  The focus of this work is the numerical approximation of time-dependent
partial differential equations associated to initial-boundary value problems.
This master dissertation is mostly concerned with the actual computation of the
solution to nonlinear stochastic evolution problems governed by
Kardar-Parisi-Zhang (KPZ) models. In addition, the dissertation aims to
contribute to corroborate, by means of a large set of numerical experiments,
that the initial-boundary value problem with periodic boundary conditions for
the equation KPZ is ill-posed and that such equation needs to be renormalized.
The approach to discretization of KPZ equation perfomed by means of the use of
hybrid and mixed finite elements with a domain decomposition procedure along
with a pertinent mollification of the noise. The obtained solution is compared
with the well known solution given by the Cole-Hopf transformation of the
stochastic heat equation with multiplicative noise. We were able to verify that
both solutions exhibit a good agreement, but there is a shift that grows as the
support of the mollifier decreases. For the numerical aproximation of the
stochastic heat equation we use a state-of-the-art numerical method for
evaluating semilinear stochastic PDE , which in turn combine spectral
techniques, Taylor's expantions and particular numerical treatment to the
underlying noise. Furthermore, a state-of-the-art renormalization procedure
introduced by Martin Hairer is used to renormalize KPZ equation that is
validated with nontrivial numerical experiments.
