Volume 14, Issue 3

In this article we develop an a posteriori error estimator for discretized optimal control problems. We are interested in estimating the error in the control variable, measured in a natural norm. We prove an error representation formula involving only quantities at hand in a second-order optimization iteration, supposing a strong form of second-order sufficient condition. Possible generalization to the control-constrained case is indicated.

In the context of multicomponent flows, we are faced with PDE systems solutions combining waves whose speeds are several orders of magnitude apart. Of these waves, only the slow kinematic ones that represent transport phenomena are of concern to us. The fast acoustic ones, although uninteresting for the physical application considered, nevertheless impose a prohibitively small time-step (via the classical CFL restriction) if treated explicitly. This is why we propose to use a hybrid finite-volume scheme in which fast waves are handled by a linearized implicit formulation and slow waves remain explicitly solved. To further decrease the computational cost, mostly due to the complexity of nonlinear thermodynamical laws, we combine this method with a fully adaptive multiresolution scheme. At each time step, a multiscale analysis followed by the thresholding of small details enables us to discretize the solution over a time-varying adaptive grid, based on the smoothness of the relevant phenomenon. Particular attention is given to the extension of the reference scheme to non uniform grid and to the prediction strategy of the adaptive grid from one time-step to another. Finally, efficiency in terms of computing time requirements is studied in conjunction with the accuracy performances.

In this article an inherently parallel algorithm to approximate the operator exponential is presented. The construction is based on the integral representation of the operator exponential and allows arbitrarily large time steps constituting a major advantage compared to classical schemes. The algorithm rests on the efficient solution of several elliptic problems depending on a complex parameter. We prove Besov regularity of the solutions to these elliptic problems. This result implies the efficiency of adaptive methods applied to the elliptic problems and leads to a complexity estimate for the complete algorithm. In the numerical experiments the efficiency of the new scheme is demonstrated by comparison to a single step method of second order.