Domain decomposition multigrid methods for nonlinear reaction–diffusion problems

In this work, we propose efficient discretizations for nonlinear evolutionary reaction–diffusion problems on general two-dimensional domains. The spatial domain is discretized through an unstructured coarse triangulation, which is subsequently refined via regular triangular grids. Following the method of lines approach, we first consider a finite element spatial discretization, and then use a linearly implicit splitting time integrator related to a suitable decomposition of the triangulation nodes. Such a procedure provides a linear system per internal stage. The equations corresponding to those nodes lying strictly inside the elements of the coarse triangulation can be decoupled and solved in parallel using geometric multigrid techniques. The method is unconditionally stable and computationally efficient, since it avoids the need for Schwarz-type iteration procedures. In addition, it is formulated for triangular elements, thus yielding much flexibility in the discretization of complex geometries. To illustrate its practical utility, the algorithm is shown to reproduce the pattern-forming dynamics of the Schnakenberg model.