Our port-reduced SCRBE approximation is equipped with efficiently computable *a posteriori* error bounds and estimators that provide certificates for the error in the approximation with respect to the underlying global FE discretization. We employ in this paper the energy-norm and compliance output bound developed in [12], and we present the main ingredients and certain extensions below. We furthermore sharpen the bounds by consideration of a multi-reference parameter bound conditioner.

The error in our approximation derives from two sources: port reduction and RB approximation. Below we first address the error due to port reduction, that is to say, the case in which the error due to RB approximation is zero. In this case the error bound presentation simplifies significantly and in particular admits a pure functional interpretation. We then subsequently perturb the equivalent algebraic interpretation to provide a bound for the general case in which the error due to RB approximation is non-zero.

### Port reduction error contribution

We assume in this subsection only that the only source of error is port reduction and hence that there is no RB-induced error. We introduce the function

\begin{array}{l}{u}^{\mathrm{PR}}\left(\mu \right)=\sum _{i=1}^{I}{b}^{f;h}\left({\mu}_{i}\right)+\sum _{p=1}^{{n}^{\Gamma}}\sum _{k=1}^{{n}_{\mathrm{A},p}^{\Gamma}}{\mathbb{U}}_{p,k}^{\mathrm{PR}}\left(\mu \right){\mathrm{\Phi}}_{p,k}\left(\mu \right)\in {X}^{h}\left(\Omega \right),\end{array}

(71)

which satisfies

\begin{array}{l}a\left({u}^{\mathrm{PR}}\right(\mu ),v;\mu )=f(v;\mu ),\phantom{\rule{1em}{0ex}}\forall v\in {\mathcal{S}}^{\mathrm{PR}}\left(\Omega \right);\end{array}

(72)

hence *u*^{PR}(*μ*) is the port-reduced approximation to *u*^{h}(*μ*) obtained in the absence of RB errors. We note that we may (as in (25)) replace the skeleton space {\mathcal{S}}^{\mathrm{PR}}\left(\Omega \right) in (72) by the skeleton space

\begin{array}{l}{\mathcal{S}}_{\mathrm{symm}}^{\mathrm{PR}}\left(\Omega \right)=\mathrm{span}\left\{{\mathrm{\Phi}}_{p,k}\right(\mu ),\phantom{\rule{1em}{0ex}}1\le k\le {n}_{\mathrm{A},p}^{\Gamma},1\le p\le {n}^{\Gamma}\}\subset {\mathcal{S}}_{\mathrm{symm}}\left(\Omega \right),\end{array}

(73)

and thus *u*^{PR}(*μ*)∈*X*^{h}(*Ω*) also satisfies

\begin{array}{l}a\left({u}^{\mathrm{PR}}\right(\mu ),v;\mu )=f(v;\mu ),\phantom{\rule{1em}{0ex}}\forall v\in {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{PR}}\left(\Omega \right);\end{array}

(74)

note that {u}^{\mathrm{PR}}\left(\mu \right)\notin {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{PR}}\left(\Omega \right) because of the source bubble terms *b*^{f;h}(*μ*_{
i
}) in (71).

We define the associated (RB-error-free) error field as

\begin{array}{ll}{e}_{0}^{h}\left(\mu \right)& \equiv {u}^{h}\left(\mu \right)-{u}^{\mathrm{PR}}\left(\mu \right)\phantom{\rule{2em}{0ex}}\\ =\sum _{p=1}^{{n}^{\Gamma}}\left(\sum _{k=1}^{{n}_{\mathrm{A},p}^{\Gamma}}\left({\mathbb{U}}_{p,k}\right(\mu )-{\mathbb{U}}_{p,k}^{\mathrm{PR}}(\mu \left)\right){\mathrm{\Phi}}_{p,k}\left(\mu \right)+\sum _{k={n}_{\mathrm{A},p}^{\Gamma}+1}^{{\mathcal{N}}_{p}^{\Gamma}}{\mathbb{U}}_{p,k}\left(\mu \right){\mathrm{\Phi}}_{p,k}\left(\mu \right)\right),\phantom{\rule{2em}{0ex}}\end{array}

(75)

(in which the subscript _{0} refers to the case of zero RB error contribution) and we note that {e}_{0}^{h}\left(\mu \right)\in {\mathcal{S}}_{\mathrm{symm}}\left(\Omega \right) because the source bubble contributions from *u*^{h}(*μ*) and *u*^{PR}(*μ*) cancel. Our goal is to develop a bound for the energy \parallel {e}_{0}^{h}\left(\mu \right){\parallel}_{\mu}, where

\begin{array}{l}\parallel \xb7{\parallel}_{\mu}\equiv \sqrt{a(\xb7,\xb7;\mu )}\end{array}

(76)

is the usual energy norm. From (25) and (74) we see that

\begin{array}{l}a\left({e}_{0}^{h}\right(\mu ),v;\mu )=f\left(v\right)-a\left({u}^{\mathrm{PR}}\right(\mu ),v;\mu ),\phantom{\rule{1em}{0ex}}\forall v\in {\mathcal{S}}_{\mathrm{symm}}\left(\Omega \right);\end{array}

(77)

this error-residual relationship is the point of departure for our error bound development.

Thanks to coercivity and symmetry of *a*(·,·;*μ*), the error field {e}_{0}^{h}\left(\mu \right) admits the equivalent definition

\begin{array}{ll}{e}_{0}^{h}\left(\mu \right)& =arg\underset{v\in {\mathcal{S}}_{\mathrm{symm}}\left(\Omega \right)}{min}\mathcal{J}(v;\mu ),\phantom{\rule{2em}{0ex}}\end{array}

(78)

where

\begin{array}{ll}\mathcal{J}(v;\mu )& \equiv \frac{1}{2}a(v,v)-\left(f\left(v\right)-a\left({u}^{\mathrm{PR}}\right(\mu ),v;\mu )\right),\phantom{\rule{2em}{0ex}}\end{array}

(79)

and furthermore \parallel {e}_{0}^{h}\left(\mu \right){\parallel}_{\mu}^{2}=a\left({e}_{0}^{h}\right(\mu ),{e}_{0}^{h}(\mu );\mu )=-2\mathcal{J}\left({e}_{0}^{h}\right(\mu );\mu ). We now relax the minimization (78) by consideration of a discontinuous (non-conforming) skeleton space

\begin{array}{l}{\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right)\equiv {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{PR}}\left(\Omega \right)\\ \oplus \mathrm{span}\left\{{\varphi}_{i,j,k}\right(\mu ),\phantom{\rule{1em}{0ex}}({n}_{\mathrm{A},i,j}^{\gamma}+1)\le k\le & {\mathcal{N}}_{i,j}^{\gamma},1\le j\le {n}_{i}^{\gamma},1\le i\le I\}\\ \phantom{\rule{2em}{0ex}}\equiv \mathrm{span}\left\{{\mathrm{\Phi}}_{i}^{\prime}\right(\mu ),\phantom{\rule{1em}{0ex}}1\le i\le {n}_{\mathrm{NC}}\},\phantom{\rule{2em}{0ex}}\end{array}

