An A Posteriori Error Estimator for a Non Homogeneous Dirichlet Problem Considering a Dual Mixed Formulation

ABSTRACT

In this paper, we describe an a posteriori error analysis for a conforming dual mixed scheme of the Poisson problem with non homogeneous Dirichlet boundary condition. As a result, we obtain an a posteriori error estimator, which is proven to be reliable and locally efficient with respect to the usual norm on $H\left(div;\Omega \right)\times L^2\left(\Omega \right)$. We remark that the analysis relies on the standard Ritz projection of the error, and take into account a kind of a quasi-Helmholtz decomposition of functions in $H\left(div;\Omega \right)$, which we have established in this work. Finally, we present one numerical example that validates the well behavior of our estimator, being able to identify the numerical singularities when they exist.

Keywords:
mixed finite element methods; a posteriori error estimator; reliability; efficiency

1 INTRODUCTION

It is well known that when the solution of a variational formulation obtained by applying a finite element method, is not smooth enough, the quality of approximation could be not good enough. This motivates us to derive an a posteriori error estimator, which is reliable and efficiency. This would allow us to establish that the estimator behaves as the error of the method, which in general is not known. Then, considering an appropriate adaptive refinement algorithm, we can obtain approximations of the formulation, of better quality, by detecting the region where this estimator is more dominant. In the context of mixed finite element methods, there are a lot of references dedicated to the a posteriori error analysis. For instance, in ¹ an a posteriori error estimator only for the flux unknown is derived, using Raviart-Thomas (RT) or Brezzi-Douglas-Marini (BDM) as its space of approximation. The analysis that yields this estimator, relies on a classical Helmholtz decomposition. On the other hand, in ¹⁰, the authors present two a posteriori error estimators for a dual mixed formulation for the Poisson problem, approximating the flux in the Raviart-Thomas space. In this case, the derivation of the estimator is obtained under a saturation assumption. This requirement is circumvented in ¹², where a reliable and efficient a posteriori error estimator for the natural norm, is derived. We remark that four different kind of a posteriori error estimators for Raviart-Thomas mixed finite elements, are provided in ²³. Concerning second order elliptic equation with mixed boundary condition, in ¹⁷ the authors developed an a posterior error analysis for the mixed finite element method with a Lagrange multiplier.

In ⁹, an a posteriori error analysis for an augmented mixed formulation of the Poisson problem with mixed boundary conditions, is developed. This is performed with the help of the Ritz projection of the error, and covers the reliability and efficiency of the estimator. It is important to remark that this technique has been successfully applied to other problems, such as the the Brinkman model in ², the Darcy flow in ⁶ and ⁷, the Stokes system in ³ and ⁵, and the Oseen equations in ⁸, for example.

In this paper, we deduce a reliable and efficient residual a posteriori error estimator for the Poisson problem with non homogeneous Dirichlet boundary condition, considering a dual mixed finite element method. To achieve this, we take into account the Ritz projection of the error, measured in the standard $H\left(div;\Omega \right)\times L^2\left(\Omega \right)$ norm. We also establish another kind of quasi Helmholtz decomposition of $H\left(div;\Omega \right)$ in the plane. We remark that in this process, no saturation assumption is required, and its extension to 3D case is not difficult. We remark that in ¹ the a posteriori error analysis is performed to a homogeneous Dirichlet problem, focusing in obtain an estimator for the $H\left(div;\Omega \right)$ norm of the flux error. Then, the results of the current work can be seen as a natural extension of what is described in ¹, since we deduce an a posteriori error estimator for the norm of the error of the flux and potential unknowns, that is reliable and efficient.

The rest of the article is organized as follows: In Section 2 we present the model problem, as well as the corresponding dual mixed formulations, at continuous and discrete levels. Next, the a posteriori error analysis with non homogeneous Dirichlet is described in Sections 3. This includes the introduction of the Ritz projection of the error, as well as the key tool for deducing a reliable a posteriori error estimator: a quasi-Helmholtz decomposition of functions in $H\left(div;\Omega \right)$. Finally, one numerical example confirming our theoretical results are reported in Section 4. We end this introduction with some notation to be used throughout the paper. Given any Hilbert space H, we denote by H ² the space of vectors of order 2 with entries in H. Finally, we use C or c, with or without subscripts, to denote generic constants, independent of the discretization parameter, that may take different values at different occurrences.

