Finite element error estimates for non-linear elliptic equations of monotone type

Abstract

Summary. In this paper we shall consider the application of the finite element method to a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient, and the derivation of error estimates for the finite element approximations. Such problems arise in many practical situations -for example, in shock-free airfoil design, seepage through coarse grained porous media, and in some glaciological problems. By making use of certain properties of the nonlinear coefficients, we shall demonstrate that the variational formulations associated with these boundary value problems are well-posed. We shall also prove that the abstract operators accompanying such problems satisfy certain continuity and monotonicity inequalities. With the aid of these inequalities and some standard results from approximation theory, we show how one may derive error estimates for the finite element approximations in the energy norm.

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

“…For related works on linear elliptic problems, see [2,1,57,41,23,58,50,51,53,52] and the discussion in Section 8. Alternative numerical approaches have also been investigated; here we only mention finite element schemes (see [36,16] and references therein), kinetic schemes (see [14,22,55] and references therein) and operator splitting schemes (see [43]). …”

Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities

“…Clearly (2 7) Under similar assumptions Chow in [6] has proved (2.4) with ô = 2 -p for p e (1,2], a x = 1 and ô = 0 for p e [2, oo), a t = 0 ; (2.5) with 8 = 0 for p e (1, 2], a 2 = 0 and 5 = p -2 for p e [2, oo ), a 2 = 1. With these one can establish some error bounds for the finite element approximation of (1.1) (see [6], [7] and [12]).…”

“…Ho we ver, the main advantage to stating the assumptions in this form is that one can obtain some sharper inequalities, see (2.4) and (2.5), than those in [6] and [16] …”

“…Associated with (WP ) is the following minimization problem : Under Assumptions (E ) it is a simple matter to establish the existence of a unique solution u to (MP ) by adapting the argument for the équation case given in [6], [7] and [12]. In addition it foliows that (WP ) and (MP ) are equivalent problems.…”

Sur l'approximation numérique des écoulements quasi-newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau

“…Dans le cas Ô = 1, f r n'est plus homogène et ces résultats ont alors été démontrés dans [3] en utilisant les techniques de [4].…”