(80)

in which the basis functions {\mathrm{\Phi}}_{i}^{\prime}\left(\mu \right), 1≤*i*≤*n*_{NC}, merely represent a re-indexing of the basis functions *Φ*_{p,k}(*μ*), 1\le k\le {n}_{p}^{\Gamma}, 1≤*p*≤*n*^{Γ}, and *ϕ*_{i,j,k}(*μ*), ({n}_{\mathrm{A},i,j}^{\gamma}+1)\le k\le {\mathcal{N}}_{i,j}^{\gamma},1\le j\le {n}_{i}^{\gamma},1\le i\le I. Note that the *ϕ*_{i,j,k}(*μ*) represent independent (non-conforming) degrees of freedom local to component *i*. The dimension of {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right) is

\begin{array}{l}{n}_{\mathrm{NC}}={n}_{\mathrm{A}}+\sum _{i=1}^{I}\sum _{j=1}^{{n}_{i}^{\gamma}}{\mathcal{N}}_{i,j}^{\gamma}-{n}_{\mathrm{A},i,j}^{\gamma}\ge {n}_{\mathrm{SC}};\end{array}

(81)

note that {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right)\supseteq {\mathcal{S}}_{\mathrm{symm}}\left(\Omega \right). We also define a non-conforming skeleton space {\mathcal{S}}^{\mathrm{NC}}\supseteq \mathcal{S}\left(\Omega \right) as

\begin{array}{l}{\mathcal{S}}^{\mathrm{NC}}\left(\Omega \right)\equiv {\mathcal{S}}^{\mathrm{PR}}\left(\Omega \right)\\ \oplus \mathrm{span}\{{\psi}_{i,j,k},\phantom{\rule{1em}{0ex}}({n}_{\mathrm{A},i,j}^{\gamma}+1)\le k\le {\mathcal{N}}_{i,j}^{\gamma},1\le j\le & {n}_{i}^{\gamma},1\le i\le I\}\\ \phantom{\rule{1em}{0ex}}\equiv \mathrm{span}\{{\Psi}_{i}^{\prime},\phantom{\rule{1em}{0ex}}1\le i\le {n}_{\mathrm{NC}}\}.\end{array}

(82)

Hence for

\begin{array}{l}{e}_{0}^{\mathrm{NC}}\left(\mu \right)\equiv arg\underset{v\in \underset{\mathrm{symm}}{\overset{\mathrm{NC}}{\mathcal{S}}}\left(\Omega \right)}{min}\mathcal{J}(v;\mu )\end{array}

(83)

(recall the “broken” definition of *a*(·,·;*μ*) in (8)) we must have

\begin{array}{l}\mathcal{J}\left({e}_{0}^{\mathrm{NC}}\right(\mu );\mu )\le \mathcal{J}\left({e}_{0}^{h}\right(\mu );\mu )\end{array}

(84)

and thus a\left({e}_{0}^{\mathrm{NC}}\right(\mu ),{e}_{0}^{\mathrm{NC}}(\mu );\mu )\ge a\left({e}_{0}^{h}\right(\mu ),{e}_{0}^{h}(\mu );\mu ). This first relaxation of (78) not only provides a bound on the energy of the error field, but also accommodates efficient bound calculation thanks to the non-conforming space {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right).

A second relaxation step is required to obtain a computationally tractable error bound. To this end we introduce a bound conditioner, the bilinear form {b}_{\mu}:{\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right)\times {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\to \mathbb{R}, defined as

\begin{array}{l}{b}_{\mu}(\xb7,\xb7)\equiv a(\xb7,\xb7;{\mu}_{\mathrm{ref}}^{\mu})\end{array}

(85)

for a reference parameter value {\mu}_{\mathrm{ref}}^{\mu}\in \mathcal{D}. Note that here, *b*_{
μ
}(·,·) depends implicitly on *μ* through the parameter-dependent reference parameter {\mu}_{\mathrm{ref}}^{\mu}. In fact, an important innovation of this paper is this multi-reference parameter bound conditioner: in the online stage, we optimally select {\mu}_{\mathrm{ref}}^{\mu} from a database of a few candidate reference parameters (through a discrete enumeration procedure); we discuss the selection of {\mu}_{\mathrm{ref}}^{\mu} further in the “Computational procedures” subsection below. We also define

\begin{array}{l}{\lambda}_{\mathrm{min}}\left(\mu \right)\equiv \underset{v\in \underset{\mathrm{symm}}{\overset{\mathrm{NC}}{\mathcal{S}}}\left(\Omega \right)}{min}\frac{a(v,v;\mu )}{{b}_{\mu}(v,v)}.\end{array}

(86)

We then introduce a modified functional

\begin{array}{l}{\mathcal{J}}_{b}(v;\mu )\equiv \frac{{\lambda}_{\mathrm{min}}\left(\mu \right)}{2}{b}_{\mu}(v,v)-\left(\phantom{\rule{0.3em}{0ex}}f\left(v\right)-a\left({u}^{\mathrm{PR}}\right(\mu ),v;\mu )\right),\phantom{\rule{1em}{0ex}}\forall v\in {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right),\end{array}

(87)

and we consider the minimization

\begin{array}{l}{\u0113}_{0}^{\mathrm{NC}}\left(\mu \right)\equiv arg\underset{v\in \underset{\mathrm{symm}}{\overset{\mathrm{NC}}{\mathcal{S}}}\left(\Omega \right)}{min}{\mathcal{J}}_{b}(v;\mu ).\end{array}

(88)

By the definition of *λ*_{min}(*μ*) in (86) it is clear that {\mathcal{J}}_{b}(v;\mu )\le \mathcal{J}(v;\mu ) for all v\in {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right). Thus in particular, since {\u0113}_{0}^{\mathrm{NC}}\left(\mu \right) is the minimizer,

\begin{array}{l}{\mathcal{J}}_{b}\left({\u0113}_{0}^{\mathrm{NC}}\right(\mu );\mu )\le {\mathcal{J}}_{b}\left({e}_{0}^{\mathrm{NC}}\right(\mu );\mu )\le \mathcal{J}\left({e}_{0}^{\mathrm{NC}}\right(\mu );\mu )\le \mathcal{J}\left({e}_{0}\right(\mu );\mu ),\end{array}

(89)

where the last inequality follows from (84). Consequently, we obtain the energy-norm error bound

\begin{array}{l}{\lambda}_{\mathrm{min}}\left(\mu \right){b}_{\mu}\left({\u0113}_{0}^{\mathrm{NC}}\right(\mu ),{\u0113}_{0}^{\mathrm{NC}}(\mu \left)\right)\ge a\left({e}_{0}^{h}\right(\mu ),{e}_{0}^{h}(\mu );\mu )\end{array}

