Dmitriy Leykekhman

Local Error Estimates for SUPG Solutions of Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems

We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimization problems governed by an advection-dominated PDE, the convergence properties of the SUPG method is substantially different from the convergence properties of the SUPG method applied for the solution of an advection-dominated PDE. The reason is that the solution of the optimal control problem involves another advection-dominated PDE, the so-called adjoint equation, whose advection field is just the negative of the advection of the governing PDEs. For the solution of the optimal control problem, a coupled system involving both the original governing PDE as well as the adjoint PDE must be solved. We show that in the presence of a boundary layer, the local error between the solution of the SUPG discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer. In the presence of interior layers, we prove optimal convergence rates for the local error in a region away from the layer between the solution of the SUPG discretized optimal control problems and the solution of the infinite dimensional problem. Numerical examples are presented to illustrate some of the theoretical results.

Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods

A model second-order elliptic equation on a general convex polyhedral domain in three dimensions is considered. The aim of this paper is twofold: First sharp Hölder estimates for the corresponding Green’s function are obtained. As an applications of these estimates to finite element methods, we show the best approximation property of the error in

Investigation of Commutative Properties of Discontinuous Galerkin Methods in PDE Constrained Optimal Control Problems

{W^1_{\infty}}

Optimal A Priori Error Estimates of Parabolic Optimal Control Problems with Pointwise Control

. In contrast to previously known results,

Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra.

{W_p^{2}}

Finite Element Pointwise Results on Convex Polyhedral Domains

regularity for p > 3, which does not hold for general convex polyhedral domains, is not required. Furthermore, the new Green’s function estimates allow us to obtain localized error estimates at a point.