Estimating and localizing the algebraic and total numerical errors using flux reconstructions

Abstract

This paper presents a methodology for computing upper and lower bounds for both the algebraic and total errors in the context of the conforming finite element discretization of the Poisson model problem and an arbitrary iterative algebraic solver. The derived bounds do not contain any unspecified constants and allow estimating the local distribution of both errors over the computational domain. Combining these bounds, we also obtain guaranteed upper and lower bounds on the discretization error. This allows to propose novel mathematically justified stopping criteria for iterative algebraic solvers ensuring that the algebraic error will lie below the discretization one. Our upper algebraic and total error bounds are based on locally reconstructed fluxes in ${\mathbf {H}}(\mathrm{div},\varOmega )$, whereas the lower algebraic and total error bounds rely on locally constructed $H^1_0(\varOmega )$-liftings of the algebraic and total residuals. We prove global and local efficiency of the upper bound on the total error and its robustness with respect to the approximation polynomial degree. Relationships to the previously published estimates on the algebraic error are discussed. Theoretical results are illustrated on numerical experiments for higher-order finite element approximations and the preconditioned conjugate gradient method. They in particular witness that the proposed methodology yields a tight estimate on the local distribution of the algebraic and total errors over the computational domain and illustrate the associated cost.

Appendices

A Details of the flux reconstruction

In this appendix we present the construction of the flux $\mathbf d ^{i}_h$. It follows [24, Section 6.2.4] (see also [10, 18]) with the difference in the construction of the algebraic residual representation $r^{i}_h$ satisfying (2.7), which allows to bound the algebraic error in Theorem 3.

For $K\in \mathcal {T}_h$, let $\mathbf {RTN}^\mathrm {}_{p'}(K) \equiv [\mathbb {P}_{p'}(K)]^d + \mathbf {x} \mathbb {P}_{p'}(K)$ be the Raviart–Thomas–Nédélec finite element space of order $p' \ge 0$. We set

$$\begin{aligned} \mathbf {RTN}^\mathrm {-1}_{p'}(\mathcal {T}_h) \equiv \left\{ \mathbf {v}_h \in [L^2(\varOmega )]^d, \mathbf {v}_h|_K \in \mathbf {RTN}^\mathrm {}_{p'}(K)\quad \forall K\in \mathcal {T}_h\right\} \end{aligned}$$

and $\mathbf {RTN}^\mathrm {}_{p'}(\mathcal {T}_h) \equiv \mathbf {RTN}^\mathrm {-1}_{p'}(\mathcal {T}_h) \cap \mathbf {H}(\mathrm {div}, \varOmega )$. We use a similar notation for these spaces on various patches. Let $\mathbf {RTN}^\mathrm {N,0}_{p'}(\mathcal {T}_{\texttt {a}})$ be the subspace of $\mathbf {RTN}^\mathrm {}_{p'}(\mathcal {T}_{\texttt {a}})$ with zero normal flux through the boundary $\partial {\omega _\texttt {a}}$ for $\texttt {a}\in {\mathcal {V}^\mathrm {int}_h}$ and through $\partial {\omega _\texttt {a}}\backslash \partial \varOmega$ for $\texttt {a}\in {\mathcal {V}^\mathrm {ext}_h}$ (corresponding to a homogeneous Neumann condition). Let $\mathbb {P}_{p'}^*(\mathcal {T}_{\texttt {a}})$ be spanned by piecewise ${p'}$th order polynomials on $\mathcal {T}_{\texttt {a}}$, with zero mean on $\mathcal {T}_{\texttt {a}}$ when $\texttt {a}\in {\mathcal {V}^\mathrm {int}_h}$.

For all vertices $\texttt {a}\in {\mathcal {V}^\mathrm {}_h}$, we first solve the following mixed finite element problems on the patches $\mathcal {T}_{\texttt {a}}$: find $\mathbf d ^{i}_{h,\texttt {a}}\in \mathbf {RTN}^\mathrm {N,0}_{p'}(\mathcal {T}_{\texttt {a}})$ and $q_{h,\texttt {a}} \in \mathbb {P}_{p'}^*(\mathcal {T}_{\texttt {a}})$, ${p'} = p$ or ${p'} = p+1$, such that

$$\begin{aligned}&(\mathbf d ^{i}_{h,\texttt {a}}, \mathbf {v}_h)_{{\omega _\texttt {a}}} - (q_{h,\texttt {a}}, \displaystyle {\nabla \cdot \,}\mathbf {v}_h)_{{\omega _\texttt {a}}} = - (\psi _{\texttt {a}}\nabla u^{i}_h, \mathbf {v}_h)_{{\omega _\texttt {a}}}, \end{aligned}$$

(A.1a)

$$\begin{aligned}&(\displaystyle {\nabla \cdot \,}\mathbf d ^{i}_{h,\texttt {a}}, \chi _h)_{{\omega _\texttt {a}}} = (f_h \psi _{\texttt {a}}- \nabla u^{i}_h\cdot \nabla \psi _{\texttt {a}}, \chi _h)_{{\omega _\texttt {a}}} - (r^{i}_h\psi _{\texttt {a}}, \chi _h )_{{\omega _\texttt {a}}} \end{aligned}$$

(A.1b)

for all $(\mathbf {v}_h, \chi _h) \in \mathbf {RTN}^\mathrm {N,0}_{p'}(\mathcal {T}_{\texttt {a}}) \times \mathbb {P}_{p'}^*(\mathcal {T}_{\texttt {a}})$. Then we set

$$\begin{aligned} \mathbf d ^{i}_h\equiv \sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}}\mathbf{d ^{i}_{h,\texttt {a}}}. \end{aligned}$$

(A.1c)

