The main goal of this paper is to analyze a family "simplest possible" initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here first step toward rigorous validation these results. Namely, we consider system corresponding self-similar solution, restricted bounded domain with smooth boundary. ...

‎In this paper‎, ‎we study spectral element approximation for a constrained‎ ‎optimal control problem in one dimension‎. ‎The equivalent a posteriori error estimators are derived for‎ ‎the control‎, ‎the state and the adjoint state approximation‎. ‎Such estimators can be used to‎ ‎construct adaptive spectral elements for the control problems.

Abstract We prove residual-type a posteriori error estimates in the maximum norm for linear scalar elliptic convection–diffusion problem that may be singularly perturbed. Similar analysis energy by Verfürth indicates dual of convective derivative must added to natural order residual estimator reliable and efficient. show situation is similar norm. In particular, we define mesh-dependent weighte...