Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements

1 Introduction

Equilibrated a posteriori error estimators have attracted much interest recently due to the guaranteed reliability bound with the reliability constant being one. This property implies that they are perfect for discretization error control on both coarse and fine meshes. Error control on coarse meshes is important but difficult for computationally challenging problems.

For the conforming finite element approximation, a mathematical foundation of equilibrated estimators is the Prager-Synge identity [31] that is valid in $H^{1} (Ω)$ (see Section 3). Based on this identity, various equilibrated estimators have been studied recently by many researchers (see, e.g., [27, 22, 29, 20, 21, 6, 3, 33, 10, 12, 13, 34, 17, 14]). The key ingredient of the equilibrated estimators for the continuous elements is local recovery of an equilibrated (locally conservative) flux in the $H (d i v; Ω)$ space through the numerical flux. By using a partition of unity, Ladevèze and Leguillon [27] initiated a local procedure to reduce the construction of an equilibrated flux to vertex patch based local calculations. For the continuous linear finite element approximation to the Poisson equation in two dimensions, an equilibrated flux in the lowest order Raviart-Thomas space was explicitly constructed in [10, 12]. This explicit approach does not lead to robust equilibrated estimator with respect to the coefficient jump without introducing a constraint minimization (see [17]). The constraint minimization on each vertex patch may be efficiently solved by first computing an equilibrated flux and then calculating a divergence free correction. For recent developments, see [14] and references therein.

The purpose of this paper is to develop and analyze equilibrated a posteriori error estimators for discontinuous elements including both nonconforming and discontinuous Galerkin elements. To do so, the first and the essential step is to extend the Prager-Synge identity to be valid for piecewise $H^{1} (Ω)$ functions. This will be done by establishing a generalized Prager-Synge inequality (see Theorem 3.1) that contains an additional term measuring the distance between $H^{1} (Ω)$ and piecewise $H^{1} (Ω)$ . Moreover, by using a Helmholtz decomposition, we will be able to show that the inequality becomes an identity in two dimensions (see Lemma 3.4). A non-optimal inequality similar to ours was obtained earlier by Braess, Fraunholz, and Hoppe in [11] for the Poisson equation with pure Dirichlet boundary condition. Based on the generalized Prager-Synge inequality and an equivalent form (see Corollary 3.2), the construction of an equilibrated a posteriori error estimator for discontinuous finite element solutions is reduced to recover an equilibrated flux in $H (div; Ω)$ and to recover either a potential function in $H^{1} (Ω)$

or a curl free vector-valued function in

H (c u r l; Ω)

Recovery of equilibrated fluxes for discontinuous elements has been studied by many researchers. For discontinuous Garlerkin (DG) methods, equilibrated fluxes in Raviart-Thomas (RT) spaces were explicitly reconstructed in [2] for linear elements and in [23] for higher order elements. For nonconforming finite element methods, existing explicit equilibrated flux recoveries in RT spaces seem to be limited to the linear Crouzeix-Raviart (CR) and the quadratic Fortin-Soulie elements by Marini [28] (see [1] in the context of estimator) and Kim [26], respectively. For higher order nonconforming elements, a local reconstruction procedure was proposed by Ainsworth and Rankin in [4] through solving element-wise minimization problems. The recovered flux is not in the RT spaces. Nevertheless, the resulting estimator provides a guaranteed upper bound.

In this paper, we will introduce a fully explicit post-processing procedure for recovering an equilibrated flux in the RT space of index $k - 1$ for the nonconforming elements of odd order of $k \geq 1$ . Currently, we are not able to extend our recovery technique to even orders. This is because our recovery procedure heavily depends on the finite element formulation and the properties of the nonconforming finite element space; moreover, structure of the nonconforming finite element spaces of even and odd orders are fundamentally different.

Recovery of a potential function in $H^{1} (Ω)$ for discontinuous elements was studied by some researchers (see, e.g., [4, 2, 11]). Local approaches for recovering equilibrated flux in [10, 12, 17, 13, 14] may be directly applied (at least in two dimensions) for computing an approximation to the gradient in the curl-free space. (As mentioned previously, this approach requires solutions of local constraint minimization problems over vertex patches.) The resulting a posteriori error estimator from either the potential or the gradient recoveries may be proved to be locally efficient. Nevertheless, to show independence of the efficiency constant on the jump, we have to assume that the distribution of the diffusion coefficient is quasi-monotone (see [30]).

