On the Adaptive Selection of the Parameter in Stabilized Finite Element Approximations

A systematic approach is developed for the selection of the stabilization parameter for stabilized finite element approximation of the Stokes problem, whereby the parameter is chosen to minimize a computable upper bound for the error in the approximation. The approach is applied in the context of both a single fixed mesh and an adaptive mesh refinement procedure. The optimization is carried out by a derivative-free optimization algorithm and is based on minimizing a new fully computable error estimator. Numerical results are presented illustrating the theory and the performance of the estimator, together with the optimization algorithm.

[1]  David J. Silvester,et al.  ANALYSIS OF LOCALLY STABILIZED MIXED FINITE-ELEMENT METHODS FOR THE STOKES PROBLEM , 1992 .

[2]  Frédéric Valentin,et al.  Consistent Local Projection Stabilized Finite Element Methods , 2010, SIAM J. Numer. Anal..

[3]  V. John,et al.  A posteriori optimization of parameters in stabilized methods for convection-diffusion problems - Part II , 2011, J. Comput. Appl. Math..

[4]  H. Weinberger,et al.  An optimal Poincaré inequality for convex domains , 1960 .

[5]  G. Stoyan.  Towards discrete velte decompositions and narrow bounds for inf-sup constants , 1999 .

[6]  Roland Becker,et al.  A finite element pressure gradient stabilization¶for the Stokes equations based on local projections , 2001 .

[7]  Rüdiger Verführt,et al.  A review of a posteriori error estimation and adaptive mesh-refinement techniques , 1996, Advances in numerical mathematics.

[8]  M. Stynes,et al.  Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems , 1996 .

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

[10]  F. Brezzi,et al.  On the Stabilization of Finite Element Approximations of the Stokes Equations , 1984 .

[11]  Mark Ainsworth,et al.  Computable error bounds for nonconforming Fortin–Soulie finite element approximation of the Stokes problem , 2012 .

[12]  F. Hecht,et al.  A POSTERIORI ANALYSIS OF A PENALTY METHOD AND APPLICATION TO THE STOKES PROBLEM , 2003 .

[13]  Jim Douglas,et al.  An absolutely stabilized finite element method for the stokes problem , 1989 .

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

[15]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[16]  Michel Fortin,et al.  Mixed Finite Elements, Compatibility Conditions, and Applications , 2008 .

[17]  Frédéric Valentin,et al.  Stabilized Finite Element Methods Based on Multiscale Enrichment for the Stokes Problem , 2006, SIAM J. Numer. Anal..

[18]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of , 1986 .

[19]  Kwang-Yeon Kim.  Flux reconstruction for the P2 nonconforming finite element method with application to a posteriori error estimation , 2012 .

[20]  Graham F. Carey,et al.  Penalty approximation of stokes flow , 1982 .

[21]  Clark R. Dohrmann,et al.  Stabilization of Low-order Mixed Finite Elements for the Stokes Equations , 2004, SIAM J. Numer. Anal..

[22]  C. Dohrmann,et al.  A stabilized finite element method for the Stokes problem based on polynomial pressure projections , 2004 .

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

[24]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[25]  L. D. Marini,et al.  A Priori Error Analysis of Residual-Free Bubbles for Advection-Diffusion Problems , 1999 .

[26]  Willy Dörfler,et al.  Reliable a posteriori error control for nonconforming finite element approximation of Stokes flow , 2005, Math. Comput..

[27]  David J. Silvester.  Optimal low order finite element methods for incompressible flow , 1994 .

[28]  Katya Scheinberg,et al.  Global Convergence of General Derivative-Free Trust-Region Algorithms to First- and Second-Order Critical Points , 2009, SIAM J. Optim..

[29]  H. Langtangen,et al.  Mixed Finite Elements , 2003 .

[30]  Katya Scheinberg,et al.  Geometry of interpolation sets in derivative free optimization , 2007, Math. Program..

[31]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[32]  Pavel B. Bochev,et al.  A Taxonomy of Consistently Stabilized Finite Element Methods for the Stokes Problem , 2004, SIAM J. Sci. Comput..

[33]  Thomas J. R. Hughes,et al.  The Stokes problem with various well-posed boundary conditions - Symmetric formulations that converge for all velocity/pressure spaces , 1987 .

[34]  Katya Scheinberg,et al.  On the convergence of derivative-free methods for unconstrained optimization , 1997 .

[35]  Martin Vohralík,et al.  A unified framework for a posteriori error estimation for the Stokes problem , 2012, Numerische Mathematik.