2 MODEL PROBLEM AND VARIATIONAL FORMULATIONS

Let Ω be a bounded and simply connected domain in ℝ² with polygonal boundary Γ. Then, given $f\in L^2\left(\Omega \right)$ and $g\in H^{1/2}\left(\Gamma \right)$, we consider the model problem: Find $u\in H^1\left(\Omega \right)$ such that −∆u = f in Ω and u = g on Γ. Since we are interested in dual mixed methods, we rewrite the Dirichlet problem as the first order system: Find (σ, u ) such that σ = −∇u in Ω, div(σ ) = f in Ω, and u = g on Γ. Hence, proceeding in the usual way, we arrive to the following dual mixed variational formulation: Find $\left(\mathit{\sigma },u\right)\in H\left(div;\Omega \right)\times L^2\left(\Omega \right)$ such that

(2.1)

where denotes the duality pairing between H ^-1/2(Γ) and H ^1/2(Γ) with respect to L ²(Γ)- inner product, and the bilinear forms $a:H\left(div;\Omega \right)\times H\left(div;\Omega \right)\to \mathrm{ℝ}$ and $b:L^2\left(\Omega \right)\times H\left(div;\Omega \right)\to \mathrm{ℝ}$, are given by $a\left(\mathit{\zeta },\mathit{\tau }\right):={\int }_{\Omega }\mathit{\zeta }·\mathit{\tau }$ and $b\left(w,\mathit{\tau }\right):={\int }_{\Omega }w\mathrm{div}\left(\mathit{\tau }\right)$, respectively. Thanks to the classical Babuška-Brezzi theory (cf. Section 5 in ¹⁶), it can be shown that there exists a unique pair $\left(\mathit{\sigma },u\right)\in H\left(div;\Omega \right)\times L^2\left(\Omega \right)$ solution of (2.1). For the discretization, we assume that Ω is a polygonal region and let ${\left\{{\mathcal{T}}_h\right\}}_{h>0}$ be a regular family of triangulations of $\overline{\Omega }$ such that $\overline{\Omega }=\cup \left\{T:T\in {\mathcal{T}}_h\right\}$. For any triangle $T\in {\mathcal{T}}_h$, we denote by h _T its diameter and define the mesh size $h:=max\left\{h_T:T\in {\mathcal{T}}_h\right\}$. In addition, given an integer ℓ ≥ 0 and a subset S of ℝ², we denote by 𝒫_ℓ(S) the space of polynomials in two variables defined in S of total degree at most ℓ, and for each $T\in {\mathcal{T}}_h$, we define the local Raviart-Thomas space of order κ ≥ 0 (cf. ²⁰), ${\mathcal{RT}}_{\kappa }\left(T\right):={\left[{\mathcal{P}}_{\kappa }\left(T\right)\right]}^2\oplus {\mathcal{P}}_{\kappa }\left(T\right)\mathit{x}\subseteq {\left[{\mathcal{P}}_{\kappa +1}\left(T\right)\right]}^2\forall \mathit{x}\in T$. Then, given an integer r ≥ 0, we define the finite element subspaces $H_{h,r}^{\mathit{\sigma }}:=\left\{{\mathit{\tau }}_h\in H\left(div;\Omega \right):{\mathit{\tau }}_h{|}_T\in {\mathcal{RT}}_r\left(T\right),\forall T\in {\mathcal{T}}_h\right\}$ and $H_{h,r}^u:=\left\{v_h\in L^2\left(\Omega \right):v_h{|}_T\in {\mathcal{P}}_r\left(T\right),\forall T\in {\mathcal{T}}_h\right\}$. Under these assumptions, and applying a discrete version of the Babuška-Brezzi theory (see Section 5 in ¹⁶), we can ensure that there exists only one $\left({\mathit{\sigma }}_h,u_h\right)\in H_{h,r}^{\mathit{\sigma }}\times H_{h,r}^u$ such that

(2.2)

Moreover, the following result is established.

Theorem 2.1.Let (σ, u ) and (σ _h , u _h ) be the solutions of (2.1) and (2.2), respectively. If$\left(\mathit{\sigma },u\right)\in {\left[H^r\left(\Omega \right)\right]}^2\times H^r\left(\Omega \right)$, and$div\left(\mathit{\sigma }\right)\in H^r\left(\Omega \right),0<r\le k+1$, then there exists C > 0, independent of the mesh size, such that

${||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{H\left(div;\Omega \right)}+{||u-u_h||}_{L^2\left(\Omega \right)}\le C{h}^r\left({\left|\mathit{\sigma }\right|}_{{\left[H^r\left(\Omega \right)\right]}^2}+{\left|u\right|}_{H^r\left(\Omega \right)}+{\left|div\left(\mathit{\sigma }\right)\right|}_{H^r\left(\Omega \right)}\right).$

