Abstract

The finite element method (FEM) is one of the most popular numerical techniques to approximate the solutions of partial differential equations (PDEs). The stability and accuracy of the finite element approximation depend on specific features of the target PDE.

It is well known that elliptic equations may possess singular solutions in many situations. These singular solutions often present a multi-scale character and pose numerous challenges both on the analysis of the PDE and on the design of the finite element scheme. In this talk, we discuss recent advances in the development of effective finite element algorithms approximating a class of singular solutions, including corner singularities with different boundary conditions and singularities from the non-smooth points on the interface in transmission problems. In particular, we establish a-priori estimates (well-posedness, regularity, and the Fredholm property) for singular solutions in weighted Sobolev spaces. Then, based on these theoretical results, we propose a simple and explicit construction of finite element spaces to recover the optimal convergence rate of the numerical solution. This systematic approach has shown great potential in solving various singular problems in physics and engineering (e.g., Schrodinger type equations with singular potentials and axisymmetric equations).

Wednesday, October 17, 2012

3:00– 4:00 P.M.

372 Science and Engineering Building

(Refreshments at 2:30-3:00 PM in the kitchen area adjacent to 368 SEB)