(90)

where the field variable {\u0113}_{0}^{\mathrm{NC}}\left(\mu \right)\in {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right) — a presumably rather good approximation to the original error field {e}_{0}^{h}\left(\mu \right)[12] — satisfies the elliptic problem {b}_{\mu}\left({\u0113}_{0}^{\mathrm{NC}}\right(\mu ),v)={\lambda}_{\mathrm{min}}{\left(\mu \right)}^{-1}\left(f\right(v;\mu )-a({u}^{\mathrm{PR}}\left(\mu \right),v;\mu \left)\right) for all v\in {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right). Equivalently, because of the Galerkin orthogonality in (19),

\begin{array}{l}{b}_{\mu}\left({\u0113}_{0}^{\mathrm{NC}}\right(\mu ),v)=\frac{1}{{\lambda}_{\mathrm{min}}\left(\mu \right)}\left(f(v;\mu )-a\left({u}^{\mathrm{PR}}\right(\mu ),v;\mu )\right),\phantom{\rule{1em}{0ex}}\forall v\in {\mathcal{S}}^{\mathrm{NC}}\left(\Omega \right).\end{array}

(91)

Thanks to incorporation of the modes related to rigid-body motion in our port space bases (presuming {n}_{\mathrm{A},p}^{\Gamma}\ge 6 on all global ports *Γ*_{
p
}, 1≤*p*≤*n*^{Γ}) we expect in general (and for a particular system, we computationally verify) that (91) is well-posed; for the simpler class of problems with scalar-valued fields we demonstrate this well-posedness in [12]. The RB-error-free bound given in (90) (together with (91)) is the basis on which we in the next subsection extend our error estimation framework to the general case of non-zero RB errors and furthermore to certain outputs of interest.

In order to implement this error bound, and to facilitate incorporation of RB-induced error contributions, we now interpret the error bound (90) in terms of algebraic quantities. To this end, we first note that, for any v\left(\mu \right)=\sum _{p=1}^{{n}^{\Gamma}}\sum _{k=1}^{{\mathcal{N}}_{p}^{\Gamma}}{\mathbb{V}}_{p,k}\left(\mu \right){\mathrm{\Phi}}_{p,k}\left(\mu \right) — that is, for any v\left(\mu \right)\in {\mathcal{S}}_{\mathrm{symm}}\left(\Omega \right) with coefficients \mathbb{V}\left(\mu \right) — we have

\begin{array}{l}a\left(v\right(\mu ),v(\mu );\mu )=\mathbb{V}{\left(\mu \right)}^{\mathrm{T}}\mathbb{A}\left(\mu \right)\mathbb{V}\left(\mu \right);\end{array}

(92)

we refer to the right-hand side of (92) as the “Schur energy” of \mathbb{V}\left(\mu \right). It shall prove convenient to introduce the zero-extended solution vectors

\begin{array}{l}{\widehat{\mathbb{U}}}_{0}^{\mathrm{PR}}\left(\mu \right)\equiv \left[\begin{array}{c}{\mathbb{U}}^{\mathrm{PR}}\left(\mu \right)\\ \mathbf{\text{0}}\end{array}\right]\in {\mathbb{R}}^{{n}_{\mathrm{SC}}},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{\widehat{\mathbb{U}}}_{0}^{\mathrm{PR},\mathit{\text{NC}}}\left(\mu \right)\equiv \left[\begin{array}{c}{\mathbb{U}}^{\mathrm{PR}}\left(\mu \right)\\ \mathbf{\text{0}}\end{array}\right]\in {\mathbb{R}}^{{n}_{\mathrm{NC}}},\end{array}

(93)

in which all but the first *n*_{A} entries are explicitly set to zero. We also define the error coefficient vector

\begin{array}{l}{\mathbb{E}}_{0}\left(\mu \right)\equiv \mathbb{U}\left(\mu \right)-{\widehat{\mathbb{U}}}_{0}^{\mathrm{PR}}\left(\mu \right)\in {\mathbb{R}}^{{n}_{\mathrm{SC}}}\end{array}

(94)

such that the error (75) can be written {e}_{0}^{h}\left(\mu \right)=\sum _{p=1}^{{n}^{\Gamma}}\sum _{k=1}^{{\mathcal{N}}_{p}^{\Gamma}}{\mathbb{E}}_{0;p,k}\left(\mu \right){\mathrm{\Phi}}_{p,k}\left(\mu \right). Note here, we tacitly interpret (without loss of generality) \mathbb{U}\left(\mu \right) such that the first *n*_{A} entries correspond to the *n*_{A} active degrees of freedom. The algebraic version of the error residual equation (77) is

\begin{array}{l}\mathbb{A}\left(\mu \right){\mathbb{E}}_{0}\left(\mu \right)={\mathbb{R}}_{0}\left(\mu \right),\end{array}

(95)

where the residual vector is given as

\begin{array}{l}{\mathbb{R}}_{0}\left(\mu \right)=\mathbb{F}\left(\mu \right)-\mathbb{A}\left(\mu \right){\widehat{\mathbb{U}}}_{0}^{\mathrm{PR}}\left(\mu \right);\end{array}

(96)

note that, thanks to (92) and the fact that {e}_{0}^{h}\left(\mu \right)\in {\mathcal{S}}_{\mathrm{symm}}\left(\Omega \right), (95) is equivalent to (77).

We now introduce a non-conforming matrix {\mathbb{A}}^{\mathrm{NC}}\left(\mu \right)\in {\mathbb{R}}^{{n}_{\mathrm{NC}}\times {n}_{\mathrm{NC}}} and vector {\mathbb{F}}^{\mathrm{NC}}\left(\mu \right)\in {\mathbb{R}}^{{n}_{\mathrm{NC}}} as

\begin{array}{ll}{\mathbb{A}}_{i,j}^{\mathrm{NC}}\left(\mu \right)& =a\left({\mathrm{\Phi}}_{j}^{\prime}\right(\mu ),{\mathrm{\Phi}}_{i}^{\prime}(\mu );\mu ),\phantom{\rule{2em}{0ex}}\end{array}

(97)

\begin{array}{ll}{\mathbb{F}}_{i}^{\mathrm{NC}}\left(\mu \right)& =f\left({\mathrm{\Phi}}_{i}^{\prime}\right(\mu );\mu )-\sum _{l=1}^{I}a\left({b}_{l}^{f;h}\right({\mu}_{l}),{\mathrm{\Phi}}_{i}^{\prime}(\mu );\mu ),\phantom{\rule{2em}{0ex}}\end{array}

(98)

for 1≤*i*,*j*≤*n*_{NC}. Note that a\left({\mathrm{\Phi}}_{j}^{\prime}\right(\mu ),{\mathrm{\Phi}}_{i}^{\prime}(\mu );\mu )=a\left({\mathrm{\Phi}}_{j}^{\prime}\right(\mu ),{\Psi}_{i}^{\prime};\mu ) because of the Galerkin orthogonality in (19), and thus {\mathbb{A}}^{\mathrm{NC}}\left(\mu \right) is indeed the non-conforming version of the Schur complement matrix \mathbb{A}\left(\mu \right) in (26); similarly, note that f\left({\mathrm{\Phi}}_{i}^{\prime}\right(\mu );\mu )-\sum _{l=1}^{I}a\left({b}_{l}^{f;h}\right({\mu}_{l}),{\mathrm{\Phi}}_{i}^{\prime}(\mu );\mu )=f({\Psi}_{i}^{\prime};\mu )-\sum _{l=1}^{I}a\left({b}_{l}^{f;h}\right({\mu}_{l}),{\Psi}_{i}^{\prime}(\mu );\mu ) because of (18) and the fact that {\mathrm{\Phi}}_{i}^{\prime}\left(\mu \right)-{\Psi}_{i}^{\prime} vanish on ports, and thus {\mathbb{F}}^{\mathrm{NC}}\left(\mu \right) is the non-conforming version of the vector \mathbb{F}\left(\mu \right) in (26).