We typically choose $f_h$ to be the $L^2(\varOmega )$-orthogonal projection of f onto the space of the piecewise polynomials of degree $p'$, and $r^{i}_h\in \mathbb {P}_{p}(\mathcal {T}_h)$; see Sect. 5.1. Since $\psi _{\texttt {a}}\in V_h$, (2.8) gives the Neumann compatibility condition of the problem (A.1a)–(A.1b),

$$\begin{aligned} (\nabla u^{i}_h, \nabla \psi _{\texttt {a}})_{\omega _\texttt {a}}= (f, \psi _{\texttt {a}})_{\omega _\texttt {a}}- (r^{i}_h, \psi _{\texttt {a}})_{\omega _\texttt {a}}. \end{aligned}$$

Consequently, we can in (A.1b) take all test functions $\chi _h \in \mathbb {P}_{p'}(\mathcal {T}_{\texttt {a}})$, which allows to show that $\mathbf d ^{i}_h$ given by (A.1) satisfies (4.2), i.e., that $\displaystyle {\nabla \cdot \,}\mathbf d ^{i}_h= f_h - r^{i}_h$ holds. Indeed, let $K \in \mathcal {T}_h$ and let $v_h \in \mathbb {P}_{p'}(K)$ be fixed. Since $\sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}}{\psi _{\texttt {a}}|_K} = 1$ and $\sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}}{ \nabla \psi _{\texttt {a}}|_K} = 0$ ($\psi _{\texttt {a}}$ form a partition of unity on K), we infer

$$\begin{aligned} (\displaystyle {\nabla \cdot \,}\mathbf d ^{i}_h, v_h)_K\!= & {} \!\sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} (\displaystyle {\nabla \cdot \,}\mathbf d ^{i}_{h,\texttt {a}}, v_h)_K \!=\! \sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} \left[ (f_h \psi _{\texttt {a}}- \nabla u^{i}_h\cdot \nabla \psi _{\texttt {a}}, v_h)_K \!-\! (r^{i}_h\psi _{\texttt {a}}, v_h )_K \right] \\= & {} (f_h, v_h)_K - \left( r^{i}_h, v_h\right) _K, \end{aligned}$$

and (4.2) is proved as $f_h - r^{i}_h\in \mathbb {P}_{p'}(\mathcal {T}_h)$.

We now briefly comment on the algorithmic construction of $\mathbf d ^{i}_h$ in (A.1). Denote by $\varvec{\varPhi }_{\texttt {a}}$ the basis of $\mathbf {RTN}^\mathrm {N,0}_{p'}(\mathcal {T}_{\texttt {a}})$, and by $\widetilde{\mathcal {X}}_{\texttt {a}}$ the basis of $\mathbb {P}_{p'}^*(\mathcal {T}_{\texttt {a}})$, Then we construct $\mathbf d ^{i}_h$ as

$$\begin{aligned} \mathbf d ^{i}_h= \sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} \varvec{\varPhi }_{\texttt {a}} \overline{{\mathsf {D}}}^i_{\texttt {a}}, \end{aligned}$$

where $\overline{{\mathsf {D}}}^i_{\texttt {a}}$ forms the part of the vector ${{\mathsf {D}}}^i_\texttt {a}$ solving the algebraic form of (A.1a)–(A.1b)

$$\begin{aligned} {{\mathbf {\mathsf{{K}}}}}_\texttt {a}{\mathsf {D}}^i_\texttt {a}= {\mathsf {E}}^i_\texttt {a}, \qquad {{\mathbf {\mathsf{{K}}}}}_\texttt {a}= \begin{bmatrix} \overline{{{\mathbf {\mathsf{{K}}}}}}_{\texttt {a}}&-\widetilde{{{\mathbf {\mathsf{{K}}}}}}_{\texttt {a}}\\ \left( \widetilde{{{\mathbf {\mathsf{{K}}}}}}_{\texttt {a}}\right) ^T&0 \end{bmatrix}, \qquad {{\mathsf {D}}}^i_\texttt {a}= \begin{bmatrix}\overline{{\mathsf {D}}}^i_{\texttt {a}}\\ \underline{{\mathsf {D}}}^i_{\texttt {a}} \end{bmatrix}. \end{aligned}$$

(A.2)

Here $\left( \overline{{{\mathbf {\mathsf{{K}}}}}}_{\texttt {a}}\right) _{kj} = (\varvec{\phi }_{j},\, \varvec{\phi }_{k})_{\omega _\texttt {a}}$ and $\big (\widetilde{{{\mathbf {\mathsf{{K}}}}}}_{\texttt {a}}\big )_{k\ell } = (\widetilde{\chi }_{\ell },\, \displaystyle {\nabla \cdot \,}\varvec{\phi }_{k})_{\omega _\texttt {a}}$ with $\varvec{\phi }_{j}, \varvec{\phi }_{k} \in \varvec{\varPhi }_{\texttt {a}}$, $\widetilde{\chi }_{\ell } \in \widetilde{\mathcal {X}}_{\texttt {a}}$. The right-hand side vector is given as

$$\begin{aligned} {\mathsf {E}}^i_\texttt {a}= {\mathsf {E}}_{\texttt {a},f} - {\mathsf {E}}_{\texttt {a},u^{i}_h} - {\mathsf {E}}_{\texttt {a},r^{i}_h} = \begin{bmatrix}0^{}\\ \underline{{\mathsf {E}}}_{\texttt {a},f^{}_{}} \end{bmatrix} -\begin{bmatrix} \overline{{\mathsf {E}}}^{}_{\texttt {a},u^{i}_h}\\ \underline{{\mathsf {E}}}_{\texttt {a},u^{i}_h} \end{bmatrix} -\begin{bmatrix}0^{}\\ \underline{{\mathsf {E}}}_{\texttt {a},r^{i}_h} \end{bmatrix}, \end{aligned}$$

where

Since $u^{i}_h= \varPsi {\mathsf {U}}^{i}$, where, recall, $\varPsi$ is the basis of $V_h$, we have $u^{i}_h|_{{\omega _\texttt {a}}} = \varPsi _\texttt {a}{\mathsf {U}}^{i}_\texttt {a}$ for $\varPsi _\texttt {a}\subset \varPsi$ a subset of basis functions that are nonvanishing on ${\omega _\texttt {a}}$ and ${\mathsf {U}}^{i}_\texttt {a}$ the associated entries of ${\mathsf {U}}^{i}$. Then

