Finite Element Methods for p-Laplace and Nonlocal Problems

<strong>Mathematics Seminar of the School of Physical Sciences -------------------------------------------------------------</strong> <strong>Title: Finite Element Methods for p-Laplace and Nonlocal Problems</strong> Speaker:<strong> Sudhakar Chaudhary</strong> (Indian Institute of Technology Delhi, New Delhi) Date:<strong> April 28, 2015</strong> <strong>Abstract: </strong>The main aim of this talk is to discuss the finite element Galerkin methods for the p-structure (p-Laplace and p-Stokes etc.) and nonlocal problems. We first present the well-posedness results at continuous and discrete levels for p-structure problems using web-spline basis functions. Based on web-spline finite elements, we show some numerical results for the solution to p-Laplace problem. Next we consider parabolic nonlocal problem with usual finite element method. The non-local term in parabolic nonlocal problem causes difficulty when the equation is solved numerically by using Newton's method. In particular after applying Newton's method we get full Galerkin matrix due to the nonlocal term. In order to avoid this difficulty we use the technique given by T. Gudi in 2012 for elliptic nonlocal problem of Kirchhoff type. We consider semi-discrete approximation of parabolic nonlocal problem by discretizing spatial variable with the help of finite element method while keeping time variable continuous. We derive the a priori error estimates for the semi-discrete approximations. These estimates are achieved by a priori bounds for the exact solution and the properties of Ritz projection. Finally, we discuss error analysis related to fully-discrete approximations by means of backward Euler scheme.