The gradient-finite element method for elliptic problems,

The coupling of the Sobolev space gradient method and the finite element method is developed. The Sobolev space gradient method reduces the solution of a quasilinear elliptic problem to a sequence of linear Poisson equations. These equations can be solved numerically by an appropriate finite element method. This coupling of the two methods will be called the gradient-finite element method (GFEM). Linear convergence of the GFEM is proved via suitable error control in the steps of the iteration. The GFEM defines an already preconditioned iteration in the sense that the theoretical ratio of convergence of the Sobolev space GM is preserved. Finally, a numerical example illustrates the method.