$$\begin{aligned} {\mathsf {E}}_{\texttt {a},u^{i}_h}&= {{\mathbf {\mathsf{{E}}}}}_{\texttt {a},\varPsi _\texttt {a}} {\mathsf {U}}^{i}_\texttt {a},&{{\mathbf {\mathsf{{E}}}}}_{\texttt {a},\varPsi _\texttt {a}}&= \begin{bmatrix} \overline{{{\mathbf {\mathsf{{E}}}}}}_{\texttt {a},\varPsi _\texttt {a}} \\ \underline{{{\mathbf {\mathsf{{E}}}}}}_{\texttt {a},\varPsi _\texttt {a}} \end{bmatrix},&\begin{array}{rl} \big (\overline{{{\mathbf {\mathsf{{E}}}}}}_{\texttt {a},\varPsi _\texttt {a}}\big )_{k j} &{}= (\psi _{\texttt {a}}\nabla \psi _j, \varvec{\phi }_k)_{{\omega _\texttt {a}}},\\ \big (\underline{{{\mathbf {\mathsf{{E}}}}}}_{\texttt {a},\varPsi _\texttt {a}}\big )_{\ell j} &{}= (\nabla \psi _j \cdot \nabla \psi _{\texttt {a}}, \widetilde{\chi }_{\ell })_{\omega _\texttt {a}}, \end{array} \end{aligned}$$

where ${\psi }_j \in \varPsi _{\texttt {a}},\, \varvec{\phi }_{k} \in \varvec{\varPhi }_{\texttt {a}},\, \widetilde{\chi }_{\ell } \in \widetilde{\mathcal {X}}_{\texttt {a}}$. Similarly, denoting by ${\mathcal {X}}_{\texttt {a}}$ the basis of $\mathbb {P}_{p}(\mathcal {T}_{\texttt {a}})$, we have for the coefficient vector $\widehat{{\mathsf {R}}}^i_\texttt {a}$ such that $r^{i}_h|_{{\omega _\texttt {a}}} = {\mathcal {X}}_{\texttt {a}} \widehat{{\mathsf {R}}}^i_\texttt {a}$,

$$\begin{aligned} {\mathsf {E}}_{\texttt {a},r^{i}_h}&= {{\mathbf {\mathsf{{E}}}}}_{\texttt {a},{\mathcal {X}}_{\texttt {a}}} \widehat{{\mathsf {R}}}^i_\texttt {a},&{{\mathbf {\mathsf{{E}}}}}_{\texttt {a},{\mathcal {X}}_{\texttt {a}}}&= \begin{bmatrix} 0 \\ \underline{{{\mathbf {\mathsf{{E}}}}}}_{\texttt {a},{\mathcal {X}}_{\texttt {a}}} \end{bmatrix},&\big (\underline{{{\mathbf {\mathsf{{E}}}}}}_{\texttt {a},{\mathcal {X}}_{\texttt {a}}}\big )_{\ell j} = (\chi _j \psi _{\texttt {a}}, \widetilde{\chi }_{\ell })_{\omega _\texttt {a}}, \end{aligned}$$

where $\chi _j \in {\mathcal {X}}_{\texttt {a}}, \,\widetilde{\chi }_{\ell } \in \widetilde{\mathcal {X}}_{\texttt {a}}$. Consequently, the vector ${\mathsf {D}}^i_\texttt {a}$ can be assembled as

$$\begin{aligned} {\mathsf {D}}^i_\texttt {a}= {{\mathbf {\mathsf{{K}}}}}_\texttt {a}^{-1} {\mathsf {E}}_{\texttt {a},f} - \left( {{\mathbf {\mathsf{{K}}}}}_\texttt {a}^{-1} \, {{\mathbf {\mathsf{{E}}}}}_{\texttt {a},\varPsi _\texttt {a}} \right) {\mathsf {U}}^{i}_\texttt {a}- \left( {{\mathbf {\mathsf{{K}}}}}_\texttt {a}^{-1} \, {{\mathbf {\mathsf{{E}}}}}_{\texttt {a},\mathcal {X}_\texttt {a}} \right) \widehat{{\mathsf {R}}}^i_\texttt {a}. \end{aligned}$$

(A.3)

This means that we can solve the system with ${{\mathbf {\mathsf{{K}}}}}_\texttt {a}$ only once with multiple right-hand sides $[{\mathsf {E}}_{\texttt {a},f}, {{\mathbf {\mathsf{{E}}}}}_{\texttt {a},\varPsi _\texttt {a}}, {{\mathbf {\mathsf{{E}}}}}_{\texttt {a},\mathcal {X}_\texttt {a}}]$ prior the start of the iterative solution of (2.5) and, at any iteration i, get the local coefficients $\overline{{\mathsf {D}}}^i_{\texttt {a}}$ of the flux reconstruction $\mathbf d ^{i}_h$ simply by matrix-vector multiplication and summing the vectors. This is particularly appealing when the error estimator is evaluated many times (e.g. when many iterations of the algebraic solver are performed). Note that assembling ${{\mathbf {\mathsf{{K}}}}}_\texttt {a}$, ${\mathsf {E}}_{\texttt {a},f}$, ${{\mathbf {\mathsf{{E}}}}}_{\texttt {a},\varPsi _\texttt {a}}$, ${{\mathbf {\mathsf{{E}}}}}_{\texttt {a},\mathcal {X}_\texttt {a}}$, $\texttt {a}\in {\mathcal {V}^\mathrm {}_h}$, and solving the systems corresponding to (A.3) can be done in parallel (indeed, the individual patch problems (A.2) are mutually independent). Also, this can be done independently of assembling the system (2.5).

