Convergence of fractional step mimetic finite difference discretizations for semilinear parabolic problems

This paper deals with the numerical solution of semilinear parabolic problems by means of efficient parallel algorithms. We first consider a mimetic finite difference method for the spatial semidiscretization. The connection of this method with an appropriate mixed finite element technique is the key to prove the convergence of the semidiscrete scheme. Next, we propose and analyze the use of a linearly implicit fractional step Runge–Kutta method as time integrator. The choice of suitable operator splittings related to an adequate decomposition of the spatial domain makes it possible to obtain totally discrete schemes that can be easily parallelized. A numerical test is shown to illustrate the theoretical results.