We further define a non-conforming reference matrix

\begin{array}{l}{\mathbb{B}}_{\mu}^{\mathrm{NC}}\equiv {\mathbb{A}}^{\mathrm{NC}}\left({\mu}_{\mathrm{ref}}^{\mu}\right),\end{array}

(99)

which corresponds to the bilinear form *b*_{
μ
}(·,·). We also introduce a non-conforming residual vector {\mathbb{R}}_{0}^{\mathrm{NC}}\left(\mu \right)\in {\mathbb{R}}^{{n}_{\mathrm{NC}}} as

\begin{array}{l}{\mathbb{R}}_{0;i}^{\mathrm{NC}}\left(\mu \right)=f\left({\mathrm{\Phi}}_{i}^{\prime}\right(\mu \left)\right)-a\left({u}^{\mathrm{PR}}\right(\mu ),{\mathrm{\Phi}}_{i}^{\prime}(\mu );\mu ),\phantom{\rule{1em}{0ex}}1\le i\le {n}_{\mathrm{NC}};\end{array}

(100)

note that {\mathbb{R}}_{0}^{\mathrm{NC}}\left(\mu \right)={\mathbb{F}}^{\mathrm{NC}}\left(\mu \right)-{\mathbb{A}}^{\mathrm{NC}}\left(\mu \right){\widehat{\mathbb{U}}}_{0}^{\mathrm{PR}}\left(\mu \right).

Next, we introduce a (unknown) coefficient vector {\stackrel{\u0304}{\mathbb{E}}}_{0}^{\mathrm{NC}}\left(\mu \right)\in {\mathbb{R}}^{{n}_{\mathrm{NC}}} such that

\begin{array}{l}{\u0113}_{0}^{\mathrm{NC}}\left(\mu \right)=\sum _{i=1}^{{n}_{\mathrm{NC}}}{\stackrel{\u0304}{\mathbb{E}}}_{0;i}\left(\mu \right){\mathrm{\Phi}}_{i}^{\prime}\left(\mu \right).\end{array}

(101)

Thus from (91), (99), and (100) we obtain

\begin{array}{l}{\stackrel{\u0304}{\mathbb{E}}}_{0}^{\mathrm{NC}}\left(\mu \right)=\frac{1}{{\lambda}_{\mathrm{min}}\left(\mu \right)}{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}{\mathbb{R}}_{0}^{\mathrm{NC}}\left(\mu \right).\end{array}

(102)

Similarly to (92), we note that for any v\left(\mu \right)=\sum _{i=1}^{{n}_{\mathrm{NC}}}{\mathbb{V}}_{i}\left(\mu \right){\mathrm{\Phi}}_{i}^{\prime}\left(\mu \right) — that is, for any v\left(\mu \right)\in {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right) — we have

\begin{array}{l}a\left(v\right(\mu ),v(\mu );\mu )=\mathbb{V}{\left(\mu \right)}^{\mathrm{T}}{\mathbb{A}}^{\mathrm{NC}}\left(\mu \right)\mathbb{V}\left(\mu \right).\end{array}

(103)

Hence in particular, since {\u0113}_{0}^{\mathrm{NC}}\left(\mu \right)\in {\mathcal{S}}_{\mathrm{symm}}^{\mathrm{NC}}\left(\Omega \right), we obtain

\begin{array}{ll}{\lambda}_{\mathrm{min}}\left(\mu \right){b}_{\mu}\left({\u0113}_{0}^{\mathrm{NC}}\right(\mu ),{\u0113}_{0}^{\mathrm{NC}}(\mu \left)\right)& ={\lambda}_{\mathrm{min}}\left(\mu \right){\stackrel{\u0304}{\mathbb{E}}}_{0}^{\mathrm{NC}}{\left(\mu \right)}^{\mathrm{T}}{\mathbb{B}}_{\mu}^{\mathrm{NC}}{\stackrel{\u0304}{\mathbb{E}}}_{0}^{\mathrm{NC}}\left(\mu \right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{{\lambda}_{\mathrm{min}}\left(\mu \right)}{\mathbb{R}}_{0}^{\mathrm{NC}}{\left(\mu \right)}^{\mathrm{T}}{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}{\mathbb{R}}_{0}^{\mathrm{NC}}\left(\mu \right).\phantom{\rule{2em}{0ex}}\end{array}

(104)

Further, since {e}_{0}^{h}\left(\mu \right)\in {\mathcal{S}}_{\mathrm{symm}}, we may invoke (92) and write

\begin{array}{l}a\left({e}_{0}\right(\mu ),{e}_{0}(\mu );\mu )={\mathbb{E}}_{0}{\left(\mu \right)}^{\mathrm{T}}\mathbb{A}\left(\mu \right){\mathbb{E}}_{0}\left(\mu \right).\end{array}

(105)

Finally, we note that *λ*_{min}(*μ*) of (86) is the smallest eigenvalue associated with the generalized eigenproblem

\begin{array}{l}{\mathbb{A}}^{\mathrm{NC}}\left(\mu \right)\mathbb{V}\left(\mu \right)=\lambda \left(\mu \right){\mathbb{B}}_{\mu}^{\mathrm{NC}}.\end{array}

(106)

The algebraic interpretation of the port reduction error bound (90) is thus

\begin{array}{l}\frac{1}{{\lambda}_{\mathrm{min}}\left(\mu \right)}{\mathbb{R}}_{0}^{\mathrm{NC}}{\left(\mu \right)}^{\mathrm{T}}{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}{\mathbb{R}}_{0}^{\mathrm{NC}}\left(\mu \right)\ge {\mathbb{E}}_{0}{\left(\mu \right)}^{\mathrm{T}}\mathbb{A}\left(\mu \right){\mathbb{E}}_{0}\left(\mu \right).\end{array}

(107)

We note that the bound (107) necessitates a solve {\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}{\mathbb{R}}_{0}^{\mathrm{NC}}\left(\mu \right) of dimension *n*_{NC}≥*n*_{SC}. However, this solve may be performed efficiently thanks to *i)* the non-conforming skeleton space {\mathcal{S}}^{\mathrm{NC}}\left(\Omega \right) which in a natural way allows component-local elimination of all degrees of freedom that do not couple at shared global ports; and *ii)* the quasi parameter-independent bound conditioner matrix {\mathbb{B}}_{\mu}^{\mathrm{NC}} associated with the bilinear form *b*_{
μ
}, which allows offline pre-factorization for all these component-local solves. And furthermore, in actual practice we invoke not *λ*_{min}(*μ*) but rather a computationally tractable eigenvalue lower bound {\stackrel{~}{\lambda}}_{\mathrm{min},\mathit{\text{LB}}}\left(\mu \right)\le {\lambda}_{\mathrm{min}}\left(\mu \right). We consider computational aspects of our error estimation framework in more detail in the “Computational procedures” subsection below.

