Finite element error estimates for an optimal control problem governed by the Burgers equation

TL;DR

This paper establishes a-priori finite element error estimates for a control problem governed by the Burgers equation, demonstrating superlinear convergence and improved rates under certain regularity assumptions, validated through numerical tests.

Contribution

It provides new error estimates for finite element approximations of Burgers control problems, including superlinear and improved convergence rates under regularity assumptions.

Findings

Superlinear $L^2$ convergence for control approximation

Improved $h^{3/2}$ error estimate under regularity assumptions

Numerical experiments confirm theoretical error estimates

Abstract

We derive a-priori error estimates for the finite-element approximation of a distributed optimal control problem governed by the steady one-dimensional Burgers equation with pointwise box constraints on the control. Here the approximation of the state and the control is done by using piecewise linear functions. With this choice, an $L^2$ superlinear order of convergence for the control is obtained; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to $h^{3/2}$. The theoretical findings are tested experimentally by means of numerical examples.