Conditions for superconvergence of HDG methods for Stokes flow

We provide an a priori error analysis of a wide class of finite element methods for the Stokes equations. The methods are based on the velocity gradient-velocity-pressure formulation of the equations and include new and old mixed and hybridizable discontinuous Galerkin methods. We show how to reduce the error analysis to the verification of some properties of an elementwise-defined projection and of the local spaces defining the methods. We also show that the projection of the errors only depends on the approximation properties of the projection. We then provide sufficient conditions for the superconvergence of the projection of the error in the approximate velocity. We give many examples of these methods and show how to systematically construct them from similar methods for the diffusion equation.

[1]  M. Fortin,et al.  E cient rectangular mixed fi-nite elements in two and three space variables , 1987 .

[2]  Michel Fortin,et al.  A mixed finite element for the stokes problem using quadrilateral elements , 1995, Adv. Comput. Math..

[3]  Francisco-Javier Sayas,et al.  Divergence-conforming HDG methods for Stokes flows , 2014, Math. Comput..

[4]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

[5]  Francisco-Javier Sayas,et al.  A projection-based error analysis of HDG methods , 2010, Math. Comput..

[6]  Ngoc Cuong Nguyen,et al.  Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics , 2012, J. Comput. Phys..

[7]  Francisco-Javier Sayas,et al.  Analysis of HDG methods for Stokes flow , 2010, Math. Comput..

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

[9]  R. Kellogg,et al.  A regularity result for the Stokes problem in a convex polygon , 1976 .

[10]  Ricardo H. Nochetto,et al.  Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations , 1989 .

[11]  Francisco-Javier Sayas,et al.  A PROJECTION-BASED ERROR ANALYSIS OF HDG METHODS , 2010 .

[12]  M. Fortin,et al.  A new mixed finite element for the Stokes and elasticity problems , 1993 .

[13]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[14]  Monique Dauge,et al.  Stationary Stokes and Navier-Stokes systems on two-or three-dimensional domains with corners , 1989 .

[15]  Bernardo Cockburn,et al.  journal homepage: www.elsevier.com/locate/cma , 2022 .

[16]  R. Stenberg.  A family of mixed finite elements for the elasticity problem , 1988 .

[17]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[18]  Rolf Stenberg,et al.  Postprocessing schemes for some mixed finite elements , 1991 .

[19]  Ke Shi,et al.  Conditions for superconvergence of HDG methods for second-order elliptic problems , 2012, Math. Comput..