### RB error contribution — *A Posteriori* error estimators

We now modify (107) in order to obtain an efficiently computable *a posteriori* error bound which is also valid in the presence of RB error contributions. First, as we in the SCRBE context only have access to an approximation of the FE Schur complement system, the residual can not be computed exactly and we thus instead compute a residual approximation together with bounds on associated RB-error-induced residual perturbation terms. Second, we introduce a lower bound (valid under an eigenvalue proximity assumption) for the eigenvalue *λ*_{min}(*μ*) which is based on the solution to a port-reduced eigenproblem, an approximate eigenproblem residual, and bounds on associated RB-error-induced eigenproblem residual perturbation terms.

Moreover, in the presence of RB error contributions the error in the Schur energy is not equal to the energy of the error in the field, and thus in addition to a bound on the former we require a bound on *additional* RB perturbation terms to obtain a bound for the latter. Further, we develop in this section, from our Schur energy error bound, a new bound on port-restricted compliance outputs. For this output bound we must take into account that we in this paper (in contrast to in [12]) employ {\mathcal{S}}^{\mathrm{PR}}\left(\Omega \right) rather than {\stackrel{~}{\mathcal{S}}}_{\mathrm{symm}}^{\mathrm{PR}}\left(\Omega \right)\subset {\stackrel{~}{\mathcal{S}}}_{\mathrm{symm}}\left(\Omega \right) (the former being a port-reduced version of the latter, which is defined in (51)) as our skeleton space. Finally, we introduce *asymptotically rigorous* error estimators, by which we reduce computational cost by neglecting typically very small quadratic RB error bound contributions.

To begin, we define the error field as

\begin{array}{l}{e}^{h}\left(\mu \right)\equiv {u}^{h}\left(\mu \right)-{\u0169}^{\mathrm{PR}}\left(\mu \right).\end{array}

(108)

It is again convenient to introduce the zero-extended solution vectors,

\begin{array}{l}{\widehat{\mathbb{U}}}^{\mathrm{PR}}\left(\mu \right)\equiv \left[\begin{array}{c}{\stackrel{~}{\mathbb{U}}}^{\mathrm{PR}}\left(\mu \right)\\ \mathbf{\text{0}}\end{array}\right]\in {\mathbb{R}}^{{n}_{\mathrm{SC}}},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{\widehat{\mathbb{U}}}^{\mathrm{PR},\mathit{\text{NC}}}\left(\mu \right)\equiv \left[\begin{array}{c}{\stackrel{~}{\mathbb{U}}}^{\mathrm{PR}}\left(\mu \right)\\ \mathbf{\text{0}}\end{array}\right]\in {\mathbb{R}}^{{n}_{\mathrm{NC}}},\end{array}

(109)

in which the solution {\stackrel{~}{\mathbb{U}}}^{\mathrm{PR}}\left(\mu \right) of (64) is extended by *n*_{SC}−*n*_{A} and *n*_{NC}−*n*_{A} zeros, respectively. We may then write

\begin{array}{l}{e}^{h}\left(\mu \right)=\sum _{i=1}^{I}\left({b}_{i}^{h;f}\right({\mu}_{i})-{\stackrel{~}{b}}_{i}^{h;f}({\mu}_{i}\left)\right)+\sum _{p=1}^{{n}^{\Gamma}}\sum _{k=1}^{{\mathcal{N}}_{p}^{\Gamma}}\left({\mathbb{U}}_{p,k}\left(\mu \right){\mathrm{\Phi}}_{p,k}\left(\mu \right)-{\widehat{\mathbb{U}}}_{p,k}^{\mathrm{PR}}\left(\mu \right){\stackrel{~}{\mathrm{\Phi}}}_{p,k}\left(\mu \right)\right),\end{array}

(110)

and we note that *e*^{h}(*μ*) is *not* a member of {\mathcal{S}}_{\mathrm{symm}}\left(\Omega \right) because of the errors in the RB bubble approximations. We also define a vector of error coefficients as

\begin{array}{l}\mathbb{E}\left(\mu \right)\equiv \mathbb{U}\left(\mu \right)-{\widehat{\mathbb{U}}}^{\mathrm{PR}}\left(\mu \right).\end{array}

(111)

We first develop a bound for the error in the Schur energy norm, \sqrt{\mathbb{E}{\left(\mu \right)}^{\mathrm{T}}\mathbb{A}\left(\mu \right)\mathbb{E}\left(\mu \right)}, through perturbations of the left-hand side of (107). We subsequently modify this bound to obtain a bound on ∥*e*^{h}(*μ*)∥_{
μ
}; note the former is not equivalent to the latter because *e*^{h}(*μ*) is not a member of {\mathcal{S}}_{\mathrm{symm}}\left(\Omega \right).

The usual error-residual relationship still holds in the presence of RB error contributions. In this case the relevant error-residual equation is

\begin{array}{l}\mathbb{A}\left(\mu \right)\mathbb{E}\left(\mu \right)=\mathbb{R}\left(\mu \right),\end{array}

(112)

where the residual vector is given as

\begin{array}{l}\mathbb{R}\left(\mu \right)=\mathbb{F}\left(\mu \right)-\mathbb{A}\left(\mu \right){\widehat{\mathbb{U}}}^{\mathrm{PR}}\left(\mu \right).\end{array}

(113)

The difference between (95) and (112) is rather subtle: the former features the residual associated with the RB-error-free solution vector {\widehat{\mathbb{U}}}_{0}^{\mathrm{PR}}\left(\mu \right) (never computationally realized), while the latter features the residual associated with the RB-error-affected SCRBE solution vector {\widehat{\mathbb{U}}}^{\mathrm{PR}}\left(\mu \right) (computed in practice). The non-conforming version of the residual is

\begin{array}{l}{\mathbb{R}}^{\mathrm{NC}}\left(\mu \right)\equiv {\mathbb{F}}^{\mathrm{NC}}\left(\mu \right)-{\mathbb{A}}^{\mathrm{NC}}\left(\mu \right){\widehat{\mathbb{U}}}^{\mathrm{PR}}\left(\mu \right).\end{array}

(114)

