Numerical Methods for Variational Inequalities in Higher Dimensions

Abstract: In this talk, I will first present a quadratic finite element method for three dimensional elliptic obstacle problem which is optimally convergent (with respect to the regularity). I will derive a priori error estimates to show the optimal convergence of the method with respect to the regularity, for this we have enriched the finite element space with element-wise bubble functions. Further, a posteriori error estimates are derived to design an adaptive mesh refinement algorithm. The result on a priori estimate will be illustrated by a numerical experiment.

Next, I will discuss on two newly proposed discontinuous Galerkin (DG) finite element methods for the elliptic obstacle problem. Using the localized behavior of DG methods, I will present a priori and a posteriori error estimates for linear and quadratic DG methods in dimension 2 and 3 without the addition of bubble functions. I will consider two discrete sets, one with integral constraints (motivated as in the previous work)and another with point constraints at quadrature points. The analysis is carried out in a unified setting which holds for several DG methods with variable polynomial degree. Later, I will propose a new and simpler residual based a posteriori error estimator for finite element approximation of the elliptic obstacle problem. The results here are two fold.

Firstly, I will address the influence of the inhomogeneous Dirichlet boundary condition in a posteriori error control of the elliptic obstacle problem. Secondly, by rewriting the obstacle problem in an equivalent form, a simpler a posteriori error bounds which are free from min/max functions will be constructed. To accomplish this, I will construct a post-processed solution ˜uh of the discrete solution uh which satisfies the exact boundary conditions although the discrete solution uh may not satisfy. I will propose two post processing methods and analyse them. I will remark that the results known in the literature are either for the homogeneous Dirichlet boundary condition or that the estimator is only weakly reliable in the case of inhomogeneous Dirichlet boundary condition. Finally, conclusions and possible extensions for the future works will also be discussed.