Crank–Nicolson finite element methods using symmetric stabilization with an application to optimal control problems subject to transient advection–diffusion equations

A new error estimate for a fully finite element discretization scheme for parabolic equations using Crank-Nicolson method

“…L 2 (H 1 )-error estimate. Under the regularity assumption u ∈ C(0, T ; H k+1 (Ω)) and u ttt ∈ L 2 (0, T ; L 2 (Ω)), the following L 2 (H 1 )-error estimate holds (see [1]):…”

“…We considered a Crank-Nicolson finite element scheme for the nonstationary heat equation. The existing literature, see for instance [1], [3], [4] and the references therein, concerning the convergence of the error state that the convergence order is h k+1 + τ 2 or h k + τ 2 in the discrete norms L ∞ (L 2 ) or L 2 (H 1 ), respectively, and [2] states that for k = 1 (piecewise linear finite elements) the order is h + τ 2 in L ∞ (H 1 ). We proved in the present contribution that the error is of order h k+1 + τ 2 in the discrete norm of W 1,∞ (L 2 ).…”

A survey of numerical methods for convection–diffusion optimal control problems

“…In addition, edge stabilized Galerkin method combined with CrankNicolson method was utilized to approximate unconstrained transient convection-diffusion optimal control problems in [12]. The optimize then discretize approach is used to derive the discrete system for the state, adjoint state and control.…”

A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

“…Many numerical methods and algorithms were designed. We refer to [1,2,20,22,23] for stabilization methods, [21,24,25] for discontinuous Galerkin methods, and [7,8] for the methods of characteristics. For more numerical methods one can refer to the references cited herein.…”