Next, we *redefine* our quasi parameter-independent (due to online reference parameter selection) bound conditioner matrix {\mathbb{B}}_{\mu}^{\mathrm{NC}} from the previous subsection as {\mathbb{B}}_{\mu}^{\mathrm{NC}}={\stackrel{~}{\mathbb{A}}}^{\mathrm{NC}}\left({\mu}_{\mathrm{ref}}^{\mu}\right); note that any SPD matrix may serve as our bound conditioner, and thus the RB approximations now present in {\mathbb{B}}_{\mu}^{\mathrm{NC}} do not necessitate modifications to the error bound expression (and therefore the {\mathbb{B}}_{\mu}^{\mathrm{NC}} of the previous subsection did not bear a subscript _{0}). Henceforth, the eigenproblem (106) is interpreted with this redefined {\mathbb{B}}_{\mu}^{\mathrm{NC}} as the right-hand side matrix, and *λ*_{min}(*μ*) is interpreted as the associated smallest eigenvalue. In the presence of RB error contributions, (107) now becomes

\begin{array}{l}\frac{{\mathbb{R}}^{\mathrm{NC}}{\left(\mu \right)}^{\mathrm{T}}{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}{\mathbb{R}}^{\mathrm{NC}}\left(\mu \right)}{{\lambda}_{\mathrm{min}}\left(\mu \right)}\ge \mathbb{E}{\left(\mu \right)}^{\mathrm{T}}\mathbb{A}\left(\mu \right)\mathbb{E}\left(\mu \right).\end{array}

(115)

To bound the error in the Schur energy, we must thus, based on residual and eigenvalue approximations, develop upper and lower bounds for the numerator and denominator, respectively, of the left-hand side of (115).

We first consider the approximation to the non-conforming residual {\mathbb{R}}^{\mathrm{NC}}\left(\mu \right). As we do not have access to {\mathbb{F}}^{\mathrm{NC}}\left(\mu \right) and {\mathbb{A}}^{\mathrm{NC}}\left(\mu \right) as defined in (97) and (98), but rather to RB-approximated versions {\stackrel{~}{\mathbb{F}}}^{\mathrm{NC}}\left(\mu \right)\approx {\mathbb{F}}^{\mathrm{NC}}\left(\mu \right) and {\stackrel{~}{\mathbb{A}}}^{\mathrm{NC}}\left(\mu \right)\approx {\mathbb{A}}^{\mathrm{NC}}\left(\mu \right), we introduce our approximation based on {\stackrel{~}{\mathbb{F}}}^{\mathrm{NC}}\left(\mu \right)\approx {\mathbb{F}}^{\mathrm{NC}}\left(\mu \right) and {\stackrel{~}{\mathbb{A}}}^{\mathrm{NC}}\left(\mu \right)\approx {\mathbb{A}}^{\mathrm{NC}}\left(\mu \right) as

\begin{array}{l}{\stackrel{~}{\mathbb{R}}}^{\mathrm{NC}}\left(\mu \right)={\stackrel{~}{\mathbb{F}}}^{\mathrm{NC}}\left(\mu \right)-{\stackrel{~}{\mathbb{A}}}^{\mathrm{NC}}\left(\mu \right){\widehat{\mathbb{U}}}^{\mathrm{PR},\mathit{\text{NC}}}\left(\mu \right)\end{array}

(116)

such that {\stackrel{~}{\mathbb{R}}}^{\mathrm{NC}}\left(\mu \right)={\mathbb{R}}^{\mathrm{NC}}\left(\mu \right)+\delta {\mathbb{R}}^{\mathrm{NC}}\left(\mu \right). Here,

\begin{array}{l}\delta {\mathbb{R}}^{\mathrm{NC}}\left(\mu \right)={\stackrel{~}{\mathbb{F}}}^{\mathrm{NC}}\left(\mu \right)-{\mathbb{F}}^{\mathrm{NC}}\left(\mu \right)+\left({\mathbb{A}}^{\mathrm{NC}}\right(\mu )-{\stackrel{~}{\mathbb{A}}}^{\mathrm{NC}}(\mu \left)\right){\widehat{\mathbb{U}}}^{\mathrm{PR},\mathit{\text{NC}}}\left(\mu \right)\end{array}

(117)

is an RB-error-induced perturbation term. We may readily from standard RB error bounds [5, 6] develop bounds on these perturbation quantities; we introduce a vector *σ*(*μ*) such that, for any \mu \in \mathcal{D},

\begin{array}{l}{\mathit{\sigma}}_{i}\left(\mu \right)\ge \left|\delta {\mathbb{R}}_{i}^{\mathrm{NC}}\right(\mu \left)\right|,\phantom{\rule{1em}{0ex}}1\le i\le {n}_{\mathrm{NC}}.\end{array}

(118)

We next consider the approximation to the eigenvalue *λ*_{min}(*μ*). Again, as we do not in practice have access to {\mathbb{A}}^{\mathrm{NC}}\left(\mu \right), and furthermore as we wish to avoid solution of a full eigenproblem of dimension *n*_{NC}, we consider an approximation {\stackrel{~}{\lambda}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right) to *λ*_{min}(*μ*) given as the smallest eigenvalue associated with the port-reduced SCRBE eigenproblem

\begin{array}{l}{\stackrel{~}{\mathbb{A}}}^{\mathrm{PR}}\left(\mu \right)\mathbb{V}\left(\mu \right)={\stackrel{~}{\lambda}}^{\mathrm{PR}}\left(\mu \right){\mathbb{B}}_{\mu}^{\mathrm{PR}}\mathbb{V}\left(\mu \right);\end{array}

(119)

here, {\mathbb{B}}_{\mu}^{\mathrm{PR}} denotes the block of {\mathbb{B}}_{\mu}^{\mathrm{NC}} associated with “Active” degrees of freedom. We denote by {\mathbb{V}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right) the eigenvector associated with {\stackrel{~}{\lambda}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right), and we assume the normalization {\mathbb{V}}_{\mathrm{min}}^{\mathrm{PR}}{\left(\mu \right)}^{\mathrm{T}}{\mathbb{B}}_{\mu}^{\mathrm{PR}}{\mathbb{V}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right)=1. We also introduce an approximate eigenproblem residual

\begin{array}{l}{\stackrel{~}{\mathbb{R}}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right)={\stackrel{~}{\mathbb{A}}}^{\mathrm{NC}}\left(\mu \right){\widehat{\mathbb{V}}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right)-{\stackrel{~}{\lambda}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right){\mathbb{B}}_{\mu}^{\mathrm{NC}}{\widehat{\mathbb{V}}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right),\end{array}

(120)

in which {\widehat{\mathbb{V}}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right)\in {\mathbb{R}}^{{n}_{\mathrm{NC}}} is a zero-expanded version of {\mathbb{V}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right)\in {\mathbb{R}}^{{n}_{\mathrm{A}}}. Note that the *exact* eigenproblem residual is given as {\mathbb{R}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right)={\mathbb{A}}^{\mathrm{NC}}{\widehat{\mathbb{V}}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right)-{\stackrel{~}{\lambda}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right){\mathbb{B}}_{\mu}^{\mathrm{NC}}{\widehat{\mathbb{V}}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right), and we may thus define a vector of RB perturbation terms \delta {\mathbb{R}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right) such that {\mathbb{R}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right)={\stackrel{~}{\mathbb{R}}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right)+\delta {\mathbb{R}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right). We may then develop bounds on these RB-error-induced perturbation quantities — we introduce a vector *σ*_{eig}(*μ*) such that, for any \mu \in \mathcal{D},

\begin{array}{l}{\mathit{\sigma}}_{\mathrm{eig},i}\left(\mu \right)\ge \left|\delta {\mathbb{R}}_{\mathrm{eig},i}^{\mathrm{NC}}\right(\mu \left)\right|,\phantom{\rule{1em}{0ex}}1\le i\le {n}_{\mathrm{NC}}.\end{array}