In this paper, we will employ a simple averaging technique over edges to recover a gradient in $H (c u r l; Ω)$ . Due to the fact that the recovered gradient is not necessarily curl free, the reliability constant of the resulting estimator is no longer one. However, it turns out that the curl free constraint is not essential and, theoretically we are able to prove that the resulting estimator has the robust local reliability as well as the robust local efficiency without the quad-monotone assumption. This is compatible with our recent result in [15] on the residual error estimator for discontinuous elements.

This paper is organized as follows. The diffusion problem and the finite element mesh are introduced in Section 2. The generalized Prager-Synge inequality for piecewise $H^{1} (Ω)$ functions are established in Section 3. Explicit recoveries of an equilibrated flux and a gradient and the resulting a posteriori error estimator for discontinuous elements are described in Section 4. Global reliability and local efficiency of the estimator are proved in Section 5. Finally, numerical results are presented in Section 6.

2 Model problem

Let $Ω$ be a bounded polygonal domain in $R^{d}, d = 2, 3$ , with Lipschitz boundary $\partial Ω = {¯ ¯¯ ¯ Γ}_{D} \cup {¯ ¯¯ ¯ Γ}_{N}$ , where ${¯ ¯¯ ¯ Γ}_{D} \cap {¯ ¯¯ ¯ Γ}_{N} = \emptyset$ . For simplicity, assume that ${meas}_{d - 1} (Γ_{D}) \neq 0$ . Considering the diffusion problem:

- \nabla \cdot (A \nabla u) = f in Ω,

(2.1)

with boundary conditions

u = 0 on Γ_{D} and - A \nabla u \cdot n = g % on Γ_{N},

where $\nabla \cdot$ and $\nabla$ are the respective divergence and gradient operators; $n$ is the outward unit vector normal to the boundary; $f \in L^{2} (Ω)$ and $g \in H^{- 1 / 2} (Γ_{N})$ are given scalar-valued functions; and the diffusion coefficient $A (x)$

is symmetric, positive definite, and piecewise constant full tensor with respect to the domain

¯ ¯¯ ¯ Ω = \cup_{i = 1}^{n} {¯ ¯¯ ¯ Ω}_{i}

. Here we assume that the subdomain,

Ω_{i}

for

i = 1, \dots, n

, is open and polygonal.

We use the standard notations and definitions for the Sobolev spaces. Let

H_{D}^{1} (Ω) = {v \in H^{1} (Ω) : v = 0 on Γ_{D}} .

Then the corresponding variational problem of (2.1) is to find $u \in H_{D}^{1} (Ω)$ such that

a (u, v) := (A \nabla u, \nabla v) = (f, v) - {⟨ g, v ⟩}_{Γ_{N}}, \forall v \in H_{D}^{1} (Ω),

(2.2)

where $(\cdot, \cdot)_{ω}$ is the $L^{2}$ inner product on the domain $ω$ . The subscript $ω$ is omitted when $ω = Ω$ .

2.1 Triangulation

Let $T = {K}$ be a finite element partition of $Ω$ that is regular, and denote by $h_{K}$ the diameter of the element $K$ . Furthermore, assume that the interfaces,

Γ = {\partial Ω_{i} \cap \partial Ω_{j} : i \neq j and i, j = 1, \dots, n},

do not cut through any element $K \in T$ . Denote the set of all edges of the triangulation $T$ by

E := E_{I} \cup E_{D} \cup E_{N},

where $E_{I}$ is the set of interior element edges, and $E_{D}$ and $E_{N}$ are the sets of boundary edges belonging to the respective $Γ_{D}$ and $Γ_{N}$ . For each $F \in E$ , denote by $h_{F}$ the length of $F$ and by $n_{F}$ a unit vector normal to $F$ . Let $K_{F}^{+}$ and $K_{F}^{-}$ be the two elements sharing the common edge $F \in E_{I}$ such that the unit outward normal of $K_{F}^{-}$ coincides with $n_{F}$ . When $F \in E_{D} \cup E_{N}$ , $n_{F}$ is the unit outward normal to $\partial Ω$ and denote by $K_{F}^{-}$ the element having the edge $F$ .

3 Generalized Prager-Synge inequality

For the conforming finite element approximation, the foundation of the equilibrated a posteriori error estimator is the Prager-Synge identity [31]. That is, let $u \in H_{D}^{1} (Ω)$ be the solution of (2.1), then

∥A1/2∇(u−w)∥2+∥A−1/2\boldmathτ+A1/2∇u∥2=∥A−1/2\boldmathτ+A1/2∇w∥2

for all $w \in H_{D}^{1} (Ω)$ and for all $\boldmathτ∈Σf(Ω)$ , where $Σ_{f} (Ω)$ is the so-called equilibrated flux space defined by

Σf(Ω)={\boldmathτ∈H(div;Ω):∇⋅\boldmathτ=f in Ω and % \boldmathτ⋅n=gN}.

