Discrete Maximal Lp Regularity for Finite Element Operators

Let ₕ > 0 be a family of elliptic finite element operators. Let $I=[0,T]$ and consider the problem uₕ'(t)-Aₕuₕ(t)=fₕ(t), t∈ I, uₕ(0)=0. In this paper, we show that for 1 < p < ∞ the solution of that problem satisfies the estimate \[ {u_h'}{L_p(I;L_p(Ω))}+{A_hu_h}{L_p(I;L_p(Ω))}≤ C{f_h}{L_p(I;L_p(Ω))}, \] where C is independent of the parameter h and fₕ. In this case ₕ > 0 is said to have discrete maximal Lₚ regularity.