(121)

We now obtain a *computable* eigenvalue lower bound in

#### Lemma 1

Let *C*>0 be such that

\begin{array}{l}\delta {\mathbb{R}}_{\mathrm{eig}}^{\mathrm{NC}}{\left(\mu \right)}^{\mathrm{T}}{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}\delta {\mathbb{R}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right)\le C\parallel \delta {\mathbb{R}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right){\parallel}_{2}^{2},\end{array}

(122)

assume that

\begin{array}{l}\left|{\lambda}_{\mathrm{min}}^{\mathrm{PR}}\right(\mu )-{\lambda}_{\mathrm{min}}(\mu \left)\right|\le \left|{\lambda}_{\mathrm{min}}^{\mathrm{PR}}\right(\mu )-\lambda (\mu \left)\right|,\end{array}

(123)

for all *λ*(*μ*)which satisfy (106) (with the redefined {\mathbb{B}}_{\mu}^{\mathrm{NC}}), and let

\begin{array}{l}{\lambda}_{\mathrm{min},\mathit{\text{LB}}}(\mu ;C)\equiv {\stackrel{~}{\lambda}}_{\mathrm{min}}^{\mathrm{PR}}\left(\mu \right)\phantom{\rule{2em}{0ex}}\\ -\sqrt{{\stackrel{~}{\mathbb{R}}}_{\mathrm{eig}}^{\mathrm{NC}}{\left(\mu \right)}^{\mathrm{T}}{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}{\stackrel{~}{\mathbb{R}}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right)+2{\mathit{\sigma}}_{\mathrm{eig}}{\left(\mu \right)}^{\mathrm{T}}\left|{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}{\stackrel{~}{\mathbb{R}}}_{\mathrm{eig}}^{\mathrm{NC}}\right(\mu \left)\right|+C\parallel {\mathit{\sigma}}_{\mathrm{eig}}\left(\mu \right){\parallel}_{2}^{2}}.\phantom{\rule{2em}{0ex}}\end{array}

(124)

Then

\begin{array}{l}{\lambda}_{\mathrm{min},\mathit{\text{LB}}}(\mu ;C)\le {\lambda}_{\mathrm{min}}\left(\mu \right).\end{array}

(125)

#### Proof

We refer to ([12], Proposition 1) for the proof, and we note that a similar residual-based eigenvalue bound has been developed in [18] for the standard eigenproblem. □

With the residual approximation \stackrel{~}{\mathbb{R}}\left(\mu \right), associated RB error bounds *σ*(*μ*), and the eigenvalue lower bound *λ*_{min,LB}(*μ*;*C*) above, we may now obtain a *computable* bound for the left-hand side of (115) and thus the error in the Schur energy norm in

#### Proposition 1

Let *C*>0 be a computable constant such that

\begin{array}{ll}\delta {\mathbb{R}}^{\mathrm{NC}}{\left(\mu \right)}^{\mathrm{T}}{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}\delta {\mathbb{R}}^{\mathrm{NC}}\left(\mu \right)& \le C\parallel \delta {\mathbb{R}}^{\mathrm{NC}}\left(\mu \right){\parallel}_{2}^{2},\phantom{\rule{2em}{0ex}}\end{array}

(126)

\begin{array}{ll}\delta {\mathbb{R}}_{\mathrm{eig}}^{\mathrm{NC}}{\left(\mu \right)}^{\mathrm{T}}{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}\delta {\mathbb{R}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right)& \le C\parallel \delta {\mathbb{R}}_{\mathrm{eig}}^{\mathrm{NC}}\left(\mu \right){\parallel}_{2}^{2}.\phantom{\rule{2em}{0ex}}\end{array}

(127)

Then define

{\mathrm{\Delta}}^{\mathbb{U}}(\mu ;C)\equiv \sqrt{\frac{{\stackrel{~}{\mathbb{R}}}^{\mathrm{NC}}{\left(\mu \right)}^{\mathrm{T}}{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}{\stackrel{~}{\mathbb{R}}}^{\mathrm{NC}}\left(\mu \right)+2\mathit{\sigma}{\left(\mu \right)}^{\mathrm{T}}\left|{\left({\mathbb{B}}_{\mu}^{\mathrm{NC}}\right)}^{-1}{\stackrel{~}{\mathbb{R}}}^{\mathrm{NC}}\right(\mu \left)\right|+C\parallel \mathit{\sigma}\left(\mu \right){\parallel}_{2}^{2}}{{\lambda}_{\mathrm{min},\mathit{\text{LB}}}(\mu ;C)}}.

(128)

Then if the assumption (123) holds, we have

\begin{array}{l}\sqrt{\mathbb{E}{\left(\mu \right)}^{\mathrm{T}}\mathbb{A}\left(\mu \right)\mathbb{E}\left(\mu \right)}\le {\mathrm{\Delta}}^{\mathbb{U}}(\mu ;C).\end{array}

(129)

#### Proof

We merely note here that the numerator in (128) is an upper bound for the numerator in (115), and that *λ*_{min,LB}(*μ*;*C*)≤*λ*_{min}(*μ*) is a lower bound for the denominator in (115). We refer to ([12], Appendix A) for the detailed proof. □

We proceed to bound the energy of the error in the field. Since *e*^{h}(*μ*) is not a member of {\mathcal{S}}_{\mathrm{symm}}\left(\Omega \right), a small modification to (128) is necessary to obtain a bound for ∥*e*^{h}(*μ*)∥_{
μ
}. To this end, we introduce additional RB perturbation terms

\begin{array}{ll}\mathrm{\Delta}{b}^{f}\left(\mu \right)& \equiv \sum _{i=1}^{I}\left({b}_{i}^{f;h}\left(\mu \right)-{\stackrel{~}{b}}_{i}^{f}\left(\mu \right)\right)\phantom{\rule{2em}{0ex}}\end{array}

(130)

\begin{array}{ll}\mathrm{\Delta}{\mathrm{\Phi}}_{\mathrm{A}}\left(\mu \right)& \equiv \sum _{p=1}^{{n}^{\Gamma}}\sum _{k=1}^{{n}_{\mathrm{A},p}^{\Gamma}}{\stackrel{~}{\mathbb{U}}}_{\mathrm{A},p,k}\left(\mu \right)\left({\mathrm{\Phi}}_{p,k}\left(\mu \right)-{\stackrel{~}{\mathrm{\Phi}}}_{p,k}\left(\mu \right)\right);\phantom{\rule{2em}{0ex}}\end{array}

