Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains

This article is concerned with the discretisation of the Stokes equations on time-dependent domains in an Eulerian coordinate framework. Our work can be seen as an extension of a recent paper by Lehrenfeld and Olshanskii (ESAIM: M2AN 53(2):585–614, 2019), where BDF-type time-stepping schemes are studied for a parabolic equation on moving domains. For space discretisation, a geometrically unfitted finite element discretisation is applied in combination with Nitsche’s method to impose boundary conditions. Physically undefined values of the solution at previous time-steps are extended implicitly by means of so-called ghost penalty stabilisations. We derive a complete a priori error analysis of the discretisation error in space and time, including optimal $$L^2(L^2)$$ L 2 ( L 2 ) -norm error bounds for the velocities. Finally, the theoretical results are substantiated with numerical examples.

Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation

In this paper, we propose $$L^2(J;H^1_0(\Omega ))$$ L 2 ( J ; H 0 1 ( Ω ) ) and $$L^2(J;L^2(\Omega ))$$ L 2 ( J ; L 2 ( Ω ) ) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.

Error estimates of finite volume method for Stokes optimal control problem

In this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal $L^{2}$ L 2 -norm error estimates. The approximate orders for the state, costate, and control variables are $O(h^{2})$ O ( h 2 ) in the sense of $L^{2}$ L 2 -norm. Furthermore, we derive $H^{1}$ H 1 -norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.

Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint

<p style='text-indent:20px;'>This paper is concerned with the Galerkin spectral approximation of an optimal control problem governed by the elliptic partial differential equations (PDEs). Its objective functional depends on the control variable governed by the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm constraint. The optimality conditions for both the optimal control problem and its corresponding spectral approximation problem are given, successively. Thanks to some lemmas and the auxiliary systems, a priori error estimates of the Galerkin spectral approximation problem are established in detail. Moreover, a posteriori error estimates of the spectral approximation problem are also investigated, which include not only <inline-formula><tex-math id="M2">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>-norm error for the state and co-state but also <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm error for the control, state and costate. Finally, three numerical examples are executed to demonstrate the errors decay exponentially fast.</p>