Regularized Least Squares Approximation over the Unit Sphere by Using Spherical Designs

In this talk, starting with some earlier results, we propose and analyze an alternate approach of optimal L2error estimates for semidiscrete Galerkin approximations to a second order linear parabolic initial and boundary value problem with rough initial data. Our analysis is based on energy arguments without using parabolic duality. Further, it follows the spirit of the proof technique used for deriving optimal error estimates for finite element approximations to parabolic problems with smooth initial data and hence, it unifies both theories, that is, one for smooth initial data and other for nonsmooth data. Moreover, the proposed technique is also extended to a semidiscrete mixed method for linear parabolic integro-differential equations with nonsmooth data for which there are some problems in deriving optimal L2-error estimates. This is a joint work with Deepjyoti Goswami.