Farhat,et al.  A discontinuous enrichment method for the finite element solution of high Péclet advection-diffusion problems , 2009 .

[26]  S. P. Oliveira,et al.  Streamline design of stability parameters for advection—diffusion problems , 2001 .

[27]  T. Hughes,et al.  Stabilized finite element methods. I: Application to the advective-diffusive model , 1992 .

[28]  Paul Houston,et al.  Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes , 2007, SIAM J. Sci. Comput..

[29]  Ted Belytschko,et al.  Concurrently coupled atomistic and XFEM models for dislocations and cracks , 2009 .

[30]  Alessandro Russo,et al.  Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion problems , 1998 .

[31]  F. Dupret,et al.  A Petrov-Galerkin method for convection-dominated problems , 2022 .

[32]  Charbel Farhat,et al.  A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime , 2003 .

[33]  Charbel Farhat,et al.  Three‐dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid‐frequency Helmholtz problems , 2006 .

[34]  C. Farhat,et al.  A discontinuous enrichment method for three‐dimensional multiscale harmonic wave propagation problems in multi‐fluid and fluid–solid media , 2008 .

[35]  Antonini Macedo,et al.  Two Level Finite Element Method and its Application to the Helmholtz Equation , 1997 .

[36]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[37]  T. Hughes,et al.  MULTI-DIMENSIONAL UPWIND SCHEME WITH NO CROSSWIND DIFFUSION. , 1979 .

[38]  Alessandro Russo,et al.  CHOOSING BUBBLES FOR ADVECTION-DIFFUSION PROBLEMS , 1994 .