Nonlocal Nonlinear Diffusion: Variational Principles, Optimal Control, and Approximation

Schytt, Marcus Johan

4. term

Mathematics, Master

2023

2023-06-02

97 pages

This thesis explores a nonlocal nonlinear thermal diffusion law inspired by peridynamics, a nonlocal reformulation of classical mechanics. The primary objective is to establish an existence theory for nonlocal equilibrium states based on the Dirichlet principle of minimum energy. By rigorously defining and analyzing nonlocal analogues of gradient and divergence operators, we prove the existence and uniqueness of nonlocal equilibrium states, characterizing them as weak solutions to a nonlocal p-Laplacian law. Dual formulations are derived using Kelvin’s principle of minimum complementary energy, and their well-posedness is established through the Ladyzhenskaya-Babuška-Brezzi theory and Fenchel-Rockafellar duality. Furthermore, we demonstrate the convergence of nonlocal equilibrium states to their local counterparts as the nonlocal interaction horizon vanishes, drawing on relevant results from Bourgain-Brezis-Mironescu and Ponce. The scope narrows down to linear diffusion, specifically focusing on the formulation and analysis of nonlocal optimal control and obstacle problems. Interestingly, the analysis of linear-quadratic optimal distributed control problems closely mirrors the corresponding local analysis. However, for locally ill-posed nonlinear control in conductivity, nonlocal analogues yield practical solutions without additional regularization. Similarly, we analyze nonlocal obstacle problems with minimal assumptions on the obstacles. We argue that nonlocal modeling proves advantageous in considering discontinuous conductivities and obstacles, which pose challenges in the local theory. Lastly, the thesis investigates the numerical approximation of equilibrium states. We demonstrate the extension of the finite element method to the nonlocal case, albeit with increased computational costs. Rigorous analysis confirms the convergence of finite element approximations to the nonlocal equilibrium state, and this is supported by a series of numerical experiments. Additionally, we apply the nonlocal finite element approximations to solve nonlocal optimal control and obstacle problems

Nonlocal modelling ; Diffusion ; PDEs ; Analysis of PDEs ; Nonlocal boundary value problems ; Nonlocal diffusion ; Optimal control ; Control in the coefficients ; Compliance minimization ; Nonlocal distributed control ; Fractional p-Laplacian ; p-Laplacian ; Nonlocal FEM ; Finite elements ; Finite element methods ; Nonlocality

Download
View record in AAU Student Projects