Table 4 Notations of function spaces and operators

\(\mathfrak{W}(T)\)

\(\{ u | u \in L_{2}(0,T;H^{1}(I)) ,\frac{\mathrm{d}u}{\mathrm{d}t} \in L_{2}(0,T;H^{-1}(I)) \}\)

\({\mathfrak {S}}_{h} \equiv \{S_{h}\}_{h}\)

A family of finite-dimensional subspaces of \(H_{0}^{1}(I)\) with parameter h<1 that tends to 0

\(r(kA_{h})\)

Padé approximation \((I_{d} + kA_{h})^{-1}\)

\({\mathcal {G}}_{\infty}^{(k)}\)

\(\{ f \in L_{2}(I) | f \perp \operatorname{Span} \langle r(kA)u((N-1)k,\cdot ) \rangle \}\)

\({\mathcal {G}}_{h}^{(k)}\)

\(\{ f \in L_{2}(I) | f \perp \operatorname{Span} \langle r(kA_{h}) \overline{P}_{h}u((N-1)k,\cdot ) \rangle \}\)

\(S_{R}^{(k)}\)

\(\{ f \in L_{2}(I) | \|f\|=R, f \in {\mathcal {G}}_{\infty}^{(k)} \}\) with R satisfying \(R > \|\overline{P}_{h}\widetilde{\sigma}_{0} \|\)