Abstract

We study the homogenization of the PDE $-A(x/\varepsilon):D^2 u_{\varepsilon} = f$ posed in a bounded convex domain subject to a Dirichlet boundary condition and the numerical approximation of the homogenized problem, where the measurable, uniformly elliptic, periodic and symmetric diffusion matrix $A$ is merely assumed to be essentially bounded and (in dimension $n>2$) to satisfy the Cordes condition. In the first part, we show existence and uniqueness of an invariant measure and prove homogenization under minimal regularity assumptions. Then, we generalize known corrector bounds and results on optimal convergence rates from the classical case of H\"{o}lder continuous coefficients to the present case. In the second part, we discuss an approximation scheme for the effective coefficient matrix based on a finite element method for the approximation of the invariant measure.

Speaker

Timo Sprekeler is a Peng Tsu Ann Assistant Professor of mathematics at the National University of Singapore. He received his BSc from TU Dortmund University (Germany) in 2016, his MASt from the University of Cambridge (UK) in 2017 (Part III of the Mathematical Tripos), and his DPhil from the University of Oxford (UK) under the supervision of Professors Endre Süli (Oxford) and Yves Capdeboscq (Paris) in 2021.

His research interest is in analysis and numerical analysis of partial differential equations with a focus on homogenization, numerical homogenization, and the finite element approximation of fully-nonlinear PDEs.