2023/03/29 by Carolina M. Bollo, Bollo, Carolina M., Claudia M. Gariboldi +3
Computer Science · Engineering · #35J88 #35R35 #49J40 #49J45 #65K15 #65N30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2303.16600
openalex publication_date 2023/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 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 (Ph α) and (Ph) of the 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 (Ph α) when the parameters α goes to infinity and the parameter h goes to zero simultaneously. We prove the convergence of the problems (Ph α) to the problem (Ph) when α→ +∞, for each h>0. We study the convergence of the problems (Ph α) and (Ph), for each α>0, when h → 0+ obtaining a commutative diagram which relates the continuous and discrete optimal control problems (Ph α),(Pα),(Ph) and (P) by taking the limits h → 0+ and α→ +∞ respectively. We also study the double convergence of (Ph α) to (P) when (h, α) →(0+,+∞) which represents the diagonal convergence in the above commutative diagram.