B Efficiency of the total error bound

We prove in this appendix the global and local efficiency of the upper bound of Theorem 1, which follows and extends the results in [24, 25, 47]. To simplify the presentation, we require that the source term f is piecewise polynomial, $f\in \mathbb {P}_{p'-1}(\mathcal {T}_h)$. Consequently, we choose $f_h = f$, and the oscillation term vanishes, $\eta _{{\text {osc}}}= 0$.

The following lemma extends [14, Theorem 3.1] and [9, p. 1191] (see also [25, Lemma 3.12]) to the inexact algebraic solver case considered in this paper. Recall the space $H^1_\mathrm {*}({{\omega _\texttt {a}}})$ introduced in (4.11).

Lemma 1

Let $\texttt {a}\in {\mathcal {V}^\mathrm {}_h}$ and let $m_{\texttt {a}}\in H^1_\mathrm {*}({{\omega _\texttt {a}}})$ be the solution of

$$\begin{aligned} (\nabla m_{\texttt {a}}, \nabla v)_{{\omega _\texttt {a}}} = \left( f, \psi _{\texttt {a}}v\right) _{{\omega _\texttt {a}}} - \left( \nabla u^{i}_h, \nabla (\psi _{\texttt {a}}v) \right) _{{\omega _\texttt {a}}} - \left( r^{i}_h, \psi _{\texttt {a}}v\right) _{{\omega _\texttt {a}}} \qquad \forall v\in H^1_\mathrm {*}({{\omega _\texttt {a}}}). \end{aligned}$$

(B.1)

Then there holds

$$\begin{aligned} \Vert \nabla m_{\texttt {a}}\Vert _{{\omega _\texttt {a}}}\le & {} C_{\mathrm {cont,PF}, {\omega _\texttt {a}}}\left( \Vert \nabla (u-u^{i}_h)\Vert _{{\omega _\texttt {a}}} + \Vert \mathbf{d}_h^{i+\nu } - \mathbf{d}_h^i\Vert _{{\omega _\texttt {a}}} \right) \\&+ {C_{\mathrm {PF}, {\omega _\texttt {a}}}h_{{\omega _\texttt {a}}}} \Vert r^{i+\nu }_h\Vert _{{\omega _\texttt {a}}}. \end{aligned}$$

Proof

From (B.1) and since, for $v\in H^1_\mathrm {*}({{\omega _\texttt {a}}})$, $\psi _{\texttt {a}}v\in H^1_0({\omega _\texttt {a}})$, we have, employing (2.2),

$$\begin{aligned} ( \nabla m_{\texttt {a}}, \nabla v)_{{\omega _\texttt {a}}} = \left( \nabla (u - u^{i}_h) , \nabla (\psi _{\texttt {a}}v) \right) _{{\omega _\texttt {a}}} - \left( r^{i}_h, \psi _{\texttt {a}}v\right) _{{\omega _\texttt {a}}}. \end{aligned}$$

The Cauchy–Schwarz inequality and the bound (4.13) give

$$\begin{aligned} \left( \nabla (u - u^{i}_h) , \nabla (\psi _{\texttt {a}}v) \right) _{{\omega _\texttt {a}}} \le \Vert \nabla (u - u^{i}_h) \Vert _{{\omega _\texttt {a}}} C_{\mathrm {cont,PF}, {\omega _\texttt {a}}}\Vert \nabla v\Vert _{{\omega _\texttt {a}}}. \end{aligned}$$

Using (4.10), the Cauchy–Schwarz inequality, and (4.12),

$$\begin{aligned} \left( r^{i}_h, \psi _{\texttt {a}}v\right) _{{\omega _\texttt {a}}}&= \big ( \displaystyle {\nabla \cdot \,}\mathbf d ^{i+\nu }_h - \displaystyle {\nabla \cdot \,}\mathbf d ^{i}_h+ r^{i+\nu }_h , \psi _{\texttt {a}}v\big )_{{\omega _\texttt {a}}} \\&= \big ( - \mathbf d ^{i+\nu }_h + \mathbf d ^{i}_h, \nabla (\psi _{\texttt {a}}v) \big )_{{\omega _\texttt {a}}} + \big ( r^{i+\nu }_h ,\psi _{\texttt {a}}v\big )_{{\omega _\texttt {a}}} \\&\le \Vert \mathbf d ^{i+\nu }_h - \mathbf d ^{i}_h\Vert _{{\omega _\texttt {a}}} C_{\mathrm {cont,PF}, {\omega _\texttt {a}}}\Vert \nabla v\Vert _{{\omega _\texttt {a}}} + \Vert r^{i+\nu }_h \Vert _{{\omega _\texttt {a}}} \Vert \psi _{\texttt {a}}\Vert _{\infty , {\omega _\texttt {a}}} \Vert v\Vert _{{\omega _\texttt {a}}} \\&\le \Vert \mathbf d ^{i+\nu }_h - \mathbf d ^{i}_h\Vert _{{\omega _\texttt {a}}} C_{\mathrm {cont,PF}, {\omega _\texttt {a}}}\Vert \nabla v\Vert _{{\omega _\texttt {a}}} + \Vert r^{i+\nu }_h \Vert _{{\omega _\texttt {a}}} C_{\mathrm {PF}, {\omega _\texttt {a}}}h_{{\omega _\texttt {a}}} \Vert \nabla v\Vert _{{\omega _\texttt {a}}} . \end{aligned}$$

Finally, using

$$\begin{aligned} \Vert \nabla m_{\texttt {a}}\Vert _{{\omega _\texttt {a}}} = \sup _{ v\in H^1_\mathrm {*}({{\omega _\texttt {a}}}), \Vert \nabla v\Vert = 1} {(\nabla m_{\texttt {a}}, \nabla v)_{{\omega _\texttt {a}}}} \end{aligned}$$

and combining the above results yields the desired bound. $\square$

The following crucial result has been shown in [9, Theorem 7] (see also [25, Corollary 3.16]) in the two-dimensional case. The three-dimensional proof is in [26, Corollary 3.3].

