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

The paper addresses numerically approximating time-dependent stochastic KPZ models, focusing on ill-posedness under periodic boundaries and the necessity of renormalization. It develops a combined mixed and hybrid finite element method with domain decomposition and noise mollification, and verifies convergence to the Cole–Hopf transformation of the stochastic heat equation, with renormalization removing divergences as mollifier support shrinks. The study systematically analyzes mollification effects, provides iterative and spectral discretizations, and demonstrates mollifier-independent renormalization limits in line with Hairer’s theory. This work advances reliable numerical methods for KPZ-type SPDEs and supplies computational evidence supporting the theoretical renormalization framework with potential applications to broader stochastic PDEs.

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.

Figures (29)

• Figure 3.1: Modelo discreto de deposição balística. Extraído de BS95.
• Figure 3.2: Modelo simples de deposição balística. Extraído de BS95.
• Figure 3.3: Efeito de crossover e rugosidade. Resultados experimentais para o modelo BD. Extraído de BS95.
• Figure 3.4: Reescalamento que leva à colisão das curvas de rugosidade obtidas para quatro escolhas diferentes de $L$ no modelo BD. Extraído de BS95.
• Figure 3.5: Modelo discreto de deposição aleatória. Extraído de BS95.