Proof. We refer to the proofs of Theorems 3.2 and 3.3 in ¹⁶, as well as the classical error estimates for the L ²-orthogonal projection onto 𝒫_r . We omit further details. □

3 A POSTERIORI ERROR ANALYSIS

In this section, we follow ⁴ (see also ⁵), and develop an a posteriori error analysis for the discrete scheme (2.2), taking into account an appropriate Ritz projection of the error and a quasi- Helmholtz decomposition. We first introduce some notations and results, concerning the Clément and Raviart-Thomas interpolation operators.

3.1 Notation and some well known results

Given $T\in {\mathcal{T}}_h$, we let E(T) be the set of its edges. By E _h we denote the set of all edges (counted once) induced by the triangulation 𝒯_h . Then, we write $E_h=E_I\cup E_{\Gamma }$, where $E_I:=\left\{e\in E_h:e\subseteq \Omega \right\}$ and $E_{\Gamma }:=\left\{e\in E_h:e\subseteq \Gamma \right\}$. Similarly, N _h will denote the list of all vertices (counted once) induced by the triangulation 𝒯_h . Then we define $N_I:=N_h\cap \Omega$ and $N_{\Gamma }:=\left\{\mathit{x}\in N_h:\mathit{x}\in \Gamma \right\}$. As a result, we have that $N_h=N_I\cup N_{\Gamma }$. In addition, for each $T\in {\mathcal{T}}_h,N\left(T\right):=\left\{\mathit{x}\in N_h:\mathit{x}\text{is a vertex of}T\right\}$, and for each $e\in E_h,N\left(e\right):=\left\{\mathit{x}\in N_h:\mathit{x}\text{is a vertex of}e\right\}$. Now, given $\mathit{x}\in N_{h}T\in {\mathcal{T}}_h$ and $e\in E_h$, we set

$\omega \left(\mathit{x}\right):=\underset{\underset{\mathit{x}\in N\left(T\right)}{T\in {\mathcal{T}}_h}}{\cup }T,\omega \left(e\right):=\underset{\mathit{x}\in N\left(e\right)}{\cup }\omega \left(\mathit{x}\right),\omega \left(T\right):=\underset{\mathit{x}\in N\left(T\right)}{\cup }\omega \left(\mathit{x}\right).$

Also, for each $T\in {\mathcal{T}}_h$, we fix a unit normal exterior vector ${\mathit{n}}_T:={\left(n_1,n_2\right)}^t$, and let ${\mathit{t}}_T:={\left(-n_2,n_1\right)}^t$ be the corresponding fixed unit tangential vector along ∂T. From now on, when no confusion arises, we simply write n and t instead of n _T and t _T , respectively. In addition, let q and τ be scalar - and vector -valued functions, respectively, that are smooth inside each element $T\in {\mathcal{T}}_h$. We denote by (q _T,e , τ _T,e ) the restriction of (q _T , τ _T ) to e. Then, given $e\in E_I$, we define the jump of q and of the tangential component of τ at x ∈ e, by

$\left[\left[q\right]\right]:=q_{T,e}-q_{T\text{'},e},\left[\left[\mathit{\tau }·\mathit{t}\right]\right]:={\mathit{\tau }}_{T,e}·{\mathit{t}}_T+{\mathit{\tau }}_{T\text{'},e}·{\mathit{t}}_{T\text{'}},$

where T and T’ are the two elements in 𝒯_h sharing the edge $e\in E_I$. On boundary edges $e\in E_{\Gamma }$, we set $\left[\left[\mathit{\tau }·\mathit{t}\right]\right]:={\mathit{\tau }}_{T,e}·{\mathit{t}}_T$, where $T\in {\mathcal{T}}_h$ is such that $\partial T\cap e\ne \cancel{0}$. Finally, given a smooth scalar field v and a vector field $\mathit{\tau }={\left({\tau }_1,{\tau }_2\right)}^t$, we define

$\mathbf{curl}\left(v\right):=\left(\begin{array}{c}\frac{\partial v}{\partial x_2}\\ -\frac{\partial v}{\partial x_1}\end{array}\right)\text{and}\mathrm{rot}\left(\mathit{\tau }\right):=\frac{\partial {\tau }_2}{\partial x_1}-\frac{\partial {\tau }_1}{\partial x_2}.$

Next, we introduce the Clément interpolation operator $I_h:H^1\left(\Omega \right)\to X_h$ (cf. ¹⁵), where $X_h:=\left\{v_h\in H^1\left(\Omega \right):v_h{|}_T\in {\mathcal{P}}_1\left(T\right),\forall T\in {\mathcal{T}}_h\right\}$. The following lemma establishes the main local approximation properties of I _h .

