Daniel Peterseim, Professor, Universitat Augsburg

Chensong Zhang, Professor

Super-localized Numerical Homogenization

16:00-17:00 July 17 (Wednesday), Meeting Number: 880-334-902 https://meeting.tencent.com/dm/khl5DQlLmmdl

Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a d-dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter H. This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximation space. This talk presents a localization technique that leads to a super-exponential decay of its basis relative to H. This suggests that basis functions with supports of width H∣logH∣^{(d−1)/d} are sufficient to preserve the optimal algebraic rates of convergence in H without pre-asymptotic effects. The collection of illustrative numerical examples includes diffusion problems with random coefficients, nonlinear eigenvalue problems under disorder, and evolution equations.

Bio: Daniel Peterseim holds the chair of Computational Mathematics at the University of Augsburg, Germany. Professor Peterseim gained his doctorate in Mathematics at the University of Zurich in 2007. Before he moved to Augsburg in 2017, he was appointed as a junior research group leader at the Humboldt-Universität zu Berlin (within the Research Center Matheon) in 2009 and as a professor for numerical simulation at the University of Bonn in 2013. His research covers aspects of computational partial differential equations with applications in engineering and physics. He is most well known for his contributions in the area of computational multiscale methods.