Compactness method in the finite element theory of nonlinear elliptic problems (Q1101193)

The authors study finite element approximation of nonlinear elliptic boundary value problems of the type \[ -\text{div} a(\cdot,u,\text{grad} u)+b(\cdot,u,\text{grad} u)=f \text{ in } \Omega,\quad u=q_ D\text{ auf } \Gamma_ D \] \(a(\cdot,u,\text{grad} u)\cdot n=q_ N\) auf \(\Gamma_ N=\partial \Omega \setminus \Gamma_ D\) in a bounded nonpolygonal domain \(\Omega\subset {\mathfrak R}^ 2.\) The corresponding form \(a(u,v)\) is assumed to be Lipschitz-continuous and pseudo-monotone (generalized condition S); this guarantees compactness. In the discretization the domain \(\Omega\) is approximated by polygonal domains; linear conforming triangular elements are used; the integrals are evaluated by numerical quadratures. Solvability of the discrete problems and convergence to an exact weak solution \(u\in H^ 1(\Omega)\) are proved. No additional assumption on the regularity of the exact weak solution is needed.