A Local Discontinuous Galerkin Method for a Class of Nonlinear Elliptic Problems with Signorini Type Boundary Conditions

Abstract

In this talk we discuss the applicability of discontinuous Galerkin methods to a class of variational inequalities arising from free boundary problems. In particular, we center our attention in the local discontinuous Galerkin (LDG) method of (Cockburn and Shu, SIAM J. Numer. Anal., 35(5): 2440–2463 (1998)) applied to a class of nonlinear elliptic equation in divergence form with Signorinitype (also called frictional) boundary conditions in part of the boundary and homogeneous Dirichlet condition on the remaining part. We derive a local description of the LDG method for this problem
and write an equivalent reduced formulation, that is equivalent to a minimization problem in a space of discontinuous piecewise polynomial functions with a discrete version of the positivity condition on the boundary. Based on this reduced formulation, we will discuss convergence and stability issues for the method.
This work has been partially supported by CONICIYT-Chile through FONDECYT Grant No. 1080168, and by the Dirección de Investigación of the Universidad de Concepción.