The multiscale perturbation method for second order elliptic equations

In the numerical solution of elliptic equations, multiscale methods typically involve two steps: the solution of families of local solutions or multiscale basis functions (an embarrassingly parallel task) associated with subdomains of a domain decomposition of the original domain, followed by the solution of a global problem. In the solution of multiphase flow problems approximated by an operator splitting method one has to solve an elliptic equation every time step of a simulation, that would require that all multiscale basis functions be recomputed. In this work, we focus on the development of a novel method that replaces a full update of local solutions by reusing multiscale basis functions that are computed at an earlier time of a simulation. The procedure is based on classical perturbation theory. It can take advantage of both an offline stage (where multiscale basis functions are computed at the initial time of a simulation) as well as of a good initial guess for velocity and pressure. The formulation of the method is carefully explained and several numerical studies are presented and discussed. They provide an indication that the proposed procedure can be of value in speeding-up the solution of multiphase flow problems by multiscale methods.