Numerical analysis of a family of simultaneous distributed-boundary mixed elliptic optimal control problems and their asymptotic behaviour through a commutative diagram and error estimates

Abstract

In this paper, we consider a family of simultaneous distributed-boundary optimal control problems (Pα) on the internal energy and the heat flux for a system governed by a mixed elliptic variational equality with a parameter α>0 (the heat transfer coefficient on a portion of the boundary of the domain) and a simultaneous distributed-boundary optimal control problem (P) governed also by an elliptic variational equality with a Dirichlet boundary condition on the same portion of the boundary. We formulate discrete approximations Phα and Ph of the optimal control problems Pα and (P) respectively, for each h>0 and for each α>0, through the finite element method with Lagrange’s triangles of type 1 with parameter h (the longest side of the triangles). The goal of this paper is to study the convergence of this family of discrete simultaneous distributed-boundary mixed elliptic optimal control problems Phα when the parameters α goes to infinity and the parameter h goes to zero simultaneously. We prove the convergence of the family of discrete problems Phα to the discrete problem Ph when α→+∞, for each h>0, in adequate functional spaces. We study the convergence of the discrete problems Phα and Ph, for each α>0, when h→0+ obtaining a commutative diagram which relates the continuous and discrete simultaneous distributed-boundary mixed elliptic optimal control problems Phα,Pα,Ph and (P) by taking the limits h→0+ and α→+∞ respectively. We also study the double convergence of Phα to (P) when (h,α)→(0+,+∞) which represents the diagonal convergence in the above commutative diagram.