A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations

A unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the Herrmann formulation within the Hellinger-Reissner principle. This quasi-optimally converging extension of PEERS (Plane Elasticity Element with Reduced Symmetry) is called Dual-Mixed Hybrid formulation (DMH). Explicit residual-based a posteriori error estimates for DMH are introduced and are mathematically shown to be locking-free, reliable, and efficient. The estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation. Numerical evidence supports that the adaptive scheme leads to optimal convergence for Lamé and Stokes benchmark problems with singularities.

[1]  J. Douglas,et al.  PEERS: A new mixed finite element for plane elasticity , 1984 .

[2]  Bernardo Cockburn,et al.  A Characterization of Hybridized Mixed Methods for Second Order Elliptic Problems , 2004, SIAM J. Numer. Anal..

[3]  O. C. Zienkiewicz,et al.  The superconvergent patch recovery (SPR) and adaptive finite element refinement , 1992 .

[4]  Susanne C. Brenner,et al.  Finite Element Methods , 2000 .

[5]  M. Fortin,et al.  Dual hybrid methods for the elasticity and the Stokes problems: a unified approach , 1997 .

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

[7]  Ilio Galligani,et al.  Mathematical Aspects of Finite Element Methods , 1977 .

[8]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[9]  Carsten Carstensen,et al.  A unifying theory of a posteriori finite element error control , 2005, Numerische Mathematik.

[10]  L. Herrmann.  Elasticity Equations for Incompressible and Nearly Incompressible Materials by a Variational Theorem , 1965 .

[11]  Rüdiger Verfürth,et al.  A posteriori error estimators for the Stokes equations II non-conforming discretizations , 1991 .

[12]  Carsten Carstensen,et al.  Averaging technique for FE – a posteriori error control in elasticity. Part I: Conforming FEM , 2001 .

[13]  Paola Causin,et al.  A dual-mixed hybrid formulation for fluid mechanics problems: Mathematical analysis and application to semiconductor process technology , 2003 .

[14]  Bernardo Cockburn,et al.  Error analysis of variable degree mixed methods for elliptic problems via hybridization , 2005, Math. Comput..

[15]  R. Verfürth.  A posteriori error estimators for the Stokes equations , 1989 .

[16]  Carsten Carstensen,et al.  Locking-free adaptive mixed finite element methods in linear elasticity , 2000 .

[17]  Gabriel N. Gatica,et al.  A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a-posteriori error estimate , 2002, Numerische Mathematik.

[18]  S. Repin,et al.  ON THE FUNCTIONAL TYPE A POSTERIORI ERROR ESTIMATES FOR THE STOKES PROBLEM. , 2004 .

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

[20]  Carsten Carstensen,et al.  A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems , 2001, Math. Comput..

[21]  D. Arnold,et al.  Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates , 1985 .

[22]  Claes Johnson,et al.  Introduction to Adaptive Methods for Differential Equations , 1995, Acta Numerica.

[23]  D. Arnold,et al.  A new mixed formulation for elasticity , 1988 .

[24]  Carsten Carstensen,et al.  A posteriori error estimates for mixed FEM in elasticity , 1998, Numerische Mathematik.

[25]  Gabriel N. Gatica,et al.  A mixed‐FEM formulation for nonlinear incompressible elasticity in the plane , 2002 .

[26]  Norbert Heuer,et al.  An implicit–explicit residual error estimator for the coupling of dual‐mixed finite elements and boundary elements in elastostatics , 2001 .

[27]  Carsten Carstensen,et al.  Uniform convergence and a posteriori error estimators for the enhanced strain finite element method , 2004, Numerische Mathematik.

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

[29]  W. Gibbs,et al.  Finite element methods , 2017, Graduate Studies in Mathematics.