Let us consider the following eigenvalue problem
$$\begin{aligned} \text {find}\,(\mu ,{\textbf {v}})\in \mathbb {R}\times \varvec{\mathcal {S}}\,\text {s.t.}\,\varvec{\mathcal {A}}({\textbf {v}},{\varvec{\phi }}) = \mu \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\phi }})\,\forall {\varvec{\phi }}\in \varvec{\mathcal {S}}. \end{aligned}$$
(24)
Here,
\(\varvec{\mathcal {A}}\) and
\(\varvec{\mathcal {B}}\) are bi-linear operators. For uniqueness of the problem, the discrete eigenvectors
\({\textbf {v}}_h\) are normalised by the condition [
56]
$$\begin{aligned} \varvec{\mathcal {B}}({\textbf {v}},{\textbf {v}}) = 1. \end{aligned}$$
(25)
Typically, discretizing the system gives the following:
$$\begin{aligned} \text {find}\,(\mu _h,{\textbf {v}}_h)\in \mathbb {R}\times \varvec{\mathcal {S}}_h^p\,\text {s.t.}\,\varvec{\mathcal {A}}({\textbf {v}}_h,{\varvec{\phi }}_h) = \mu \varvec{\mathcal {B}}({\textbf {v}}_h,{\varvec{\phi }}_h)\,\forall {\varvec{\phi }}_h\in \varvec{\mathcal {S}}_h^p, \end{aligned}$$
(26)
where the eigenpairs
\(\hat{{\textbf {v}}}_h = (\mu _h,{\textbf {v}}_h)\) are the solutions of the eigenvalue problem. In addition, the adjoint eigenvalue problem is defined by the eigenvalue problem [
69]:
$$\begin{aligned} \text {find}\,(\eta ,{\varvec{\psi }})\in \mathbb {R}\times \varvec{\mathcal {S}}\,\text {s.t.}\,\varvec{\mathcal {A}}({\varvec{\psi }},{\varvec{\phi }}) = \eta \varvec{\mathcal {B}}({\varvec{\psi }},{\varvec{\phi }})\,\forall ({\varvec{\phi }})\in \varvec{\mathcal {S}}, \end{aligned}$$
(27)
Of for which the normalization similar to Eq. (
25) is used for the dual eigenvectors
\({\varvec{\psi }}\)$$\begin{aligned} \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\psi }}) = 1. \end{aligned}$$
(28)
To derive the DWR method for the eigenvalue problem in Eq. (
24), the functional
\(\varvec{\mathcal {V}}(\cdot ,\cdot )\) is defined, such that the following problem should be solved
$$\begin{aligned} \begin{aligned}&\text {Find}\,\hat{{\textbf {v}}}=(\mu ,{\textbf {v}})\in \mathbb {R}\times \varvec{\mathcal {S}}\,\text {s.t.}\\&\quad \varvec{\mathcal {V}}(\hat{{\textbf {v}}},\hat{{\varvec{\phi }}}) = \mu \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\phi }}) - \varvec{\mathcal {A}}({\textbf {v}},{\varvec{\phi }}) + \tau \left( \varvec{\mathcal {B}}({\textbf {v}},{\textbf {v}})-1\right) = 0,\\&\quad \forall {\hat{{\varvec{\phi }}}}=(\tau ,{\phi })\in \mathbb {R}\times \varvec{\mathcal {S}}, \end{aligned} \end{aligned}$$
(29)
where the normalisation condition from Eq. (
25) is enforced weakly. The discrete counterpart of this equation reads:
$$\begin{aligned} \begin{aligned}&\text {Find}\,\hat{{\textbf {v}}}_h=(\mu _h,{\textbf {v}}_h)\in \mathbb {R}\times \varvec{\mathcal {S}}_h^p\,\text {s.t.}\\&\quad \varvec{\mathcal {V}}(\hat{{\textbf {v}}}_h,\hat{{\varvec{\phi }}}_h) = \mu \varvec{\mathcal {B}}({\textbf {v}}_h,{\varvec{\phi }}_h) - \varvec{\mathcal {A}}({\textbf {v}}_h,{\varvec{\phi }}_h) + \tau _h \left( \varvec{\mathcal {B}}({\textbf {v}}_h,{\textbf {v}}_h)-1\right) = 0,\\&\quad \forall \hat{{\varvec{\phi }}}_h=(\tau _h,{\varvec{\phi }}_h)\in \mathbb {R}\times \varvec{\mathcal {S}}_h^p. \end{aligned} \end{aligned}$$
(30)
Furthermore, a goal-function for the eigenvalues is defined as follows:
$$\begin{aligned} \varvec{\mathcal {L}}(\hat{{\textbf {v}}}) = \mu = \mu \varvec{\mathcal {B}}({\textbf {v}},{\textbf {v}}), \end{aligned}$$
(31)
giving
$$\begin{aligned} \Delta \varvec{\mathcal {L}}(\hat{{\varvec{\phi }}}_h) = \mu - \mu _h. \end{aligned}$$
(32)
Using the non-linear functional
\(\varvec{\mathcal {V}}\) and the goal functional
\(\varvec{\mathcal {L}}\), the same derivations as in Sect.
