The narrow escape problem consists of deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. The asymptotic formula explicitly exhibit the nonlinear interaction of many small absorbing targets. 

 It is clear that the physically meaningful quantity is not the potential itself, but its difference or gradient. However, the theory is based on the potential itself. In this talk the speaker will introduce new concept of relative potential.

In this talk, we first introduce the weakly over-penalized symmetric interior penalty (WOPSIP) method for second order elliptic problems, which belongs to the family of discontinuous Galerkin methods. Similar to the classical nonconforming P1 finite element method, this method satisfies the same types of error estimates as the standard conforming finite element method in both the energy norm and the L2 norm. Moreover, the WOPSIP method is more flexible than the classical nonconforming P1 finite element method in the sense that it can be implemented on meshes with hanging nodes.

Secondly, we discuss two-level additive Schwarz preconditioners for the WOPSIP method. The key ingredient of the two-level additive Schwarz preconditioner is the construction of the subdomain solvers and the coarse solver. In our approach, we consider different choices of coarse problems and intergrid transfer operators. It is shown that the condition number estimates previously obtained for classical finite element methods also hold for the WOPSIP method. In addition, we present numerical results that illustrate the parallel performance of these preconditioners.