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

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.