Abstract
We present some resolvent estimates of elliptic differential and finite‐element operators in pairs of function spaces, for which the first space in a pair is endowed with stronger norm. In this work we deal with estimates in (Lebesgue, Lebesgue), (Hölder, Lebesgue), and (Hölder, Hölder) pairs of norms. In particular, our results are useful for the stability and error analysis of semidiscrete and fully discrete approximations to parabolic partial differential problems with rough and distribution‐valued data.
Highlights
The main objective is to get some new resolvent estimates of elliptic finite-element operators which are intended for use in applications of the finite-element method to parabolic initial boundary value problems
We start with showing resolvent estimates for differential operators
Continuous and discrete, we make an emphasis on deriving such estimates in pairs of function spaces, for which the first space in a pair is endowed with stronger norm
Summary
The known resolvent estimates allow one to show that e−tAh is uniformly bounded with respect to h ∈ We present, as mentioned above, resolvent estimates for the elliptic differential operator A in pairs of function spaces It will be convenient, given φ ∈ (0, π /2), to denote. An interpolation argument (see, e.g., Triebel [35, Theorem 1.3.3]), applied to (3.4) and (3.3) with p = ∞ (the last one clearly holds for all v ∈ Ꮿ0), implies for any fixed ξ ∈ [0, 1), with the aid of (2.3) (for ξ = 0 this is a direct consequence of (3.3) with p = ∞), Phv ξ ≤ C|v|ξ for v ∈ Ꮿξ.
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