Lemma 3.1.There exist constants c₁, c₂ > 0, independent of h, such that for all$v\in H^1\left(\Omega \right)$, there holds

${||v-I_h\left(v\right)||}_{H^m\left(T\right)}\le c_1{h}_T^{1-m}{\left|v\right|}_{H^1\left(\omega \left(T\right)\right)},\forall m\in \left\{0,1\right\},\forall T\in {\mathcal{T}}_h,$

and

${||v-I_h\left(v\right)||}_{L^2\left(e\right)}\le c_2{h}_e^{1/2}{\left|v\right|}_{H^1\left(\omega \left(e\right)\right)},\forall e\in E_h,$

where h_edenotes the length of the side$e\in E_h$.

Proof. We refer to ¹⁵. □

On the other hand, we also need to introduce the Raviart-Thomas interpolation operator (see ¹¹, ²⁰), ${\pi }_h^r:{\left[H^1\left(\Omega \right)\right]}^2\to H_h^{\mathit{\sigma }}$, which given $\mathit{\tau }\in {\left[H^1\left(\Omega \right)\right]}^2,{\pi }_h^r\mathit{\tau }\in H_h^{\mathit{\sigma }}$ is characterized by the following conditions:

$\forall e\in E_h:\forall q\in {\mathcal{P}}_r\left(e\right):{\int }_e{\pi }_h^r\left(\mathit{\tau }\right)·\mathit{n}q={\int }_e\mathit{\tau }·\mathit{n}q,\text{when}r\ge 0,$ (3.1)

and

$\forall T\in {\mathcal{T}}_h:\forall \mathit{\rho }\in {\left[{\mathcal{P}}_{r-1}\left(T\right)\right]}^2:{\int }_T{\pi }_h^r\left(\mathit{\tau }\right)·\mathit{\rho }={\int }_T\mathit{\tau }·\mathit{\rho },\text{when}r\ge 1,$ (3.2)

The operator ${\pi }_h^r$ satisfies the following approximation properties.

Lemma 3.2.There exist constants c₃, c₄, c₅> 0, independent of h, such that for all$T\in {\mathcal{T}}_h$

$\forall \mathit{\tau }\in {\left[H^m\left(\Omega \right)\right]}^2:{||\mathit{\tau }-{\pi }_h^r\left(\mathit{\tau }\right)||}_{{\left[L^2\left(T\right)\right]}^2}\le c_3{h}_T^m{\left|\mathit{\tau }\right|}_{{\left[H^m\left(T\right)\right]}^2}1\le m\le r+1,$ (3.3)

for all $\mathit{\tau }\in {\left[H^m\left(\Omega \right)\right]}^2$ with $div\left(\tau \right)\in H^m\left(\Omega \right)$ ,

${||div\left(\mathit{\tau }-{\pi }_h^r\left(\mathit{\tau }\right)\right)||}_{L^2\left(T\right)}\le c_4{h}_T^m{\left|div\left(\mathit{\tau }\right)\right|}_{H^m\left(T\right)},0\le m\le r+1,$ (3.4)

and for any $\mathit{\tau }\in {\left[H^1\left(\Omega \right)\right]}^2$

$\forall e\in E_h:{||\mathit{\tau }·\mathit{n}-{\pi }_h^r\left(\mathit{\tau }\right)·\mathit{n}||}_{L^2\left(e\right)}\le c_5{h}_e^{1/2}{||\mathit{\tau }||}_{{\left[H^1\left(T_e\right)\right]}^2},$ (3.5)

where$T_e\in {\mathcal{T}}_h$, such that it contains e on its boundary.Proof. See e.g. ¹¹ or ²⁰. □

In addition, the interpolation operator ${\pi }_h^r$ can also be defined as a bounded linear operator from the larger space ${\left[H^s\left(\Omega \right)\right]}^2\cap H\left(div;\Omega \right)$ into $H_h^{\mathit{\sigma }}$, for all $s\in (1/2,1]$ (see, e.g. Theorem 3.16 in ¹⁹). In this case, there holds the following interpolation error estimate