Lemma 2

Let $\mathbf{d}^{i}_{h,a}$ be given by (A.1) with ${p'=p+1}$ and let $m_{\texttt {a}}$ be given by (B.1). Let $f\in \mathbb {P}_{p}(\mathcal {T}_h)$. Then there exists a constant $C_{\mathrm {st}, {\omega _\texttt {a}}}>0$ depending only on the shape of elements of the patch $\mathcal {T}_{\texttt {a}}$ but not on their diameters such that

$$\begin{aligned} \Vert \psi _{\texttt {a}}\nabla u^{i}_h+ \mathbf{d}^{i}_{h,a}\Vert _{{\omega _\texttt {a}}} \le C_{\mathrm {st}, {\omega _\texttt {a}}}\Vert \nabla m_{\texttt {a}}\Vert _{{\omega _\texttt {a}}}. \end{aligned}$$

(B.2)

The constant $C_{\mathrm {st}, {\omega _\texttt {a}}}$ is not computable. It can, however, be bounded from above considering a finite-dimensional subspace of $H^1_\mathrm {*}({{\omega _\texttt {a}}})$ and solving the discrete version of the problem (B.1); see [25, Lemma 3.23]. Hereafter we denote

$$\begin{aligned} C_{\mathrm {cont,PF}}\equiv \max _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} C_{\mathrm {cont,PF}, {\omega _\texttt {a}}}, \quad C_{\mathrm {PF}}\equiv \max _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} C_{\mathrm {PF}, {\omega _\texttt {a}}}, \quad C_{\mathrm {st}}\equiv \max _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} C_{\mathrm {st}, {\omega _\texttt {a}}}. \end{aligned}$$

We now state the main result on the global efficiency of the estimators of Theorem 1, both for the global stopping criteria in the sense of [24, 34] and for the secure stopping criterion in the sense of (6.3), relying on the estimator $\mu ^{i}_{{\text {total}}}$ of Theorem 2:

Theorem 7

(Global efficiency) Let the estimators of Theorem 1 satisfy the global stopping criteria

$$\begin{aligned} C_{\mathrm {F}}h_{\varOmega }\Vert r^{i+\nu }_h\Vert&\le \gamma _{{\text {rem}}} \Vert \mathbf{d}_h^{i+\nu } - \mathbf{d}_h^i\Vert , \end{aligned}$$

(B.3a)

$$\begin{aligned} \Vert \mathbf{d}_h^{i+\nu } - \mathbf{d}_h^i\Vert&\le \gamma _{{\text {alg}}} \Vert \nabla u^{i}_h+ \mathbf{d}_h^i\Vert \end{aligned}$$

(B.3b)

with positive parameters $\gamma _{{\text {rem}}}$, $\gamma _{{\text {alg}}}$ such that

$$\begin{aligned} \gamma _{{\text {alg}}} C_{\mathrm {st}}\left( C_{\mathrm {cont,PF}}+ \gamma _{{\text {rem}}} \frac{C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}}h_{{\omega _\texttt {a}}}}{C_{\mathrm {F}}h_{\varOmega }} \right) \le \frac{1}{2(d+1)}. \end{aligned}$$

(B.4)

Alternatively, instead of (B.3)–(B.4), let

$$\begin{aligned} C_{\mathrm {F}}h_{\varOmega }\Vert r^{i+\nu }_h\Vert&\le \gamma _{{\text {rem}}} \Vert \mathbf{d}_h^{i+\nu } - \mathbf{d}_h^i\Vert , \end{aligned}$$

(B.5a)

$$\begin{aligned} \Vert \mathbf{d}_h^{i+\nu } - \mathbf{d}_h^i\Vert&\le \frac{\gamma _{{\text {alg}}}}{(1+\gamma _{{\text {alg}}}^2)^{1/2}} \;\mu ^{i}_{{\text {total}}} \end{aligned}$$

(B.5b)

without any requirement on $\gamma _{{\text {rem}}}$, $\gamma _{{\text {alg}}}$, supposing only

$$\begin{aligned} \frac{C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}}h_{{\omega _\texttt {a}}}}{C_{\mathrm {F}}h_{\varOmega }} \le C_{\mathrm {cont,PF}}\end{aligned}$$

that is typically satisfied, apart possibly the coarsest meshes. Let the assumptions of Lemma 2 hold. Then the upper bound of Theorem 1 is globally efficient,

$$\begin{aligned} \eta ^{i,\nu }_{{\text {total}}}\le C_{{\text {glob. eff.}}} \Vert \nabla (u - u^{i}_h)\Vert \end{aligned}$$

with the global efficiency constant

$$\begin{aligned} C_{{\text {glob. eff.}}} \equiv (1+ \gamma _{{\text {alg}}} + \gamma _{{\text {alg}}} \gamma _{{\text {rem}}}) 2(d+1)C_{\mathrm {st}}C_{\mathrm {cont,PF}}. \end{aligned}$$

Recall that $\mathcal {V}_K$ stands for the vertices of the element K and consider the functions $m_{h,\texttt {a}}$ specified in Theorem 2. Then the local version of Theorem 7 proving the local efficiency under the local stopping criteria is as follows:

Theorem 8

(Local efficiency) Let, for an element $K \in \mathcal {T}_h$, the estimators of Theorem 1 satisfy the local stopping criteria

$$\begin{aligned} C_{\mathrm {F}}h_{\varOmega }\Vert r^{i+\nu }_h\Vert _{K'}&\le \gamma _{{\text {rem}},K} \Vert \mathbf{d}_h^{i+\nu } - \mathbf{d}_h^i\Vert _{K'}&\forall K' \in \mathcal {T}_h \text{ such } \text{ that } K' \cap K \ne \emptyset , \end{aligned}$$

(B.6a)

$$\begin{aligned} \Vert \mathbf{d}_h^{i+\nu } - \mathbf{d}_h^i\Vert _{{\omega _\texttt {a}}}&\le \gamma _{{\text {alg}},K} \Vert \nabla u^{i}_h+ \mathbf{d}_h^i\Vert _K&\forall \texttt {a}\in {\mathcal {V}_K} \end{aligned}$$