Here, $H (d i v; Ω) \subset L^{2} (Ω)^{d}$ denotes the space of all vector-valued functions whose divergence are in $L^{2} (Ω)$ . The Prager-Synge identity immediately leads to

∥A1/2∇(u−w)∥2≤inf\boldmathτ∈Σf(Ω)∥A−1/2\boldmathτ+A1/2∇w∥2.

(3.1)

Choosing $w \in H_{D}^{1} (Ω)$ to be the conforming finite element approximation, then (3.1) implies that

ητ:=∥A−1/2\boldmathτ+A1/2∇w∥,∀\boldmathτ∈Σf(Ω)

(3.2)

is a reliable estimator with the reliability constant being one.

We now proceed to establish a generalization of (3.1) for piecewise $H^{1} (Ω)$ functions with applications to nonconforming and discontinuous Galerkin finite element approximations. To this end, denote the broken $H^{1} (Ω)$ space with respect to $T$ by

H^{1} (T) = {v \in L^{2} (Ω) : v |_{K} \in H^{1} (K), \forall K \in T} .

Define $\nabla_{h}$ be the discrete gradient operator on $H^{1} (T)$ such that for any $v \in H^{1} (T)$

(\nabla_{h} v) |_{K} = \nabla (v |_{K}), \forall K \in T .

Theorem 3.1.

Let $u \in H_{D}^{1} (Ω)$ be the solution of (2.1). In both two and three dimensions, for all $w \in H^{1} (T)$ , we have

∥A1/2∇h(u−w)∥2≤inf\boldmathτ∈Σf(Ω)∥A−1/2\boldmathτ+A1/2∇hw∥2+infv∈H1D(Ω)∥A1/2∇h(v−w)∥2.

(3.3)

Proof.

Let $w \in H^{1} (T)$ , for all $\boldmathτ∈Σf(Ω)$ and for all $v \in H_{D}^{1} (Ω)$ , it follows from integration by parts and the Cauchy-Schwarz and Young’s inequalities that

$2(∇h(u−w),A∇u+\boldmathτ)$	$=$	$2(∇(u−v),A∇u+\boldmathτ)+2(∇h(v−w),A∇u+\boldmathτ)$	(3.4)
	$=$	$2(∇h(v−w),A∇u+\boldmathτ)$
	$\leq$	$∥A1/2∇h(v−w)∥2+∥A1/2∇u+A−1/2% \boldmathτ∥2.$

It is easy to see that

∥A1/2∇hw+A−1/2\boldmathτ∥2=∥A1/2∇h(u−w)∥2+∥A1/2∇u+A−1/2\boldmathτ∥2−2(∇h(u−w),A∇u+\boldmathτ),

which, together with (3.4), implies

			$∥ A^{1 / 2} \nabla_{h} (u - w) ∥^{2}$
		$=$	$∥A1/2∇hw+A−1/2\boldmathτ∥2−∥A1/2∇u+A−1/2\boldmathτ∥2+2(∇h(u−w),A∇u+\boldmathτ)$
		$\leq$	$∥A1/2∇hw+A−1/2\boldmathτ∥2+∥A1/2∇h(v−w)∥2$

for all $\boldmathτ∈Σf(Ω)$ and all $v \in H_{D}^{1} (Ω)$ . This implies the validity of (3.3) and, hence, the theorem. ∎

A suboptimal result for the Poisson equation ( $A = I$ ) with pure Dirichlet boundary condition is proved in [11] by Braess, Fraunholz, and Hoppe:

∥∇h(u−w)∥≤inf\boldmathτ∈Σf(Ω)∥∇w+\boldmathτ∥+2infv∈H10(Ω)∥∇h(v−w)∥.

Let $H (c u r l; Ω) \subset L^{2} (Ω)^{d}$ be the space of all vector-valued functions whose curl are in $L^{2} (Ω)$ , and denote its curl free subspace by

˚HD(curl;Ω)={\boldmathτ∈H(curl;Ω):∇×\boldmathτ=0 in Ω and \boldmathτ⋅t=0 on ΓD},

where $t$ denotes the tangent vector(s).

Corollary 3.2.

