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 ($P_α$) on the internal energy and the heat flux for a system governed by a mixed elliptic variational equality with a parameter $α>0$ 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 $\left(P_{h α}\right)$ and $\left(P_h\right)$ of the problems $\left(P_α\right)$ 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 $\left(P_{h α}\right)$ when the parameters $α$ goes to infinity and the parameter $h$ goes to zero simultaneously. We prove the convergence of the problems $\left(P_{h α}\right)$ to the problem $\left(P_h\right)$ when $α\rightarrow +\infty$, for each $h>0$. We study the convergence of the problems $\left(P_{h α}\right)$ and $\left(P_h\right)$, for each $α>0$, when $h \rightarrow 0^+$ obtaining a commutative diagram which relates the continuous and discrete optimal control problems $\left(P_{h α}\right),\left(P_α\right),\left(P_h\right)$ and $(P)$ by taking the limits $h \rightarrow 0^+$ and $α\rightarrow +\infty$ respectively. We also study the double convergence of $\left(P_{h α}\right)$ to $(P)$ when $(h, α) \rightarrow(0^+,+\infty)$ which represents the diagonal convergence in the above commutative diagram.