An adaptive least-squares FEM for the Stokes equations with optimal convergence rates

This paper introduces the first adaptive least-squares finite element method (LS-FEM) for the Stokes equations with optimal convergence rates based on the newest vertex bisection with lowest-order Raviart-Thomas and conforming $$P_1$$P1 discrete spaces for the divergence least-squares formulation in 2D. Although the least-squares functional is a reliable and efficient error estimator, the novel refinement indicator stems from an alternative explicit residual-based a posteriori error control with exact solve. Particular interest is on the treatment of the data approximation error which requires a separate marking strategy. The paper proves linear convergence in terms of the levels and optimal convergence rates in terms of the number of unknowns relative to the notion of a non-linear approximation class. It extends and generalizes the approach of Carstensen and Park (SIAM J. Numer. Anal. 53:43–62 2015) from the Poisson model problem to the Stokes equations.

[1]  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..

[2]  Long Chen,et al.  Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations , 2011, Math. Comput..

[3]  Carsten Carstensen,et al.  Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis , 2004, Numerische Mathematik.

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

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

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

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

[8]  Carsten Carstensen,et al.  An a posteriori error estimate for a first-kind integral equation , 1997, Math. Comput..

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

[10]  Ping Wang,et al.  Least-Squares Methods for Incompressible Newtonian Fluid Flow: Linear Stationary Problems , 2004, SIAM J. Numer. Anal..

[11]  Thomas A. Manteuffel,et al.  LOCAL ERROR ESTIMATES AND ADAPTIVE REFINEMENT FOR FIRST-ORDER SYSTEM LEAST SQUARES (FOSLS) , 1997 .

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

[13]  Michael Feischl,et al.  Convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data☆ , 2014, J. Comput. Appl. Math..

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

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

[16]  Carsten Carstensen,et al.  Quasi-optimal Adaptive Pseudostress Approximation of the Stokes Equations , 2013, SIAM J. Numer. Anal..

[17]  Lei Tang,et al.  Efficiency Based Adaptive Local Refinement for First-Order System Least-Squares Formulations , 2011, SIAM J. Sci. Comput..

[18]  Carsten Carstensen,et al.  Axioms of adaptivity , 2013, Comput. Math. Appl..

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

[20]  Ronald A. DeVore,et al.  Fast computation in adaptive tree approximation , 2004, Numerische Mathematik.

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

[22]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[23]  Pavel B. Bochev,et al.  Mathematical Foundations of Least-Squares Finite Element Methods , 2009 .

[24]  C. Carstensen,et al.  L2 best approximation of the elastic stress in the Arnold–Winther FEM , 2016 .

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

[26]  Michael Feischl,et al.  Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd , 2013 .

[27]  Carsten Carstensen,et al.  Convergence and Optimality of Adaptive Least Squares Finite Element Methods , 2015, SIAM J. Numer. Anal..

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