Let $u \in H_{D}^{1} (Ω)$ be the solution of (2.1). In both two and three dimensions, for all $w \in H^{1} (T)$ , we have

∥A1/2∇h(u−w)∥2≤inf\boldmathτ∈Σf(Ω)∥A−1/2\boldmathτ+A1/2∇hw∥2+inf% \boldmathγ∈˚HD(curl;Ω)∥A1/2(% \boldmathγ−∇hw)∥2.

(3.5)

Proof.

The result of (3.5) is an immediate consequence of (3.3) and the fact that $\nabla H_{D}^{1} (Ω) = {˚ H}_{D} (c u r l; Ω)$ . ∎

In the remaining section, we prove that, in two dimensions, the inequality (3.3) in Theorem 3.1 is indeed an equality. For each $F \in E$ , in two dimensions, assume that $n_{F} = (n_{_{1, F}}, n_{_{2, F}})$ , then denote by $t_{F} = (- n_{_{2, F}}, n_{_{1, F}})$ the unit vector tangent to $F$ and by $s_{_{F}}$ and $e_{_{F}}$ the start and end points of $F$ , respectively, such that $e_{_{F}} - s_{_{F}} = h_{F} t_{F}$ . Let

H = {v \in H^{1} (Ω) : \int_{Ω} v d x = 0 and \frac{\partial v}{\partial t} = 0 on Γ_{N}} .

For a vector-valued function $\boldmathτ=(τ1,τ2)∈H(curl;Ω)$ , define the curl operator by

∇×\boldmathτ=∂τ2∂x−∂τ1∂y.

For a scalar-valued function $v \in H^{1} (Ω)$ , define the formal adjoint operator of the curl by

\nabla^{⊥} v = (\frac{\partial v}{\partial y}, - \frac{\partial v}{\partial x}) .

For a fixed $w \in H^{1} (T)$ , there exist unique $ϕ \in H_{D}^{1} (Ω)$ and $ψ \in H$ for the following Helmholtz decomposition (see, e.g., [4]) such that

A \nabla_{h} (u - w) = A \nabla ϕ + \nabla^{⊥} ψ,

(3.6)

and $ϕ$ and $ψ$ satisfy

(A \nabla ϕ, \nabla v) = (A \nabla_{h} (u - w), \nabla v) \forall v \in H_{D}^{1} (Ω),

and

(A^{- 1} \nabla^{⊥} ψ, \nabla^{⊥} w) = (\nabla_{h} (u - w), \nabla^{⊥} w) \forall w \in H,

respectively. It is easy to see that $\nabla ϕ$ and $\nabla^{⊥} ψ$ are orthogonal with respect to the $L^{2}$ inner product, which yields

∥ A^{1 / 2} \nabla_{h} (u - w) ∥^{2} = ∥ A^{1 / 2} \nabla ϕ ∥^{2} + ∥ A^{- 1 / 2} \nabla^{⊥} ψ ∥^{2} .

(3.7)

Lemma 3.3.

Let $w$ be a fixed function in $H^{1} (T)$ and $ϕ$ and $ψ$ be the corresponding Helmholtz decomposition of $w$ given in (3.6). We have

inf\boldmathτ∈Σf(Ω)∥A−1/2% \boldmathτ+A1/2∇hw∥=∥A1/2∇ϕ∥ and infv∈H1D(Ω)∥A1/2∇h(v−w)∥=∥A−1/2∇⊥ψ∥.

(3.8)

Proof.

For any $\boldmathτ∈Σf(Ω)$ , (3.6) and integration by parts give

∥A1/2∇ϕ∥2=(A∇h(u−w),∇ϕ)=(A∇u+% \boldmathτ,∇ϕ)−(\boldmathτ+A∇hw,∇ϕ)=−(\boldmathτ+A∇hw,∇ϕ),

which, together with the Cauchy-Schwarz inequality and the choice $\boldmathτ=∇⊥ψ−A∇u∈Σf(Ω)$ , yields the first equality in (3.8) as follows:

∥A1/2∇ϕ∥≤inf\boldmathτ∈Σf(Ω)∥A−1/2\boldmathτ+A1/2∇hw∥≤∥A1/2∇h(u−w)−A−1/2∇⊥ψ∥=∥A1/2∇ϕ∥.

Now we proceed to prove the second equality in (3.8). For any $v \in H_{D}^{1} (Ω)$ , by (3.6) and integration by parts, we have

∥ A^{- 1 / 2} \nabla^{⊥} ψ ∥^{2} = (\nabla_{h} (u - w), \nabla^{⊥} ψ) = (\nabla_{h} (v - w), \nabla^{⊥} ψ) .

