A dual Petrov-Galerkin finite element method for the convection-diffusion equation

We present a minimum-residual finite element method (based on a dual PetrovGalerkin formulation) for convectiondiffusion problems in a higher order, adaptive, continuous Galerkin setting. The method borrows concepts from both the Discontinuous PetrovGalerkin (DPG) method by Demkowicz and Gopalakrishnan (2011) and the method of variational stabilization by Cohen, Dahmen, and Welper (2012), and it can also be interpreted as a variational multiscale method in which the fine-scales are defined through a dual-orthogonality condition. A key ingredient in the method is the proper choice of dual norm used to measure the residual, and we present two choices which are observed to be robust in both convection and diffusion-dominated regimes, as well as a proof of stability for quasi-uniform meshes and a method for the weak imposition of boundary conditions. Numerically obtained convergence rates in 2D are reported, and benchmark numerical examples are given to illustrate the behavior of the method.

[1]  Weifeng Qiu,et al.  An analysis of the practical DPG method , 2011, Math. Comput..

[2]  A. H. Schatz,et al.  Crosswind Smear and Pointwise Errors in Streamline Diffusion Finite Element Methods , 1987 .

[3]  Leszek Demkowicz,et al.  A Class of Discontinuous Petrov–Galerkin Methods. Part I: The Transport Equation , 2010 .

[4]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[5]  Martin Stynes,et al.  Convection-diffusion-reaction problems, SDFEM/SUPG and a priori meshes , 2007, Int. J. Comput. Sci. Math..

[6]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[7]  J. A. Cottrell.  Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method , 2007 .

[8]  Wolfgang Dahmen,et al.  Adaptive Petrov-Galerkin Methods for First Order Transport Equations , 2011, SIAM J. Numer. Anal..

[9]  Victor M. Calo,et al.  A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D , 2011, J. Comput. Phys..

[10]  Leszek Demkowicz,et al.  A class of discontinuous Petrov-Galerkin methods. Part III , 2012 .

[11]  L. Demkowicz.  One and two dimensional elliptic and Maxwell problems , 2006 .

[12]  Nathan V. Roberts,et al.  The DPG method for the Stokes problem , 2014, Comput. Math. Appl..

[13]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[14]  Leszek Demkowicz,et al.  A class of discontinuous Petrov–Galerkin methods. II. Optimal test functions , 2011 .

[15]  Wolfgang Dahmen,et al.  Adaptivity and variational stabilization for convection-diffusion equations∗ , 2012 .

[16]  J. Hesthaven,et al.  On the constants in hp-finite element trace inverse inequalities , 2003 .

[17]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[18]  Leszek F. Demkowicz,et al.  Analysis of the DPG Method for the Poisson Equation , 2011, SIAM J. Numer. Anal..

[19]  Rob Stevenson,et al.  A Petrov-Galerkin discretization with optimal test space of a mild-weak formulation of convection-diffusion equations in mixed form , 2015 .

[20]  Leszek F. Demkowicz,et al.  A primal DPG method without a first-order reformulation , 2013, Comput. Math. Appl..

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

[22]  Norbert Heuer,et al.  A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms , 2014, Comput. Math. Appl..

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

[24]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[25]  Giancarlo Sangalli,et al.  Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods , 2007, SIAM J. Numer. Anal..

[26]  Garrett. Barter,et al.  Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method , 2008 .

[27]  Kenneth Eriksson,et al.  Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems , 1993 .

[28]  Norbert Heuer,et al.  Robust DPG Method for Convection-Dominated Diffusion Problems , 2013, SIAM J. Numer. Anal..

[29]  M. Stynes,et al.  Numerical methods for singularly perturbed differential equations : convection-diffusion and flow problems , 1996 .

[30]  P. Raviart,et al.  Primal hybrid finite element methods for 2nd order elliptic equations , 1977 .

[31]  Thomas J. R. Hughes,et al.  Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements , 2013, Numerische Mathematik.

[32]  Leszek Demkowicz,et al.  Numerical simulations of cloaking problems using a DPG method , 2013 .

[33]  Barry Koren,et al.  Least‐squares spectral element method applied to the Euler equations , 2008 .

[34]  Rob Stevenson,et al.  A robust Petrov-Galerkin discretisation of convection-diffusion equations , 2014, Comput. Math. Appl..

[35]  Christoph Schwab,et al.  The p and hp versions of the finite element method for problems with boundary layers , 1996, Math. Comput..

[36]  Weifeng Qiu,et al.  First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems , 2013, Comput. Math. Appl..