Numerical solution of elliptic distributed optimal control problems with boundary value tracking

TL;DR

This paper develops finite element methods for solving elliptic boundary control problems with boundary value tracking, providing error estimates and efficient solvers validated by numerical experiments.

Contribution

It introduces a tensor-product finite element discretization for elliptic boundary control problems with boundary tracking, including error analysis and fast solution techniques.

Findings

Optimal discretization error estimates derived

Fast solvers for the discretized problems developed

Numerical experiments confirm theoretical results

Abstract

We consider some boundary value tracking optimal control problem constrained by a Neumann boundary value problem for some elliptic partial differential equation where the control acts as right-hand side. This optimal control problem can be reformulated asa state-based variational problem that is the starting point for the finite element discretizion. In this paper, we only consider atensor-product finite element discretizion for which optimal discretization error estimates and fast solvers can be derived.Numerical experiments illustrate the theoretical results quantitatively.