The second equality in (3.8) is then a consequence of the Cauchy-Schwartz inequality and the choice of $v = u - ϕ \in H_{D}^{1} (Ω)$ :

∥ A^{- 1 / 2} \nabla^{⊥} ψ ∥ \leq inf v \in H_{D}^{1} (Ω) ∥ A^{1 / 2} \nabla_{h} (v - w) ∥ \leq ∥ A^{1 / 2} \nabla_{h} (u - ϕ - w ∥ = ∥ A^{- 1 / 2} \nabla^{⊥} ψ ∥ .

This completes the proof of the lemma. ∎

Lemma 3.4.

Let $u \in H_{D}^{1} (Ω)$ be the solution of (2.1). In two dimensions, for all $w \in H^{1} (T)$ , we have

∥A1/2∇h(u−w)∥2=inf\boldmathτ∈Σf(Ω)∥A−1/2\boldmathτ+A1/2∇hw∥2+infv∈H1D(Ω)∥A1/2∇h(v−w)∥2.

(3.9)

Proof.

The identity (3.9) is a direct consequence of (3.7) and Lemma 3.3. ∎

Remark 3.5.

It is easy to see that if $w \in H_{D}^{1} (Ω)$ in Lemma 3.4, i.e., $w$ is conforming, the second part on the right of (3.9) vanishes. It is thus natural to refer $inf\boldmathτ∈Σf(Ω)∥A−1/2% \boldmathτ+A1/2∇hw∥2$ or $∥ A^{1 / 2} \nabla ϕ ∥$ as the conforming error and $inf v \in H_{D}^{1} (Ω) ∥ A^{1 / 2} \nabla_{h} (v - w) ∥^{2}$ or $∥ A^{- 1 / 2} \nabla^{⊥} ψ ∥$ as the nonconforming error.

For each $K \in T$ , denote by $Λ_{K}$ and $λ_{K}$

the maximal and minimal eigenvalues of

A_{K} = A |_{K}

, respectively. For each

F \in E

, let

Λ_{F}^{\pm} = Λ_{K_{F}^{\pm}}

λ_{_{F}}^{\pm} = λ_{K_{F}^{\pm}}

, and

λ_{F} = min {λ_{F}^{+}, λ_{F}^{-}}

F \in E_{I}

and

λ_{F} = λ_{F}^{-}

F \in E_{D} \cup E_{N}

. To this end, let

Λ_{T} = max K \in T Λ_{K} and λ_{T} = min K \in T λ_{K} .

Assume that each local matrix $A_{K}$

is similar to the identity matrix in the sense that its maximal and minimal eigenvalues are almost of the same size. More precisely, there exists a moderate size constant

κ > 0

such that

\frac{Λ_{K}}{λ_{K}} \leq κ, \forall K \in T .

Nevertheless, the ratio of global maximal and minimal eigenvalues, $Λ_{T} / λ_{T}$ , is allowed to be very large.

For a function $w \in H^{1} (T)$ , denote its traces on $F$ by $w |_{F}^{-} := (w |_{K_{F}^{-}}) |_{F}$ and $w |_{F}^{+} := (w |_{K_{F}^{+}}) |_{F}$ and the jump of $w$ across the edge $F$ by

[[w]] |_{F} = {\begin{matrix} w |_{F}^{-} - w |_{F}^{+}, & \forall F \in E_{I}, w |_{F}^{-}, & \forall F \in E_{D} \cup E_{N} . \end{matrix}

In the following lemma, we show the relationship between the nonconforming error and the residual based error of solution jump on edges. It is noted that the constant is robust with respect to the coefficient jump.

Lemma 3.6.

Let $w$ be a fixed function in $H^{1} (T)$ . In two dimensions, there exists a constant $C_{r}$ that is independent of the jump of the coefficient such that

inf\boldmathτ∈˚HD(curl;Ω)∥A1/2(\boldmathτ−∇hw)∥≤Cr⎛⎝∑F∈EI∪EDλFh−1F∥[[w]]∥20,F⎞⎠1/2.

(3.10)

Proof.

Let $ψ$ be given in the Helmholtz decomposition in (3.6), then integration by parts gives

∥ A^{- 1 / 2} \nabla^{⊥} ψ ∥^{2} = (\nabla_{h} (u - w), \nabla^{⊥} ψ) = - \sum F \in E_{I} \cup E_{D} \int_{F} [[w]] (\nabla^{⊥} ψ \cdot n_{F}) d s .