3.1 can be followed to find a system of equations to solve the DWR eigenvalue problem. The Gateaux derivative of
\(\varvec{\mathcal {V}}\), denoted by
\(\varvec{\mathcal {V}}'\) is given by:
$$\begin{aligned} \varvec{\mathcal {V}}'(\hat{{\textbf {v}}},\hat{{\varvec{\phi }}},\hat{{\varvec{\psi }}}) = \eta \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\psi }}) + \mu \varvec{\mathcal {B}}({\varvec{\psi }},{\varvec{\phi }}) - \varvec{\mathcal {A}}({\varvec{\psi }},{\varvec{\phi }}) + \tau ( \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\psi }}) + \varvec{\mathcal {B}}({\varvec{\psi }},{\textbf {v}}) ), \end{aligned}$$
(33)
where the derivatives
\(\varvec{\mathcal {A}}'({\varvec{\psi }},{\varvec{\phi }})\) and
\(\varvec{\mathcal {B}}'({\varvec{\psi }},{\varvec{\phi }})\) are equal to the bi-linear operators
\(\varvec{\mathcal {A}}({\textbf {u}},{\varvec{\phi }})\) and
\(\varvec{\mathcal {B}}({\textbf {u}},{\varvec{\phi }})\) themselves. Furthermore, the solution around which the linerisation is performed is denoted by
\(\hat{{\textbf {v}}}=(\mu ,{\textbf {v}})\), the test functions are denoted by
\(\hat{{\varvec{\phi }}}=(\tau ,{\varvec{\phi }})\) and the trial functions are denoted by
\(\hat{{\varvec{\psi }}}=(\eta ,{\varvec{\psi }})\). Furthermore, the linearisation of the goal functional Eq. (
31) is
$$\begin{aligned} \varvec{\mathcal {L}}'(\hat{{\textbf {v}}},\hat{{\varvec{\psi }}}) = \eta \varvec{\mathcal {B}}({\textbf {v}},{\textbf {v}}) + \mu \left[ \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\psi }}) + \varvec{\mathcal {B}}({\varvec{\psi }},{\textbf {v}}) \right] , \end{aligned}$$
(34)
such that the adjoint eigenvalue problem, analoguously to Eq. (
21), given by
$$\begin{aligned} \text {Find}\,\hat{{\varvec{\phi }}}= & {} (\tau , {\varvec{\phi }})\in \mathbb {R}\times \varvec{\mathcal {S}}\,\text {s.t.}\, \varvec{\mathcal {V}}'(\hat{{\textbf {v}}},\hat{{\varvec{\phi }}},\hat{{\varvec{\psi }}}) = \varvec{\mathcal {L}}'(\hat{{\textbf {v}}},\hat{{\varvec{\psi }}})\nonumber \\ \forall \hat{{\varvec{\psi }}}= & {} (\eta ,{\psi })\in \mathbb {R}\times \varvec{\mathcal {S}}, \end{aligned}$$
(35)
becomes [
56,
69]:
$$\begin{aligned} \begin{aligned}&\text {Find}\,\hat{{\varvec{\phi }}}=(\tau , {\varvec{\phi }})\in \mathbb {R}\times \varvec{\mathcal {S}}\,\text {s.t.}\\&\quad \eta \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\phi }}) + \mu \varvec{\mathcal {B}}({\varvec{\psi }},{\varvec{\phi }}) - \varvec{\mathcal {A}}({\varvec{\psi }},{\varvec{\phi }}) + \tau ( \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\psi }}) \\&\quad + \varvec{\mathcal {B}}({\varvec{\psi }},{\textbf {v}}) ) = \eta \varvec{\mathcal {B}}({\textbf {v}},{\textbf {v}}) \\&\quad +\mu \left[ \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\psi }}) + \varvec{\mathcal {B}}({\varvec{\psi }},{\textbf {v}}) \right] ,\\&\quad \forall \hat{{\varvec{\psi }}}=(\eta , {\psi
