Finite element approximation of a nonlinear elliptic equation arising from bimaterial problems in elastic-plastic mechanics (Q1587931)

The author considers a quasilinear Dirichlet problem for \(\operatorname{div} (k \operatorname{grad} u) + f = 0\) in a two-dimensional Lipschitz domain \(\Omega\) which consists of two subdomains. Here \(k\) depends on \(|\operatorname {grad} u |\) and may be different for the two subdomains. To study the accuracy of a conforming linear finite element method on a regular triangulation, and defining \[ a(u,w):=\int_\Omega k(|\operatorname{grad} u|)\operatorname{grad} u\cdot\operatorname{grad} w, \] as the distance of \(u\) and \(v\) the expression \(a(u,u-v)-a(v,u-v)\) is taken and shown to be equivalent to a certain quasi-norm. This leads to optimal error bounds for sufficiently regular solutions.