(B.6b)

with positive parameters $\gamma _{{\text {rem}},K}$, $\gamma _{{\text {alg}},K}$ such that

$$\begin{aligned} \gamma _{{\text {alg}},K} C_{\mathrm {st}}\left( C_{\mathrm {cont,PF}}+ \gamma _{{\text {rem}},K} \frac{C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}_K}}h_{{\omega _\texttt {a}}}}{C_{\mathrm {F}}h_{\varOmega }} \right) \le \frac{1}{2(d+1)}. \end{aligned}$$

(B.7)

Alternatively, instead of (B.6)–(B.7), let, for all $\texttt {a}\in {\mathcal {V}_K}$,

$$\begin{aligned} C_{\mathrm {F}}h_{\varOmega }\Vert r^{i+\nu }_h\Vert _{\omega _\texttt {a}}&\le \gamma _{{\text {rem}},K} \Vert \mathbf{d}_h^{i+\nu } - \mathbf{d}_h^i\Vert _{\omega _\texttt {a}}, \end{aligned}$$

(B.8a)

$$\begin{aligned} \Vert \mathbf{d}_h^{i+\nu } - \mathbf{d}_h^i\Vert _{\omega _\texttt {a}}&\le \frac{\gamma _{{\text {alg}},K}}{(1+\gamma _{{\text {alg}},K}^2)^{1/2}} \frac{ \Vert \nabla m_{h,\texttt {a}}\Vert _{\omega _\texttt {a}}}{C_{\mathrm {cont,PF}, {\omega _\texttt {a}}}}, \end{aligned}$$

(B.8b)

without any requirement on $\gamma _{{\text {rem}},K}$, $\gamma _{{\text {alg}},K}$, supposing only

$$\begin{aligned} \frac{C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}_K}}h_{{\omega _\texttt {a}}}}{C_{\mathrm {F}}h_{\varOmega }} \le C_{\mathrm {cont,PF}}\end{aligned}$$

that is typically satisfied, apart possibly the coarsest meshes. Let the assumptions of Lemma 2 hold. Then we have the local efficiency of the upper bound,

$$\begin{aligned} \Vert \mathbf{d}_h^{i+\nu } - \mathbf{d}_h^i\Vert _K + C_{\mathrm {F}}h_{\varOmega }\Vert r^{i+\nu }_h\Vert _K + \Vert \nabla u^{i}_h+ \mathbf{d}_h^i\Vert _K \le C_{{\text {loc. eff.}}, K} \sum _{\texttt {a}\in {\mathcal {V}_K}}\Vert \nabla (u - u^{i}_h)\Vert _{{\omega _\texttt {a}}} \end{aligned}$$

with the local efficiency constant

$$\begin{aligned} C_{{\text {loc. eff.}}, K} \equiv (1 + \gamma _{{\text {alg}},K} + \gamma _{{\text {alg}},K} \gamma _{{\text {rem}},K}) 2 C_{\mathrm {st}}C_{\mathrm {cont,PF}}. \end{aligned}$$

Proof of Theorem 7

From the flux construction (A.1) of $\mathbf d ^{i}_h$, using (B.2),

$$\begin{aligned} \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert ^2&= \sum _{K\in \mathcal {T}_h}{\Big \Vert \sum _{\texttt {a}\in {\mathcal {V}_K}} (\psi _{\texttt {a}}\nabla u^{i}_h+ \mathbf d ^{i}_{h,\texttt {a}}) \Big \Vert _K^2} \\&\le (d+1) \sum _{K\in \mathcal {T}_h} \sum _{\texttt {a}\in {\mathcal {V}_K}} \Vert \psi _{\texttt {a}}\nabla u^{i}_h+ \mathbf d ^{i}_{h,\texttt {a}}\Vert ^2_K\\&= (d+1) \sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} \Vert \psi _{\texttt {a}}\nabla u^{i}_h+ \mathbf d ^{i}_{h,\texttt {a}}\Vert ^2_{{\omega _\texttt {a}}} \\&\le (d+1) C_{\mathrm {st}}^2 \sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} \Vert \nabla m_{\texttt {a}}\Vert ^2_{{\omega _\texttt {a}}}, \end{aligned}$$

as any element $K\in \mathcal {T}_h$ has $d+1$ vertices. From Lemma 1, we have

$$\begin{aligned}&\bigg [\sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} \Vert \nabla m_{\texttt {a}}\Vert _{{\omega _\texttt {a}}}^2 \bigg ]^{1/2} \le \bigg [ \sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} C_{\mathrm {cont,PF}, {\omega _\texttt {a}}}^2 \Vert \nabla (u-u^{i}_h)\Vert ^2_{{\omega _\texttt {a}}} \bigg ]^{1/2}\\&\quad + \bigg [ \sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} C_{\mathrm {cont,PF}, {\omega _\texttt {a}}}^2 \Vert \mathbf d ^{i+\nu }_h - \mathbf d ^{i}_h\Vert ^2_{{\omega _\texttt {a}}} \bigg ]^{1/2} + \bigg [ \sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} {C_{\mathrm {PF}, {\omega _\texttt {a}}}^2 (h_{{\omega _\texttt {a}}})^2} \Vert r^{i+\nu }_h\Vert ^2_{{\omega _\texttt {a}}} \bigg ]^{1/2}. \end{aligned}$$

Therefore, using $\left[ \sum _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}} \Vert z\Vert ^2_{{\omega _\texttt {a}}} \right] ^{1/2} = (d+1)^{1/2} \Vert z\Vert$,

$$\begin{aligned}&\Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert \le {} (d+1) C_{\mathrm {st}}C_{\mathrm {cont,PF}}\Vert \nabla (u-u^{i}_h)\Vert \nonumber \\&\qquad + (d+1) C_{\mathrm {st}}C_{\mathrm {cont,PF}}\Vert \mathbf d ^{i+\nu }_h - \mathbf d ^{i}_h\Vert \;+\; (d+1) C_{\mathrm {st}}C_{\mathrm {PF}}{\max _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}}h_{{\omega _\texttt {a}}} } \Vert r^{i+\nu }_h\Vert .\qquad \end{aligned}$$

