2024/02/07 by Richard Löscher, Olaf Steinbach
Engineering · Computer Science · #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering #Computational Fluid Dynamics and Aerodynamics
paper · doi:10.1137/22m1532962
.We consider space-time tracking-type distributed optimal control problems for the wave equation in the space-time domain \(Q:= Ω × (0,T) ⊂ℝn+1\), where the control is assumed to be in the energy space \([H0;,01,1(Q)]^*\), rather than in \(L2(Q)\), which is more common. While the latter ensures a unique state in the Sobolev space \(H1,10;0,(Q)\), this does not define a solution isomorphism. Hence, we use an appropriate state space \(X\) such that the wave operator becomes an isomorphism from \(X\) onto \([H0;,01,1(Q)]^*\). Using space-time finite element spaces of piecewise linear continuous basis functions on completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error \(‖\widetildeu\varrho h-u‖L2(Q)\) between the computed space-time finite element solution \(\widetildeu\varrho h\) and the target function \(u\) with respect to the regularization parameter \(\varrho\), and the space-time finite element mesh size \(h\), depending on the regularity of the desired state \(u\). These estimates lead to the optimal choice \(\varrho =h2\) in order to define the regularization parameter \(\varrho\) for a given space-time finite element mesh size \(h\) or to determine the required mesh size \(h\) when \(\varrho\) is a given constant representing the costs of the control. The theoretical results will be supported by numerical examples with targets of different regularities, including discontinuous targets. Furthermore, an adaptive space-time finite element scheme is proposed and numerically analyzed.Keywordsdistributed optimal control problemwave equationspace-time finite element methodsa priori error estimatesadaptivityMSC codes49M4135L0565M1565M60