Sobolev gradient preconditioning for elliptic reaction-diffusion problems with some nonsmooth nonlinearities

The Sobolev gradient approach is an efficient way to construct preconditioned iterations for solving nonlinear problems. We extend this technique to be applicable for elliptic equations describing stationary states of reaction-diffusion problems if the nonlinearities have certain lack of differentiability. We derive convergence results of the Sobolev gradient method on an abstract level and then for our elliptic problem under different assumptions. Numerical tests show convergence as expected. (C) 2019 Elsevier B.V. All rights reserved.