(B.9)

From the stopping criteria (B.3),

$$\begin{aligned} \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert \le {}&(d+1) C_{\mathrm {st}}C_{\mathrm {cont,PF}}\Vert \nabla (u-u^{i}_h)\Vert \\ {}&+ (d+1) \gamma _{{\text {alg}}} C_{\mathrm {st}}\left( C_{\mathrm {cont,PF}}+ \gamma _{{\text {rem}}} \frac{C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}}h_{{\omega _\texttt {a}}}}{C_{\mathrm {F}}h_{\varOmega }}\right) \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert , \end{aligned}$$

and from (B.4),

$$\begin{aligned} \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert \le 2 (d+1) C_{\mathrm {st}}C_{\mathrm {cont,PF}}\Vert \nabla (u-u^{i}_h)\Vert . \end{aligned}$$

Finally, we get the assertion for the stopping criteria (B.3),

$$\begin{aligned} \eta ^{i,\nu }_{{\text {total}}}= & {} \Vert \mathbf d ^{i+\nu }_h - \mathbf d ^{i}_h\Vert + C_{\mathrm {F}}h_{\varOmega }\Vert r^{i+\nu }_h\Vert + \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert \\\le & {} (1+\gamma _{{\text {alg}}}+\gamma _{{\text {alg}}}\gamma _{{\text {rem}}}) \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert \le C_{{\text {glob. eff.}}} \Vert \nabla (u-u^{i}_h)\Vert . \end{aligned}$$

The efficiency under the stopping criteria (B.5) actually does not request any restrictive assumptions of the form (B.4). Using (B.5b) and the bound of Theorem 2,

$$\begin{aligned} \Vert \mathbf d ^{i+\nu }_h - \mathbf d ^{i}_h\Vert \le \frac{\gamma _{{\text {alg}}}}{(1+\gamma _{{\text {alg}}}^2)^{1/2}} \Vert \nabla (u-u^{i}_h)\Vert . \end{aligned}$$

Now a combination with (B.9) and (B.5a) gives

$$\begin{aligned}&\Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert \le (d+1) C_{\mathrm {st}}C_{\mathrm {cont,PF}}\Vert \nabla (u-u^{i}_h)\Vert \\&\quad + (d+1) \frac{\gamma _{{\text {alg}}}}{(1+\gamma _{{\text {alg}}}^2)^{1/2}} C_{\mathrm {st}}\left( C_{\mathrm {cont,PF}}+ \gamma _{{\text {rem}}} \frac{C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}}h_{{\omega _\texttt {a}}}}{C_{\mathrm {F}}h_{\varOmega }}\right) \Vert \nabla (u-u^{i}_h)\Vert , \end{aligned}$$

so that the assertion for the stopping criteria (B.5) follows with the constant

$$\begin{aligned}&(d\!+\!1) C_{\mathrm {st}}\left( C_{\mathrm {cont,PF}}\!+\! \frac{\gamma _{{\text {alg}}}}{(1\!+\!\gamma _{{\text {alg}}}^2)^{1/2}} C_{\mathrm {cont,PF}}\!+\! \gamma _{{\text {rem}}} \frac{\gamma _{{\text {alg}}}}{(1\!+\!\gamma _{{\text {alg}}}^2)^{1/2}} \frac{C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}^\mathrm {}_h}}h_{{\omega _\texttt {a}}}}{C_{\mathrm {F}}h_{\varOmega }} \right) \\&\quad \le (1+ \gamma _{{\text {alg}}} + \gamma _{{\text {alg}}} \gamma _{{\text {rem}}}) (d+1)C_{\mathrm {st}}C_{\mathrm {cont,PF}}\le \frac{C_{{\text {glob. eff.}}}}{2}. \end{aligned}$$

$\square$

Proof of Theorem 8

For the proof of the local efficiency, we first note that

$$\begin{aligned} \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert _K \le \sum _{\texttt {a}\in {\mathcal {V}_K}} \Vert \psi _{\texttt {a}}\nabla u^{i}_h+ \mathbf d ^{i}_{h,\texttt {a}}\Vert _{{\omega _\texttt {a}}} \le \sum _{\texttt {a}\in {\mathcal {V}_K}} C_{\mathrm {st}, {\omega _\texttt {a}}}\Vert \nabla m_{\texttt {a}}\Vert _{{\omega _\texttt {a}}}. \end{aligned}$$

From Lemma 1,

$$\begin{aligned}&\Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert _K \le C_{\mathrm {st}}C_{\mathrm {cont,PF}}\sum _{\texttt {a}\in {\mathcal {V}_K}} \Vert \nabla (u-u^{i}_h)\Vert _{{\omega _\texttt {a}}} \nonumber \\&\quad + C_{\mathrm {st}}C_{\mathrm {cont,PF}}\sum _{\texttt {a}\in {\mathcal {V}_K}} \Vert \mathbf d ^{i+\nu }_h - \mathbf d ^{i}_h\Vert _{{\omega _\texttt {a}}} + C_{\mathrm {st}}C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}_K}}h_{{\omega _\texttt {a}}} \sum _{\texttt {a}\in {\mathcal {V}_K}} \Vert r^{i+\nu }_h\Vert _{{\omega _\texttt {a}}}.\qquad \qquad \end{aligned}$$

(B.10)