(131)

we also introduce an RB error bound [6]*κ*(*μ*) such that, for any \mu \in \mathcal{D},

\begin{array}{l}\kappa \left(\mu \right)\ge \parallel \mathrm{\Delta}{b}^{f}\left(\mu \right)+\mathrm{\Delta}{\mathrm{\Phi}}_{\mathrm{A}}\left(\mu \right){\parallel}_{\mu}.\end{array}

(132)

We then introduce our bound for the energy of the error field in

#### Proposition 2

Define Δ^{u}(*μ*;*C*) as

\begin{array}{l}{\mathrm{\Delta}}^{u}(\mu ;C)\equiv \sqrt{{\left({\mathrm{\Delta}}^{\mathbb{U}}(\mu ;C)\right)}^{2}+\kappa {\left(\mu \right)}^{2}}.\end{array}

(133)

*where* *κ*(*μ*)*is given in* (132). *Then if the assumption* (123) *holds, we have*

\begin{array}{l}\parallel {e}^{h}\left(\mu \right){\parallel}_{\mu}\le {\mathrm{\Delta}}^{u}(\mu ;C).\end{array}

(134)

#### Proof

We refer to ([12], Appendix A) for the proof. □

Next, we develop a bound for the error in *port-restricted* compliance outputs. To this end we introduce a matrix {\mathit{\sigma}}_{\mathbb{A}}\left(\mu \right)\in {\mathbb{R}}^{{n}_{\mathrm{A}}\times {n}_{\mathrm{A}}} such that

\begin{array}{l}{\mathit{\sigma}}_{\mathbb{A},i,j}\left(\mu \right)\ge \left|{\mathbb{A}}_{i,j}\right(\mu )-{\stackrel{~}{\mathbb{A}}}_{i,j}(\mu \left)\right|,\phantom{\rule{2em}{0ex}}1\le i,j\le {n}_{\mathrm{A}}.\end{array}

(135)

We then state

#### Proposition 3

Let

\begin{array}{l}{\mathrm{\Delta}}^{s}(\mu ;C)\equiv {\left({\mathrm{\Delta}}^{\mathbb{U}}(\mu ;C)\right)}^{2}+\left|{\stackrel{~}{\mathbb{U}}}^{\mathrm{PR}}\right(\mu \left){|}^{\mathrm{T}}{\mathit{\sigma}}_{\mathbb{A}}\right(\mu \left)\right|{\stackrel{~}{\mathbb{U}}}^{\mathrm{PR}}\left(\mu \right)|\end{array}

(136)

(in which |·| denotes entry-wise absolute value and not vector modulus). Assume that the source *f*(·;*μ*) is restricted to ports such that {b}_{i}^{f;h}\left({\mu}_{i}\right)=0, 1≤*i*≤*I*. The error in a port-restricted compliance output {\stackrel{~}{s}}^{\mathrm{PR}}\left(\mu \right)=f\left({\u0169}^{\mathrm{PR}}\right(\mu );\mu ) can then be bounded as

\begin{array}{l}\left|{s}^{h}\right(\mu )-{\stackrel{~}{s}}^{\mathrm{PR}}(\mu \left)\right|\le {\mathrm{\Delta}}^{s}(\mu ;C)\end{array}

(137)

#### Proof

We provide here a full proof as in the present paper (skeleton space {\mathcal{S}}^{\mathrm{PR}}\left(\Omega \right)) the proof is different from a related proof in [12] (skeleton space {\stackrel{~}{\mathcal{S}}}_{\mathrm{symm}}\left(\Omega \right)).

We first note that

\begin{array}{l}{e}^{h}\left(\mu \right)=\mathrm{\Delta}{b}^{\phantom{\rule{2.77626pt}{0ex}}f}\left(\mu \right)+\mathrm{\Delta}{\mathrm{\Phi}}_{\mathrm{A}}\left(\mu \right)+\sum _{p=1}^{{n}^{\Gamma}}\sum _{k=1}^{{\mathcal{N}}_{p}^{\Gamma}}{\mathbb{E}}_{p,k}\left(\mu \right){\mathrm{\Phi}}_{p,k}\left(\mu \right);\end{array}

(138)

note in the port-restricted output case considered here, Δ*b*^{f}(*μ*)=0. For the compliance output error, we may then write (using symmetry of *a*(·,·;*μ*))

\begin{array}{ll}{s}^{h}\left(\mu \right)& -{\stackrel{~}{s}}^{\mathrm{PR}}\left(\mu \right)=a\left({u}^{h}\right(\mu ),{e}^{h}(\mu );\mu )\\ \phantom{\rule{0.5em}{0ex}}=a\left({e}^{h}\right(\mu ),{u}^{h}(\mu );\mu )=a\left({e}^{h}\right(\mu ),{e}^{h}(\mu );\mu )+a\left({e}^{h}\right(\mu ),{\u0169}^{\mathrm{PR}}(\mu );\mu ),\end{array}

(139)

and thus by (138) (and again symmetry of *a*(·,·;*μ*))

\begin{array}{ll}{s}^{h}\left(\mu \right)-{\stackrel{~}{s}}^{\mathrm{PR}}\left(\mu \right)& =\mathbb{E}{\left(\mu \right)}^{\mathrm{T}}\mathbb{A}\left(\mu \right)\mathbb{E}\left(\mu \right)+a\left(\mathrm{\Delta}{\mathrm{\Phi}}_{\mathrm{A}}\right(\mu ),\mathrm{\Delta}{\mathrm{\Phi}}_{\mathrm{A}}(\mu );\mu )\\ +2\sum _{p=1}^{{n}^{\Gamma}}\sum _{k=1}^{{\mathcal{N}}_{p}^{\Gamma}}a\left({\mathrm{\Phi}}_{p,k}\right(\mu ),\mathrm{\Delta}{\mathrm{\Phi}}_{\mathrm{A}}(\mu );\mu )+a\left({e}^{h}\right(\mu ),{\u0169}^{\mathrm{PR}}(\mu );\mu ).\end{array}

(140)

We note that *e*^{h}(*μ*) is *not* Galerkin-orthogonal to {\u0169}^{\mathrm{PR}}\left(\mu \right) because {\u0169}^{\mathrm{PR}}\left(\mu \right) (even in the case {b}_{i}^{f;h}\left({\mu}_{i}\right)=0) is not a member of the skeleton test space {\mathcal{S}}^{\mathrm{PR}}\left(\Omega \right). We thus do not obtain equality between the compliance output error and the squared energy of the error field in (139). This is the key difference between the compliance output error bound result here and in [12]; in [12], we invoke the skeleton space {\stackrel{~}{\mathcal{S}}}_{\mathrm{symm}}^{\mathrm{PR}}\left(\Omega \right)\subset {\stackrel{~}{\mathcal{S}}}^{\mathrm{PR}}\left(\Omega \right) (the latter is defined in (51)) of which {\u0169}^{\mathrm{PR}}\left(\mu \right)*is* a member (for port-restricted compliance such that {\stackrel{~}{b}}_{i}^{f}\left(\mu \right)=0), and thus we directly obtain this equality.

We next note that \mathrm{\Delta}{\mathrm{\Phi}}_{\mathrm{A}}\left(\mu \right){|}_{{\Omega}_{}}