Finite element approximationof a nonlinear elliptic equation arising from bimaterial problemsin elastic-plastic mechanics

Summary. In [13], a nonlinear elliptic equation arising from elastic-plastic mechanics is studied. A well-posed weak formulation is established for the equation and some regularity results are further obtained for the solution of the boundary problem. In this work, the finite element approximation of this boundary problem is examined in the framework of [13]. Some error bounds for this approximation are initially established in an energy type quasi-norm, which naturally arises in degenerate problems of this type and proves very useful in deriving sharper error bounds for the finite element approximation of such problems. For sufficiently regular solutions optimal error bounds are then obtained for some fully degenerate cases in energy type norms.

[1]  Alexander Ženíšek.  The finite element method for nonlinear elliptic equations with discontinuous coefficients , 1990 .

[2]  R. Glowinski,et al.  Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires , 1975 .

[3]  C. M. Elliott,et al.  Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces , 1987 .

[4]  Tadeusz Iwaniec,et al.  Regularity of p-harmonic functions on the plane. , 1989 .

[5]  John W. Barrett,et al.  Finite element approximation of the p-Laplacian , 1993 .

[6]  John W. Barrett,et al.  A further remark on the regularity of the solutions of the p -Laplacian and its applications to their finite element approximations , 1993 .

[7]  S. Chow.  Finite element error estimates for non-linear elliptic equations of monotone type , 1989 .

[8]  Miloslav Feistauer,et al.  Finite element approximation of nonlinear elliptic problems with discontinuous coefficients , 1989 .

[9]  M. Chipot.  Finite Element Methods for Elliptic Problems , 2000 .

[10]  John W. Barrett,et al.  Higher-order regularity for the solutions of some degenerate quasilinear elliptic equations in the plane , 1993 .

[11]  John W. Barrett,et al.  Finite element approximation of some degenerate monotone quasilinear elliptic systems , 1996 .

[12]  W. B. Liu,et al.  Quasi-norm Error Bounds for the Nite Element Approximation of a Non-newtonian Ow , 1994 .

[13]  John R. Rice,et al.  Mathematical analysis in the mechanics of fracture , 1968 .

[14]  J. Rice,et al.  Plane strain deformation near a crack tip in a power-law hardening material , 1967 .