Partitioning methods for reaction–diffusion problems

We consider the numerical solution of reaction–diffusion systems using linear finite elements on a space grid changing in time. For the integration with respect to the time variable a W-method with several variants of implicit/explicit partitioning is used. For grid adaption an algorithm featuring a flexible refinement and coarsening control is proposed. The partitioned W-methods keep the stability of implicit schemes but reduce the size of the linear systems to be solved. We combine local partitioning with partitioning between the diffusion and reaction terms, leading to a large variety of methods. The efficiency of several partitioning methods is compared in numerical tests. The calculations show an increase of efficiency if partitioned schemes are used instead of a fully implicit W-method. We include a numerical comparison of three linear solvers. Optimal truncation of the iteration process is discussed.