MAXIMUM L2-CONVERGENCE RATES OF THE CRANK–NICOLSON SCHEME TO THE STOKES INITIAL VALUE PROBLEM

摘要

Let A denote the Stokes operator and DAα the domain of its fractional powers Aα. We consider the homogeneous Stokes initial value problem with initial data u(0) = u0 ∈ DA1+ε , ε ∈ (0, 1). For Stokes-like equations the range ε ∈ (0, 1 4 ) is of special interest, as any solution derived from ε ≥ 14 would necessarily have to satisfy an additional, in practice unverifiable compatibility condition at time t = 0. Approximating any strong solution u ∈ C0([0,∞),DA1+ε ) in time direction on a finite time interval [0, T] with a Crank–Nicolson scheme, we show convergence of order O( τ2 t1?ε ) which is maximal for the assumed regularity of the data and reflects the loss of regularity as t → 0. The error estimates are derived by energy and semigroup methods combined with a parabolic duality argument.