# Combined use of mixed and hybrid finite elements method with domain decomposition and spectral methods for a study of renormalization for the KPZ model

## TL;DR

This paper develops a numerical approach combining mixed and hybrid finite elements, domain decomposition, and spectral methods to study the KPZ equation, demonstrating the need for renormalization and validating solutions through numerical experiments.

## Contribution

It introduces a novel numerical discretization method for KPZ equations and applies Hairer's renormalization, providing new insights into their ill-posedness and solution behavior.

## Key findings

- KPZ equation with periodic boundary conditions is ill-posed without renormalization.
- Numerical solutions agree with Cole-Hopf transformation solutions, with shifts depending on mollifier support.
- Hairer's renormalization procedure effectively regularizes the KPZ equation in numerical simulations.

## Abstract

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.