Thus, under the stopping criteria (B.6),

$$\begin{aligned}&\Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert _K \le {} C_{\mathrm {st}}C_{\mathrm {cont,PF}}\sum _{\texttt {a}\in {\mathcal {V}_K}} \Vert \nabla (u-u^{i}_h)\Vert _{{\omega _\texttt {a}}} \\&\quad + \,(d+1) C_{\mathrm {st}}\gamma _{{\text {alg}},K} \left( C_{\mathrm {cont,PF}}+ \gamma _{{\text {rem}},K} \frac{C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}_K}}h_{{\omega _\texttt {a}}}}{C_{\mathrm {F}}h_{\varOmega }} \right) \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert _K. \end{aligned}$$

From (B.7), we further obtain

$$\begin{aligned} \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert _K \le 2 C_{\mathrm {st}}C_{\mathrm {cont,PF}}\sum _{\texttt {a}\in {\mathcal {V}_K}} \Vert \nabla (u-u^{i}_h)\Vert _{{\omega _\texttt {a}}}, \end{aligned}$$

so that finally

$$\begin{aligned}&\Vert \mathbf d ^{i+\nu }_h - \mathbf d ^{i}_h\Vert _K + C_{\mathrm {F}}h_{\varOmega }\Vert r^{i+\nu }_h\Vert _K + \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert _K \\&\quad \le (1+\gamma _{{\text {alg}},K}+\gamma _{{\text {alg}},K}\gamma _{{\text {rem}},K}) \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert \\&\quad \le C_{{\text {loc. eff.}}, K} \sum _{\texttt {a}\in {\mathcal {V}_K}}\Vert \nabla (u - u^{i}_h)\Vert _{{\omega _\texttt {a}}}. \end{aligned}$$

Let $\widetilde{m}_{\texttt {a}}\in H^1_\mathrm {*}({{\omega _\texttt {a}}})$ be the solution of

$$\begin{aligned} (\nabla \widetilde{m}_{\texttt {a}}, \nabla v)_{{\omega _\texttt {a}}} = \left( f, \psi _{\texttt {a}}v\right) _{{\omega _\texttt {a}}} - \left( \nabla u^{i}_h, \nabla (\psi _{\texttt {a}}v) \right) _{{\omega _\texttt {a}}} \qquad \forall v\in H^1_\mathrm {*}({{\omega _\texttt {a}}}), \end{aligned}$$

in the continuous counterpart to $m_{h,\texttt {a}}$ of Theorem 2 and similarly to (B.1). The fact that $m_{h,\texttt {a}}$ is a projection of $\widetilde{m}_{\texttt {a}}$ from $H^1_\mathrm {*}({{\omega _\texttt {a}}})$ onto $W^{\texttt {a}}_h$ gives $\Vert \nabla m_{h,\texttt {a}}\Vert _{\omega _\texttt {a}}\le \Vert \nabla \widetilde{m}_{\texttt {a}}\Vert _{\omega _\texttt {a}}$. Proceeding as in the proof of Lemma 1 with $r^{i}_h= 0$, we get the inequality $\Vert \nabla \widetilde{m}_{\texttt {a}}\Vert _{{\omega _\texttt {a}}} \le C_{\mathrm {cont,PF}, {\omega _\texttt {a}}}\Vert \nabla (u-u^{i}_h)\Vert _{{\omega _\texttt {a}}}$, so that

$$\begin{aligned} \Vert \nabla m_{h,\texttt {a}}\Vert _{\omega _\texttt {a}}\le C_{\mathrm {cont,PF}, {\omega _\texttt {a}}}\Vert \nabla (u-u^{i}_h)\Vert _{\omega _\texttt {a}}. \end{aligned}$$

Thus, under the secure local stopping criterion (B.8b), we obtain

$$\begin{aligned} \Vert \mathbf d ^{i+\nu }_h - \mathbf d ^{i}_h\Vert _{\omega _\texttt {a}}\le \frac{\gamma _{{\text {alg}},K}}{(1+\gamma _{{\text {alg}},K}^2)^{1/2}} \Vert \nabla (u-u^{i}_h)\Vert _{\omega _\texttt {a}}, \end{aligned}$$

and, employing (B.10) and (B.8a),

$$\begin{aligned} \Vert \nabla u^{i}_h+\mathbf d ^{i}_h\Vert _K\le & {} C_{\mathrm {st}}C_{\mathrm {cont,PF}}\sum _{\texttt {a}\in {\mathcal {V}_K}} \Vert \nabla (u-u^{i}_h)\Vert _{{\omega _\texttt {a}}}\\&+\, C_{\mathrm {st}}\frac{\gamma _{{\text {alg}},K}}{(1+\gamma _{{\text {alg}},K}^2)^{1/2}} \left( C_{\mathrm {cont,PF}}+ \gamma _{{\text {rem}},K} \frac{C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}_K}}h_{{\omega _\texttt {a}}}}{C_{\mathrm {F}}h_{\varOmega }} \right) \\&\times \sum _{\texttt {a}\in {\mathcal {V}_K}} \Vert \nabla (u-u^{i}_h)\Vert _{\omega _\texttt {a}}. \end{aligned}$$

The claim in this case thus follows from

$$\begin{aligned}&C_{\mathrm {st}}\left( C_{\mathrm {cont,PF}}\!+\! \frac{\gamma _{{\text {alg}},K}}{(1\!+\!\gamma _{{\text {alg}},K}^2)^{1/2}} C_{\mathrm {cont,PF}}\!+\! \gamma _{{\text {rem}},K} \frac{\gamma _{{\text {alg}},K}}{(1\!+\!\gamma _{{\text {alg}},K}^2)^{1/2}} \frac{C_{\mathrm {PF}}\max _{\texttt {a}\in {\mathcal {V}_K}}h_{{\omega _\texttt {a}}}}{C_{\mathrm {F}}h_{\varOmega }} \right) \\&\qquad \le (1+ \gamma _{{\text {alg}},K} + \gamma _{{\text {alg}},K} \gamma _{{\text {rem}},K}) C_{\mathrm {st}}C_{\mathrm {cont,PF}}\le \frac{C_{{\text {loc. eff.}}, K}}{2}. \end{aligned}$$

$\square$