Iterative Galerkin discretizations for strongly monotone problems

In this article we investigate the use of fixed point iterations to solve the Galerkin approximation of strictly monotone problems. As opposed to Newton’s method, which requires information from the previous iteration in order to linearize the iteration matrix (and thereby to recompute it) in each step, the alternative method used in this article exploits the monotonicity properties of the problem, and only needs the iteration matrix calculated once for all iterations of the fixed point method. We outline the abstract a priori and a posteriori analyses for the iteratively obtained solutions, and apply this to a finite element approximation of a second-order elliptic quasilinear boundary value problem. We show both theoretically, as well as numerically, how the number of iterations of the fixed point method can be restricted in dependence of the mesh size, or of the polynomial degree, to obtain optimal convergence. Using the a posteriori error analysis we also devise an adaptive algorithm for the generation of a sequence of Galerkin spaces (adaptively refined finite element meshes in the concrete example) to minimize the number of iterations on each space.