Finite element approximation of elliptic homogenization problems in nondivergence-form

Thesis

This thesis focuses on the construction of finite element numerical homogenization schemes for both linear and selected fully-nonlinear elliptic partial differential equations in nondivergence-form.

In the first part of the thesis, we study periodic homogenization problems of the form A(x/ε):D² u_ε = f subject to a homogeneous Dirichlet boundary condition. We provide a qualitative W^2,p theory and obtain optimal gradient and Hessian bounds with correction terms taken into account in the L^p-norm. Consequently, we find that (u_ε)_ε>0 converges strongly in the W^1,p-norm to the solution of the corresponding effective problem, and that the optimal rate for this convergence is O(ε). Based on these quantitative homogenization results, we propose and rigorously analyze a finite element-type numerical homogenization scheme for the approximation of the solution to the effective problem and the solution u_ε to the original problem in the H¹ and H² Sobolev-norms. We extend the scheme to the framework of nonuniformly oscillating coefficients and provide a variety of numerical experiments illustrating the theoretical results.

In the second part of the thesis, we propose and rigorously analyze numerical homogenization schemes for the fully-nonlinear Hamilton--Jacobi--Bellman (HJB) and HJB--Isaacs (HJBI) equations. More precisely, we are interested in the approximation of the effective Hamiltonian which determines the effective equation. Our numerical schemes are based on finite element approximations for suitable corrector problems arising in the periodic homogenization of these equations. We present a mixed finite element scheme as well as discontinuous Galerkin and C⁰ interior penalty finite element approaches. Several numerical experiments accompany the theoretical results and illustrate the performance of the numerical schemes.

Authors: Sprekeler, T

• DOI: 10.5287/ora-wvdyzdm54
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: finite element methods; nondivergence-form elliptic PDE; Homogenization
• Subjects: Mathematics