Without loss of generality, assume that $λ_{F}^{-} \leq λ_{F}^{+}$ for each $F \in E_{I}$ . It follows from Lemma 2.4 in [15] and the Cauchy-Schwarz inequality that

	$\sum F \in E_{I} \cup E_{D} \int_{F} [[w]] (\nabla^{⊥} ψ \cdot n_{F}) d s$	$\leq$	$C \sum F \in E_{I} \cup E_{D} h_{F}^{- 1 / 2} ∥ [[w]] ∥_{0, F} ∥ \nabla^{⊥} ψ ∥_{0, K_{F}^{-}}$
		$\leq$	$C {⎛ ⎝ \sum F \in E_{I} \cup E_{D} λ_{F} h_{F}^{- 1} ∥ [[w]] ∥_{0, F}^{2} ⎞ ⎠}_{F}^{1 / 2} ∥ A^{- 1 / 2} \nabla^{⊥} ψ ∥,$

which, together with the above equality, yields

∥ A^{- 1 / 2} \nabla^{⊥} ψ ∥ \leq C {⎛ ⎝ \sum F \in E_{I} \cup E_{D} λ_{F} h_{F}^{- 1} ∥ [[w]] ∥_{0, F}^{2} ⎞ ⎠}_{F}^{1 / 2} .

This completes the proof of the lemma. ∎

4 Error estimators and indicators

4.1 NC finite element approximation

For the convenience of readers, in this subsection we introduce the nonconforming finite element space and its properties.

Let $P_{k} (K)$ and $P_{k} (F)$ be the spaces of polynomials of degree less than or equal to $k$ on the element $K$ and $F$ , respectively. Define the nonconforming finite element space of order $k (k \geq 1)$ on the triangulation $T$ by

U^{k} (T) = {v \in L^{2} (Ω) : v |_{K} \in P_{k} (K), \forall K \in T and \int_{_{F}} [[v]] p d s = 0, \forall p \in P_{k - 1} (F), \forall F \in E_{I}}

(4.1)

and its subspace by

U_{D}^{k} (T) = {v \in U^{k} (T) : \int_{_{F}} v p d s = 0, \forall p \in P_{k - 1} (F) and \forall F \in E_{D}} .

The spaces defined above are exactly the same as those defined in [19] for $k = 1$ , [24] for $k = 2$ , [18] for $k = 4$ and $6$ , [4] for general odd order, and [32, 5] for general order. Then the nonconforming finite element approximation of order $k$ is to find $u_{_{T}} \in U_{D}^{k} (T)$ such that

a_{h} (u_{_{T}}, v) := (A \nabla_{h} u_{_{T}}, \nabla_{h} v) = (f, v) - {⟨ g, v ⟩}_{Γ_{N}}, \forall v \in U_{_{D}}^{k} (T) .

(4.2)

Below we describe basis functions of $U^{k} (T)$ and their properties. To this end, for each $K \in T$ , let $m_{k} = dim (P_{k - 3} (K))$ for $k > 3$ and $m_{k} = 0$ for $k \leq 3$ . Denote by ${x_{j}, j = 1, \dots, m_{k}}$ the set of all interior Lagrange points in $K$ with respect to the space $P_{k} (K)$ and by $P_{j, K} \in P_{k - 3} (K)$ the nodal basis function corresponding to $x_{j}$ , i.e.,

P_{j, K} (x_{i}) = δ_{i j} for i = 1, \dots, m_{k},

where $δ_{i j}$ is the Kronecker delta function. For each $0 \leq j \leq k - 1$ , let $L_{j, F}$ be the $j$ th order Gauss-Legendre polynomial on $F$ such that $L_{j, F} (e_{_{F}}) = 1$ . Note that $L_{j, F}$ is an odd or even function when $j$ is odd or even. Hence, $L_{j, F} (s_{F}) = - 1$ for odd $j$ and $L_{j, F} (s_{F}) = 1$ for even $j$ .

