With Prof. Hui Li's gracious invitation, Mr. Pengfei Liu from California Institute of Technology will visit Centre of Structural Monitoring and Control (SMC) at Harbin Institute of Technology and make a presentation on August 19, 2015. We sincerely welcome the interested faculties and students to join in.
The lecture room: 402, School of Civil Engineering
Time: August 19, 10:00am-11:00am
Topic: Optimal Local Multi-scale Basis for Elliptic Equations with Rough Coefficients.

ABSTRACT:
We introduce a multi-scale finite element method for second order linear elliptic equations with arbitrarily rough coefficient. We propose a local oversampling method to construct basis functions that have optimal local approximation property. Our methodology is based on the compactness of the solution operator restricted on local regions of the spatial domain, and does not depend on any scale-separation or periodicity assumption of the coefficient. We focus on a special type of basis functions that are harmonic on each element and have optimal approximation property. We first reduce our problem to approximating the trace of the solution space on each edge of the underlying mesh, and then achieve this goal through the singular value decomposition of an oversampling operator. Rigorous error estimates can be obtained through thresholding in constructing the basis functions. Numerical results for several problems with multiple spatial scales and high contrast inclusions are presented to demonstrate the compactness of the local solution space and the capacity of our method in identifying and exploiting this compact structure to achieve computational savings. This is a joint work with Prof. Thomas Y. Hou.

Biosketch of Mr. Pengfei Liu

Education Background
2008-2012 B.S. Peking University Department of Mathematics.
2012-now Ph.D. Caltech, Applied and Computational Mathematics.

Research Interest
Finite-time singularity in Fluid Equations;  Numerical Methods for PDEs with Rough Coefficients and Uncertainty.