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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, a superlinear order of convergence for the control is obtained in the (Formula presented.) -norm; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to (Formula presented.) , extending the results in Rösch (Optim. Methods Softw. 21(1): 121–134, 2006). The theoretical findings are tested experimentally by means of numerical examples.