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We analyze the classical discontinuous Galerkin method for a general parabolic equation. Symmetric error estimates for schemes of arbitrary order are presented. The ideas we develop allow us to relax many assumptions freqently required in previous work. For example, we allow different discrete spaces to be used at each time step and do not require the… (More)

Numerical schemes to compute approximate solutions of the evolutionary Stokes and Navier-Stokes equations are studied. The schemes are discontinuous in time and conforming in space and of arbitrarily high order. Fully-discrete error estimates are derived and dependence of the viscosity constant is carefully tracked. It is shown that the errors are bounded… (More)

The velocity tracking problem for the evolutionary Navier–Stokes equations in two dimensions is studied. The controls are of distributed type and are submitted to bound constraints. First and second order necessary and sufficient conditions are proved. A fully discrete scheme based on the discontinuous (in time) Galerkin approach, combined with conforming… (More)

We consider fully discrete finite element approximations of a Robin optimal boundary control problem, constrained by linear parabolic PDEs with rough initial data. Conforming finite element methods for spatial discretization combined with discontinuous time-stepping Galerkin schemes are being used for the space-time discretization. Error estimates are… (More)

We consider fully discrete finite element approximations of a distributed optimal control problem, constrained by the evolutionary Stokes equations. Conforming finite element methods for spatial discretization combined with discontinuous time-stepping Galerkin schemes are being used for the space-time discretization. Error estimates are proved under weak… (More)

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