Isotropic and anisotropic a posteriori error estimation of the mixed finite element method for second order operators in divergence form.

This paper presents an a posteriori residual error estimator for the mixed FEM of second order operators using isotropic or anisotropic meshes in R^d, d=2 or 3. The reliability and efficiency of our estimator is established without any regularity assumptions on the solution of our problem.

[1]  Carsten Carstensen,et al.  A posteriori error estimate for the mixed finite element method , 1997, Math. Comput..

[2]  Gerd Kunert,et al.  A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes , 1999 .

[3]  G. Kunert Toward anisotropic mesh construction and error estimation in the finite element method , 2002 .

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

[5]  Ricardo G. Durán,et al.  The Maximum Angle Condition for Mixed and Nonconforming Elements: Application to the Stokes Equations , 1999, SIAM J. Numer. Anal..

[6]  Jens Markus Melenk,et al.  hp-Finite Element Methods for Singular Perturbations , 2002 .

[7]  M. Krízek,et al.  On the maximum angle condition for linear tetrahedral elements , 1992 .

[8]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[9]  M. Dauge Elliptic boundary value problems on corner domains , 1988 .

[10]  S. Nicaise,et al.  The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges , 1998 .

[11]  J. Lions Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal , 1973 .

[12]  Gerd Kunert,et al.  Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes , 2000, Numerische Mathematik.

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

[14]  Serge Nicaise,et al.  General Interface Problems-II , 1994 .

[15]  T. Apel Anisotropic Finite Elements: Local Estimates and Applications , 1999 .

[16]  Serge Nicaise,et al.  Numerische Simulation Auf Massiv Parallelen Rechnern a Posteriori Error Estimation for the Stokes Problem: Anisotropic and Isotropic Discretizations , 2022 .

[17]  Dietrich Braess,et al.  A Posteriori Error Estimators for the Raviart--Thomas Element , 1996 .

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

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

[20]  Dominique Leguillon,et al.  Computation of singular solutions in elliptic problems and elasticity , 1987 .

[21]  Simona Perotto,et al.  New anisotropic a priori error estimates , 2001, Numerische Mathematik.

[22]  Jean E. Roberts,et al.  Mixed and hybrid methods , 1991 .

[23]  Serge Nicaise,et al.  SOME MIXED FINITE ELEMENT METHODS ON ANISOTROPIC MESHES , 2001 .

[24]  M. Costabel,et al.  Singularities of Maxwell interface problems , 1999 .

[25]  Serge Nicaise,et al.  Polygonal interface problems:higher regularity results , 1990 .

[26]  Kunibert G. Siebert,et al.  An a posteriori error estimator for anisotropic refinement , 1996 .

[27]  A. Alonso Error estimators for a mixed method , 1996 .