A Discontinuous Galerkin Time-Stepping Scheme for the Velocity Tracking Problem

Eduardo Casas,Konstantinos Chrysafinos

doi:10.1137/110829404

Eduardo Casas, Konstantinos Chrysafinos

Open Access

https://doi.org/10.1137/110829404

Copy DOI

Abstract

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 finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, $\tau$ and $h$, respectively, satisfy $\tau \leq Ch^2$, then $L^2$ error estimates of order $O(h)$ are proved for the difference between the locally optimal controls and their discrete approximations.

Highlights

In this paper we prove some error estimates for the numerical approximation of a distributed optimal control problem governed by the evolution Navier–Stokes equations with pointwise control constraints
Yu denotes the solution of the two-dimensional evolution Navier–Stokes equations yt − νΔy + (y · ∇)y + ∇p = f + u in ΩT = (0, T ) × Ω, div y = 0 in ΩT, y(0) = y0 in Ω, y = 0 on ΣT = (0, T ) × Γ, and Uad is the set of feasible controls, defined for −∞ ≤ αj < βj ≤ +∞, j = 1, 2, by Uad = {u ∈ L2(0, T ; L2(Ω)) : αj ≤ uj(t, x) ≤ βj a.e. (t, x) ∈ ΩT, j = 1, 2}
We focus on the lowest case of polynomial approximation in time, due to the low regularity imposed by the nature of our optimal control problem

Summary

Introduction

In this paper we prove some error estimates for the numerical approximation of a distributed optimal control problem governed by the evolution Navier–Stokes equations with pointwise control constraints. Yu denotes the solution of the two-dimensional evolution Navier–Stokes equations (1.1). Yt − νΔy + (y · ∇)y + ∇p = f + u in ΩT = (0, T ) × Ω, div y = 0 in ΩT , y(0) = y0 in Ω, y = 0 on ΣT = (0, T ) × Γ, and Uad is the set of feasible controls, defined for −∞ ≤ αj < βj ≤ +∞, j = 1, 2, by Uad = {u ∈ L2(0, T ; L2(Ω)) : αj ≤ uj(t, x) ≤ βj a.e.

EDUARDO CASAS AND KONSTANTINOS CHRYSAFINOS

We also consider a subspace Yh of Zh defined by

Findings

Nτ τn