Special Seminar: Jie Shen (Purdue)

Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality".

We shall construct a new sparse spectral method for high-dimensional problems, and present, in particular, rigorous error estimates as well as efficient numerical algorithms for elliptic equations.

We shall also propose a new weighted weak formulation for the Fokker-Planck equation of FENE dumbbell model, and prove its well-posedness in weighted Sobolev spaces. Based on the new formulation, we are able to design simple, efficient, and unconditionally stable semi-implicit Fourier-Jacobi schemes for the Fokker-Planck equation of the FENE dumbbell model.

It is hoped that the combination of the two new approaches would make it possible to directly simulate the five or six dimensional Navier-Stokes Fokker-Planck system.