Quasi-optimal Adaptive Pseudostress Approximation of the Stokes Equations

The pseudostress-velocity formulation of the stationary Stokes problem allows some quasi-optimal Raviart--Thomas mixed finite element formulation for any polynomial degree. The adaptive algorithm employs standard residual-based explicit a posteriori error estimation from Carstensen, Kim, and Park [SIAM J. Numer. Anal., 49 (2011), pp. 2501--2523] for the lowest-order Raviart--Thomas finite element functions in a simply connected Lipschitz domain. This paper proves optimal convergence rates in terms of the number of unknowns of the adaptive mesh-refining algorithm based on the concept of approximation classes. The proofs use some novel equivalence to first-order nonconforming Crouzeix--Raviart discretization plus a particular Helmholtz decomposition of deviatoric tensors.

[1]  Hella Rabus.  A Natural Adaptive Nonconforming FEM Of Quasi-Optimal Complexity , 2010, Comput. Methods Appl. Math..

[2]  Carsten Carstensen,et al.  Convergence analysis of an adaptive nonconforming finite element method , 2006, Numerische Mathematik.

[3]  ROB STEVENSON,et al.  The completion of locally refined simplicial partitions created by bisection , 2008, Math. Comput..

[4]  Gary R. Consolazio,et al.  Finite Elements , 2007, Handbook of Dynamic System Modeling.

[5]  Long Chen,et al.  Convergence and optimality of adaptive mixed finite element methods , 2010, Math. Comput..

[6]  Shipeng Mao,et al.  Adaptive nonconforming finite elements for the Stokes equations , 2010 .

[7]  Thomas J. R. Hughes,et al.  Encyclopedia of computational mechanics , 2004 .

[8]  Gabriel N. Gatica,et al.  Augmented Mixed Finite Element Methods for the Stationary Stokes Equations , 2008, SIAM J. Sci. Comput..

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

[10]  Shipeng Mao,et al.  Quasi-Optimality of Adaptive Nonconforming Finite Element Methods for the Stokes Equations , 2011, SIAM J. Numer. Anal..

[11]  Carsten Carstensen,et al.  Error reduction and convergence for an adaptive mixed finite element method , 2006, Math. Comput..

[12]  Gabriel N. Gatica,et al.  A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows , 2011 .

[13]  Carsten Carstensen,et al.  Optimal adaptive nonconforming FEM for the Stokes problem , 2013, Numerische Mathematik.

[14]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

[15]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[16]  Zhiqiang Cai,et al.  A Multigrid Method for the Pseudostress Formulation of Stokes Problems , 2007, SIAM J. Sci. Comput..

[17]  R. Hoppe,et al.  ERROR REDUCTION AND CONVERGENCE FOR ANADAPTIVE MIXED FINITE ELEMENT , 2022 .

[18]  V. Thomée,et al.  Error estimates for some mixed finite element methods for parabolic type problems , 1981 .

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

[20]  Panayot S. Vassilevski,et al.  Mixed finite element methods for incompressible flow: Stationary Stokes equations , 2010 .

[21]  Shun Zhang,et al.  Mixed Finite Element Methods for Incompressible Flow: Stationary Navier-Stokes Equations , 2010, SIAM J. Numer. Anal..

[22]  Carsten Carstensen,et al.  An optimal adaptive mixed finite element method , 2011, Math. Comput..

[23]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[24]  Dongho Kim,et al.  A Priori and A Posteriori Pseudostress-velocity Mixed Finite Element Error Analysis for the Stokes Problem , 2011, SIAM J. Numer. Anal..

[25]  Jun Hu,et al.  Convergence and Optimality of the Adaptive Nonconforming Linear Element Method for the Stokes Problem , 2012, Journal of Scientific Computing.

[26]  Shipeng Mao,et al.  A Convergent Nonconforming Adaptive Finite Element Method with Quasi-Optimal Complexity , 2010, SIAM J. Numer. Anal..

[27]  G. Gatica,et al.  Analysis of a velocity–pressure–pseudostress formulation for the stationary Stokes equations ☆ , 2010 .

[28]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[29]  Zhiqiang Cai,et al.  Pseudostress–velocity formulation for incompressible Navier–Stokes equations , 2010 .

[30]  Rob P. Stevenson,et al.  Optimality of a Standard Adaptive Finite Element Method , 2007, Found. Comput. Math..

[31]  L. D. Marini.  An Inexpensive Method for the Evaluation of the Solution of the Lowest Order Raviart–Thomas Mixed Method , 1985 .

[32]  Christian Kreuzer,et al.  Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method , 2008, SIAM J. Numer. Anal..