Subgrid stabilization of Galerkin approximations of linear monotone operators

This paper presents a stabilized Galerkin technique for approximating monotone linear operators in a Hilbert space. The key idea consists in introducing an approximation space that is broken up into resolved scales and subgrid scales so that the bilinear form associated with the problem satisfies a uniform inf-sup condition with respect to this decomposition. An optimal Galerkin approximation is obtained by introducing an artificial diffusion on the subgrid scales.

[1]  Franco Brezzi,et al.  Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.) , 1993 .

[2]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[3]  K. W. Morton,et al.  Finite element methods for hyperbolic equations , 1987 .

[4]  Caskey,et al.  GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS I . THE BASIC EXPERIMENT , 1962 .

[5]  J. Guermond.  Stabilization of Galerkin approximations of transport equations by subgrid modelling , 1999 .

[6]  J. Smagorinsky,et al.  GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS , 1963 .

[7]  C. Bardos.  Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport , 1970 .

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

[9]  R. Temam,et al.  Modelling of the interaction of small and large eddies in two dimensional turbulent flows , 1988 .

[10]  M. Fortin,et al.  A stable finite element for the stokes equations , 1984 .

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

[12]  Jérôme Pousin,et al.  Inégalité de Poincaré courbe pour le traitement variationnel de l'équation de transport , 1996 .

[13]  Guohui Zhou,et al.  How accurate is the streamline diffusion finite element method? , 1997, Math. Comput..

[14]  R. Temam,et al.  Nonlinear Galerkin methods: The finite elements case , 1990 .

[15]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[16]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[17]  F. Brezzi,et al.  A relationship between stabilized finite element methods and the Galerkin method with bubble functions , 1992 .

[18]  P. Moin,et al.  A dynamic subgrid‐scale eddy viscosity model , 1990 .

[19]  Franco Brezzi,et al.  $b=\int g$ , 1997 .

[20]  Alessandro Russo,et al.  Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles , 1996 .

[21]  Eitan Tadmor,et al.  Legendre pseudospectral viscosity method for nonlinear conservation laws , 1993 .