APDE: Seminars

Seminars in Applied and Computational Analysis

Norm-Resolvent Convergence in Perforated Domains

For several different types of boundary conditions (Dirichlet, Neumann and Robin), we prove norm-resolvent convergence in L2 for the Laplacian in an open domain perforated epsilon-periodically by spherical holes. The limit operator is of the form -Δ+m on the unperforated domain, where m is a positive constant. This is an improvement of previous results [Cioranescu & Murat. A Strange Term Coming From Nowhere, Progress in Nonlinear Differential Equations and Their Applications, 31, (1997)], [S. Kaizu. The Robin Problems on Domains with Many Tiny Holes. Proc. Japan Acad., 61, Ser. A (1985)], which show strong resolvent convergence. In particular, our result implies convergence of the spectrum of the operator for the perforated domain problem.

Low-regularity Fourier integrators for the nonlinear Schrödinger equation

A large toolbox of numerical schemes for the nonlinear Schrödinger equation has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever “non-smooth’’ phenomena enter the scene such as for problems at low-regularity and high oscillations. Classical schemes fail to capture the oscillatory parts within the solution which leads to severe instabilities and loss of convergence. In this talk I present a new class of Fourier integrators for the nonlinear Schrödinger equation at low-regularity. The key idea in the construction of the new schemes is to tackle and hardwire the underlying structure of resonances into the numerical discretization.​ These terms are the cornerstones of theoretical analysis of the long time behaviour of differential equations and their numerical discretizations (cf. modulated Fourier Expansion; Hairer, Lubich & Wanner) and offer the new schemes strong geometric structure at low regularity.​



Numerical homogenisation

Abstract not available

Graduate Seminars (WE 15:00, MR5)