Discontinuous Galerkin approximations for an optimal control problem of three-dimensional Navier-Stokes-Voigt equations (Q2194041)

Some error estimates are proved for the numerical approximation of a distributed optimal control problem \[ \min \underbrace{J(y,u)}_{u\in U_{\alpha,\beta}} = \frac{\alpha_T}{2}\int_\Omega |y(x,T)-y_T(x)|^2dx+ \frac{\alpha_Q}{2}\iint_Q |y(x,t)-y_Q(x,t)|^2dxdt + \frac{\gamma}{2}\iint_Q |u(x,t)|^2dxdt \] governed by the 3D Navier-Stokes-Voigt equations \begin{align*} y_t -\nu \Delta y -\alpha^2 \Delta y_t + (y\cdot \nabla )y + \nabla p &= u, \; x\in \Omega, \,t > 0, \\ \nabla \cdot y & = 0, \; x\in \Omega,\, t > 0, \\ y(x,T) & = 0, \; x\in \partial \Omega,\, t > 0, \\ y(x,t) & = y_0(x,t), \,x \in \Omega, \end{align*} with pointwise set of admissible control constraints defined by \[ U_{\alpha,\beta} = \{ u\in L^2(Q): \alpha_j \le u_j(x,t) \le \beta_j \; a.e. (x,t) \in Q, \; j=1,2,3 \}, \] where \(\Omega\) be an open bounded domain in \(\mathbb{R}^3\) with boundary \(\partial \Omega\) and \(Q=\Omega \times (0,T)\) is the space-time cylinder. The velocity and pressure are denoted by \(y(x,t)\) and \(p(x,t)\), respectively, \(\nu > 0\) is the kinematic viscosity coefficient, and \(\alpha\ne 0\) is the length-scale parameter characterizing the elasticity of the fluid. 3D Navier-Stokes-Voigt equations are discretized in time with the discontinuous time-stepping Galerkin scheme for the piecewise constant time combined with standard conforming finite element subspaces for the discretization in space. It is proven that the space-time error estimates of order \({\mathcal O} (\sqrt{\tau} + h)\), where \(\tau\) and \(h\) are respectively the time and space discretization parameters.