Tuesday, November 10, 2015 at 3:00 PM in SAS 4201
Chunmei Wang, Georgia Tech
Weak Galerkin Finite Element Methods for PDEs
Weak Galerkin (WG) is a new finite element method for partial differential equations where the differential operators (e.g., gradient, divergence, curl, Laplacian etc) in the variational forms are approximated by weak forms as generalized distributions. The WG discretization procedure often involves the solution of inexpensive problems defined locally on each element. The solution from the local problems can be regarded as a reconstruction of the corresponding differential operators. The fundamental difference between the weak Galerkin finite element method and other existing methods is the use of weak functions and weak derivatives (i.e., locally reconstructed differential operators) in the design of numerical schemes based on existing variational forms for the underlying PDE problems. Weak Galerkin is, therefore, a natural extension of the conforming Galerkin finite element method. Due to its great structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations by providing the needed stability and accuracy in approximation.
In this talk, the speaker will introduce a general framework for WG methods by using the second order elliptic problem as an example. Furthermore, the speaker will present WG finite element methods for several model PDEs, including the linear elasticity problem, a fourth order problem arising from fluorescence tomography, and the second order problem in nondivergence form. The talk should be accessible to graduate students with adequate training in computational mathematics.
Tuesday, November 17, 2015 at 3:00 PM in SAS 4201
Jingwei Hu, Purdue University
Fast algorithms for the quantum Boltzmann collision operator
The quantum Boltzmann equation describes the non-equilibrium dynamics of a system comprised of a large number of quantum particles such as bosons or fermions. The most prominent feature of this equation is a high-dimensional integral operator modeling particle collisions, whose nonlinear and nonlocal structure poses a great challenge for numerical simulation. I will introduce two fast algorithms for the quantum Boltzmann collision operator. The first one is a quadrature based solver specifically designed for the collision operator in reduced energy space. Compared to cubic complexity of a direct evaluation, our algorithm runs in only linear complexity (optimal up to a logarithmic factor). The second one accelerates the computation of the full phase space collision operator. It is a spectral method based on a low rank expansion of the collision kernel. Numerical examples including an application to semiconductor device modeling are presented to illustrate the efficiency and accuracy of the proposed algorithms.
Wednesday, November 18, 2015 at 3:00 PM in SAS 4201
Qi Wang, U of South Carolina
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