For odd $k$

, the set of degrees of freedom of

U^{k} (T)

(see Lemma 2.1 in [4]) can be given by

\int_{K} v P_{j, K} d x, j = 1, \dots, m_{k}

(4.3)

for all $K \in T$ and

\int_{F} v L_{j, F} d s, j = 0, \dots, k - 1

(4.4)

for all $F \in E$ . Define the basis function $ϕ_{i, K} \in U^{k} (T)$ satisfying

{\begin{matrix} \int_{K^{'}} ϕ_{i, K} P_{j, K^{'}} d x = δ_{i j} δ_{K K^{'}}, & \forall j = 1, \dots, m_{k}, & \forall K^{'} \in T, \int_{F} ϕ_{i, K} L_{j, F} d s = 0, & \forall j = 0, \dots, k - 1, & \forall F \in E, \end{matrix}

(4.5)

for $i = 1, \dots, m_{k}$ and $K \in T$ , and the basis function $ϕ_{i, F} \in U^{k} (T)$ satisfying

(4.6)

for $i = 0, \dots, k - 1$ and $F \in E$ . Then the nonconforming finite element space is the space spanned by all these basis functions, i.e.,

U^{k} (T) = span {ϕ_{i, K} : K \in T}_{i, K}^{m_{k}} \oplus span {ϕ_{i, F} : F \in E}_{i, F}^{k - 1} .

Lemma 4.1.

For all $K \in T$ , the basis functions ${ϕ_{j, K}}_{j = 1}^{m_{k}}$ have support on $K$ and vanish on the boundary of $K$ , i.e.,

ϕ_{j, K} \equiv 0 on \partial K .

Proof.

Obviously, (4.5) implies that $support {ϕ_{j, K}} \in ¯ ¯¯¯ ¯ K$ . To show that $ϕ_{j, K} |_{\partial K} \equiv 0$ , considering each edge $F \in E_{K}$ , the second equation of (4.5) indicates that there exists $a_{_{F}} \in R$ such that

ϕ_{j, K} |_{F} = a_{_{F}} L_{k, F} .

Note that $L_{k, F}$ is an odd function on $F$ and that values of $L_{k, F}$ at two end-points of $F$ are $- 1$ and $1$ , respectively. Now the continuity of $ϕ_{j, K}$ in $K$ implies that $a_{_{F}} = 0$ and, hence, $ϕ_{j, K} \equiv 0$ on $\partial K$ . ∎

For each $K$ , denote by $E_{K}$ the set of all edges of $K$ . For each $F \in E$ , denote by $ω_{F}$ the union of all elements that share the common edge $F$ ; and define a sign function $χ_{F}$ on the set $E_{K_{F}^{+}} \cup E_{K_{F}^{-}} ∖ {F}$ (when $F$ is a boundary edge, let $E_{K_{F}^{+}} = \emptyset$ ) such that

Lemma 4.2.

For all $F \in E$ , the basis functions ${ϕ_{j, F}}_{j = 0}^{k - 1}$ have support on ${¯ ¯ ¯ ω}_{F}$ , and their restrictions on $E_{K_{F}^{+}} \cup E_{K_{F}^{-}}$ has the following representation:

(4.7)

when $j$ is odd, and

ϕ_{j, F} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{1}{∥ L_{j, F} ∥_{0, F}^{2}} L_{j, F}, & on & F, \frac{χ_{F} (F^{'})}{∥ L_{j, F} ∥_{0, F}^{2}} L_{k, F^{'}}, & on & F^{'} \in E_{K_{F}^{+}} \cup E_{K_{F}^{-}} ∖ {F} \end{matrix}

(4.8)

when $j$ is even.

Proof.

By (4.6), it is easy to see that support of $ϕ_{j, F}$ is ${¯ ¯ ¯ ω}_{F}$ . Since $ϕ_{j, F} |_{F}^{\pm} \in P_{k} (F)$ , there exist constants $a_{i, F}^{\pm}$ such that

ϕ_{j, F} |_{_{F}}^{\pm} = k \sum i = 0 a_{i, F}^{\pm} L_{i, F} .

Using (4.6) and the orthogonality of ${L_{i, F}}_{i = 0}^{k}$ , it is obvious that

a_{i, F}^{\pm} = {\begin{matrix} ∥ L_{j, F} ∥_{0, F}^{- 2}, & for i = j, 0, & for 0 \leq i \leq k - 1 and i \neq j \end{matrix}

and, hence,

ϕ_{j, F} |_{F}^{\pm} = \frac{1}{∥ L_{j, F} ∥_{0, F}^{2}} L_{j, F} + a_{k, F}^{\pm} L_{k, F} .

(4.9)

By (4.6), it is also easy to see that there exists constant $a_{j, F, F^{'}}$ for each $F^{'} \in E_{K_{F}^{+}} \cup E_{K_{F}^{-}} ∖ {F}$ such that

ϕ_{j, F} |_{F^{'}} = a_{j, F, F^{'}} L_{k, F^{'}} .