})\in \mathbb {R}\times \varvec{\mathcal {S}}. \end{aligned} \end{aligned}$$
(36)
This equation can be simplified to obtain the following [
56,
69]:
$$\begin{aligned} \begin{aligned}&\text {Find}\,\hat{{\varvec{\phi }}}=(\tau , {\varvec{\phi }})\in \mathbb {R}\times \varvec{\mathcal {S}}\,\text {s.t.}\\&\quad \mu \varvec{\mathcal {B}}({\varvec{\psi }},{\varvec{\phi }}) - \varvec{\mathcal {A}}({\varvec{\psi }},{\varvec{\phi }}) + \eta \left[ \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\phi }}) - \varvec{\mathcal {B}}({\textbf {v}},{\textbf {v}}) \right] \\&\quad + (\tau - \mu ) \left[ \varvec{\mathcal {B}}({\textbf {v}},{\varvec{\psi }}) + \varvec{\mathcal {B}}({\varvec{\psi }},{\textbf {v}}) \right] = 0, \\&\quad \forall \hat{{\varvec{\psi }}}=(\eta , {\psi })\in \mathbb {R}\times \varvec{\mathcal {S}}. \end{aligned} \end{aligned}$$
(37)
Using the normalizations from Eqs. (
25) and (
28) and the fact that Eq. (
27) solves the same equation as Eq. (
24), it follows that Eq. (
37) is solved by Eq. (
27) [
69].
Using Eqs. (
20), (
32) and (
15) with
\(\varvec{\mathcal {W}}=\varvec{\mathcal {V}}\) according to Eq. (
29) and with
\({\varvec{\psi }}\) denoting the dual eigenvector and
\(\eta\) the dual eigenvalue, the error estimation according to the DWR method for an eigenvalue problem is
$$\begin{aligned} \Delta \varvec{\mathcal {L}}(\hat{{\textbf {v}}}_h)= & {} \varvec{\mathcal {A}}({\textbf {v}}_h,{\varvec{\psi }}-{\varvec{\psi }}_h)- \mu _h \varvec{\mathcal {B}}({\textbf {v}}_h,{\varvec{\psi }}-{\varvec{\psi }}_h) \nonumber \\{} & {} +(\eta -\eta _h)(\varvec{\mathcal {B}}({\textbf {v}}_h,{\textbf {v}}_h)-1), \end{aligned}$$
(38)
for
\(\hat{{\textbf {v}}}_h=(\mu _h,{\textbf {v}}_h)\in \mathbb {R}\times \varvec{\mathcal {S}}_h^p\),
\(\hat{{\varvec{\psi }}}_h=(\eta _h,{\varvec{\psi }}_h)\in \mathbb {R}\times \varvec{\mathcal {S}}_h^p\) and
\(\hat{{\varvec{\psi }}}=(\eta ,{\varvec{\psi }})\in \mathbb {R}\times \varvec{\mathcal {S}}\). The exact adjoint solution
\(\hat{{\varvec{\psi }}}_h\) is again approximated by solving Eq. (
27) on an
enriched space
\(\tilde{\varvec{\mathcal {S}}}_h^p\subset \varvec{\mathcal {S}}\),
\(\tilde{\varvec{\mathcal {S}}}_h^p\supset \varvec{\mathcal {S}}_h^p\), providing
\((\tilde{\eta }_h,\tilde{{\varvec{\psi }}_h})\in \mathbb {R}\times \tilde{\varvec{\mathcal {S}}}_h^p\). In [
38] different choices for constructing
\(\tilde{\varvec{\mathcal {S}}}_h^p\) are given, including an
h-refinement and a
p-refinement. As in the work of [
33], the second approach is used in the present paper, with the same mesh as for
\(\varvec{\mathcal {S}}_h^p\), but with a higher order and with the same regularity, i.e.
\(\tilde{\varvec{\mathcal {S}}}_h^p = \varvec{\mathcal {S}}_h^{p+1}\), as it introduces less degrees of freedom compared to an
h-refinement.