This project continues investigation of the complexity of differential and integral equations. New problems to be studied include the following: (1) Problems with singular data. Special attention is paid to problems on domains having corner singularities, whose coefficient data has known singularities, and problems for which the forcing function has a singularity of a given type. (2) Exterior problems. The research concentrates on the Helmholtz problem on exterior domains, which arise in scattering theory. (3) Further work on mixed derivative spaces. The research concentrates on problems, as well as the possible optimality of finite element methods. (4) Stochastic differential equations. These problems arise in many areas, such as science, engineering, and finance. Nothing is known about their complexity. The complexity of optimal algorithms for such problems is being studied.