(4.10)

Since $L_{k, F^{'}}$ is an odd function for all $F^{'} \in E_{K_{F}^{+}} \cup E_{K_{F}^{-}} ∖ {F}$ and $ϕ_{j, F}$ is continuous in $K_{F}^{+}$ and $K_{F}^{-}$ , (4.10) implies that

ϕ_{j, F} |_{K} (s_{_{F}}) = ϕ_{j, F} |_{K} (e_{_{F}}), K \in {K_{F}^{+}, K_{F}^{-}} .

(4.11)

Combining the facts that $L_{j, F} (e_{_{F}}) = - L_{j, F} (s_{_{F}}) = 1$ for odd $j$ and that $L_{j, F} (e_{_{F}}) = L_{j, F} (s_{_{F}}) = 1$ for even $j$ , (4.9), and (4.11), we have

a_{k, F}^{\pm} = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} - \frac{1}{∥ L_{j, F} ∥_{0, F}^{2}}, & for odd j, 0, & for even j, \end{matrix}

which, together with (4.9), leads to the formulas of $ϕ_{j, F} |_{F}$ in (4.7) and (4.8). Finally, for each $F^{'} \in E_{K_{F}^{+}} \cup E_{K_{F}^{-}} ∖ {F}$ , $a_{_{j, F, F^{'}}}$ in (4.10) can be directly computed based on the continuity of $ϕ_{j, F}$ in $K_{F}^{+}$ and $K_{F}^{-}$ . This completes the proof of the lemma.

∎

Remark 4.3.

As a consequence of Lemma 4.2, the basis function $ϕ_{j, F}$ is continuous on the edge $F$ , i.e., $[[ϕ_{j, F}]] {∣ ∣}_{F} = 0$ for all $j = 0, \dots, k - 1$ ; moreover, $ϕ_{j, F}$ vanishes at end points of $F$ , i.e., $ϕ_{j, F} (s_{F}) = ϕ_{j, F} (e_{F}) = 0$ , for odd $j$ .

Lemma 4.4.

Let $F$ be an edge of $K$ . Assume that $p \in P_{k - 1} (F)$ . Then we have that

\int_{\partial K} p ϕ_{j, F} d s = \int_{F} p ϕ_{j, F} d s .

(4.12)

Moreover, if $\int_{F} p ϕ_{j, F} d s = 0$ for all $j = 0, \dots, k - 1$ , then $p \equiv 0$ on $F$ .

Proof.

Since ${L_{j, F}}_{j = 0}^{k}$ are orthogonal polynomials on $F$ , Lemma 4.4 is a direct consequence of Lemma 4.2. ∎

4.2 Equilibrated flux recovery

In this subsection, we introduce a fully explicit post-processing procedure for recovering an equilibrated flux. To this end, define $f_{k - 1} \in L^{2} (Ω)$ by

f_{k - 1} |_{K} = Π_{K} (f), \forall K \in T,

where $Π_{K}$ is the $L^{2}$ projection onto $P_{k - 1} (K)$ . For simplicity, assume that the Neumann data $g$ is a piecewise polynomial of degree less than or equal to $k - 1$ , i.e., $g |_{F} \in P_{k - 1} (F)$ for all $F \in E_{N}$ .

Denote the $H (div; Ω)$ conforming Raviart-Thomas (RT) space of index $k - 1$ with respect to $T$ by

RTk−1(T)={\boldmathτ∈H(div;Ω):\boldmathτ|K∈RTk−1(K),∀K∈T},

where $R T^{k - 1} (K) = P_{k - 1} (K)^{d} + x P_{k - 1} (K)$ . Let

Σk−1f(T)={\boldmathτ∈RTk−1:∇⋅\boldmathτ=fk−1inΩand\boldmathτ⋅n

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Corollary 3.2.

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Lemma 3.3.

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Lemma 3.4.

Proof.

Remark 3.5.

Lemma 3.6.

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Lemma 4.2.

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4.2 Equilibrated flux recovery

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1 Introduction

2 Model problem

2.1 Triangulation

3 Generalized Prager-Synge inequality

Theorem 3.1.

Proof.

Corollary 3.2.

Proof.

Lemma 3.3.

Proof.

Lemma 3.4.

Proof.

Remark 3.5.

Lemma 3.6.

Proof.

4 Error estimators and indicators

4.1 NC finite element approximation

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Remark 4.3.

Lemma 4.4.

Proof.

4.2 Equilibrated flux recovery