A posteriori error estimation for variational problems with uniformly convex functionals

The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the form inf {F(v) + G(Λv)}, where F: V → R is a convex lower semicontinuous functional, G: Y → R is a uniformly convex functional, V and Y are reflexive Banach spaces, and Λ: V → Y is a bounded linear operator. We show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and, thus, can be justified via the duality theory.

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

[2]  Sergey Repin,et al.  A posteriori error estimation for nonlinear variational problems by duality theory , 2000 .

[3]  Solomon G. Mikhlin,et al.  Error Analysis in Numerical Processes , 1991 .

[4]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[5]  Kenneth Eriksson,et al.  An adaptive finite element method for linear elliptic problems , 1988 .

[6]  Pierre Ladevèze,et al.  Error Estimate Procedure in the Finite Element Method and Applications , 1983 .

[7]  S. Repin,et al.  A posteriori error estimates for approximate solutions of variational problems with functionals of power growth , 2000 .

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

[9]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[10]  Philippe Blanchard,et al.  Variational Methods in Mathematical Physics , 1992 .

[11]  Sergey Repin,et al.  A posteriori error estimation for elasto-plastic problems based on duality theory☆ , 1996 .

[12]  Claes Johnson,et al.  Adaptive finite element methods in computational mechanics , 1992 .

[13]  V. Thomée,et al.  The stability in _{} and ¹_{} of the ₂-projection onto finite element function spaces , 1987 .

[14]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

[15]  R. Temam,et al.  Analyse convexe et problèmes variationnels , 1974 .

[16]  S. Repin,et al.  Error estimates for stresses in the finite element analysis of the two-dimensional elasto-plastic problems , 1995 .

[17]  Leszek Demkowicz,et al.  Toward a universal h-p adaptive finite element strategy , 1989 .

[18]  Pierre Ladevèze,et al.  ERROR ESTIMATION AND MESH OPTIMIZATION FOR CLASSICAL FINITE ELEMENTS , 1991 .

[19]  Vidar Thomée,et al.  An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem , 1990 .

[20]  Leszek Demkowicz,et al.  A posteriori error analysis in finite elements: the element residual method for symmetrizable problems with applications to compressible Euler and Navier-Stokes equations , 1990 .

[21]  Ivo Babuška,et al.  A Posteriori Error Analysis of Finite Element Solutions for One-Dimensional Problems , 1981 .

[22]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[23]  J. Oden,et al.  A unified approach to a posteriori error estimation using element residual methods , 1993 .

[24]  Jean-Pierre Aubin,et al.  SOME ASPECTS OF THE METHOD OF THE HYPERCIRCLE APPLIED TO ELLIPTIC VARIATIONAL PROBLEMS. , 1969 .

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

[26]  D. Kelly,et al.  The self‐equilibration of residuals and complementary a posteriori error estimates in the finite element method , 1984 .

[27]  J. Douglas,et al.  The stability inLq of theL2-projection into finite element function spaces , 1974 .

[28]  S. L. Sobolev,et al.  Some Applications of Functional Analysis in Mathematical Physics , 1991 .

[29]  Sergey Repin,et al.  A posteriori error estimation for nonlinear variational problems , 1997 .

[30]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[31]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[32]  Sergey Repin,et al.  A posteriori error estimates for approximate solutions to variational problems with strongly convex functionals , 1999 .

[33]  S. L. Sobolev,et al.  Applications of functional analysis in mathematical physics , 1963 .

[34]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[35]  Ivo Babuška,et al.  The problem of the selection of an a posteriori error indicator based on smoothening techniques , 1993 .

[36]  S. Repin.  ERRORS OF FINITE ELEMENT METHOD FOR PERFECTLY ELASTO-PLASTIC PROBLEMS , 1996 .

[37]  R. Bank,et al.  Some a posteriori error estimators for elliptic partial differential equations , 1985 .

[38]  Sergey Repin,et al.  A unified approach to a posteriori error estimation based on duality error majorants , 1999 .

[39]  S. Repin,et al.  A priori error estimates of variational-difference methods for Hencky plasticity problems , 1997 .