A multiscale recursive numerical method for semilinear parabolic problems

Resumo

We present a multiscale recursive numerical method in the context of time-dependent initial-boundary
value problems for semilinear parabolic equations with discontinuous and high-contrast coefficients. We consider

a backward Euler scheme for the temporal discretization along with an extension of the Recursive Mixed Multi-
scale Method based on domain decomposition technique, recently introduced in the literature by Ferraz [1], for

the spatial discretization of the semilinear parabolic operator. Thus, at each time step, the spatial and temporal
discretizations lead to large linearized systems of equations that involve solving local multiscale boundary value
problems followed by the solution of a family of decomposed interface problems that showed excellent scalability.
We will also briefly discuss some ideas of the proposed recursive multiscale approach for non-linear parabolic
problems, by considering efficient approximation strategies along with the reuse of the multiscale basis functions

and parallelization. Numerical examples with both homogeneous medium and heterogeneous high-contrast coeffi-
cients for semilinear problems are considered to show the behavior of the multiscale approach and our findings.