$\forall T\in {\mathcal{T}}_h:{||\mathit{\tau }-{\pi }_h^r\left(\mathit{\tau }\right)||}_{{\left[L^2\left(T\right)\right]}^2}\le C{h}_T^s\left\{{||\mathit{\tau }||}_{{\left[H^s\left(T\right)\right]}^2}+{||div\left(\mathit{\tau }\right)||}_{L^2\left(T\right)}\right\}.$

Taking into account (3.1) and (3.2), it is not difficult to show that

$div\left({\pi }_h^r\left(\mathit{\tau }\right)\right)=P_h^r\left(div\left(\mathit{\tau }\right)\right),$ (3.6)

where $P_h^r:L^2\left(\Omega \right)\to H_h^u$ is the L² −orthogonal projector. On the other hand, it is well known (see, e.g. ¹⁴) that for each $v\in H^m\left(\Omega \right)$, with $0\le m\le r+1$, there exists C > 0, independent of h, such that

$\forall T\in {\mathcal{T}}_h:{||v-P_h^r\left(v\right)||}_{L^2\left(T\right)}\le C{h}_T^m{\left|\mathit{v}\right|}_{H^m\left(T\right)}.$ (3.7)

3.2 Reliability of the estimator

Let $\left(\mathit{\sigma },u\right)\in \mathit{\Sigma }:=H\left(div;\Omega \right)\times L^2\left(\Omega \right)$ and $\left({\mathit{\sigma }}_h,u_h\right)\in H_{h,r}^{\mathit{\sigma }}\times H_{h,r}^u\subseteq \mathit{\Sigma }$ be the unique solution to problems (2.1) and (2.2), respectively. We provide Σ with its usual inner product

which induces the norm

${||\left(\mathit{\tau },v\right)||}_{\mathit{\Sigma }}:={\left({||\mathit{\tau }||}_{H\left(div;\Omega \right)}^2+{||v||}_{L^2\left(\Omega \right)}^2\right)}^{1/2}\forall \left(\mathit{\tau },v\right)\in \mathit{\Sigma }.$

Next, we consider the Ritz projection of the error with respect to as the unique element $\left(\overline{\mathit{\sigma }},\overline{u}\right)\in \mathit{\Sigma }$, such that

(3.8)

where the global bilinear form $A:\mathit{\Sigma }\times \mathit{\Sigma }\to \mathrm{ℝ}$ arises from the variational formulation (2.1), after adding its equations, that is

$A\left(\left(\mathit{\rho },w\right),\left(\mathit{\tau },v\right)\right):=a\left(\mathit{\rho },\mathit{\tau }\right)-b\left(w,\mathit{\tau }\right)-b\left(v,\mathit{\rho }\right)\forall \left(\mathit{\rho },w\right),\left(\mathit{\tau },v\right)\in \mathit{\Sigma }.$

We remark that the existence and uniqueness of $\left(\overline{\mathit{\sigma }},\overline{u}\right)\in \mathit{\Sigma }$ is guaranteed by the Lax-Milgram Lemma. Moreover, we point out that the properties of the bilinear forms a(·,·) and b(·,·) implies that A(·,·) satisfies a global inf-sup condition, i.e., there exist α > 0 such that

$\alpha {||\left(\mathit{\zeta },w\right)||}_{\mathit{\Sigma }}\le \underset{\theta \ne \left(\mathit{\tau },v\right)\in \mathit{\Sigma }}{sup}\frac{A\left(\left(\mathit{\zeta },w\right),\left(\mathit{\tau },v\right)\right)}{{||\left(\mathit{\tau },v\right)||}_{\mathit{\Sigma }}},\forall \left(\mathit{\zeta },w\right)\in \mathit{\Sigma }.$

This particularity allows us to bound the error in terms of the solution of its Ritz projection, as follows:

$\alpha {||\left(\mathit{\sigma }-{\mathit{\sigma }}_h,u-u_h\right)||}_{\mathit{\Sigma }}\le \underset{\theta \ne \left(\mathit{\tau },v\right)\in \mathit{\Sigma }}{sup}\frac{A\left(\left(\mathit{\sigma }-{\mathit{\sigma }}_h,u-u_h\right),\left(\mathit{\tau },v\right)\right)}{{||\left(\mathit{\tau },v\right)||}_{\mathit{\Sigma }}}={||\left(\overline{\mathit{\sigma }},\overline{u}\right)||}_{\mathit{\Sigma }}.$ (3.9)

Then, according to (3.9), and with the purpose of obtaining a reliable a posteriori error estimate for the discrete scheme (2.2), it is enough to bound from above the Ritz projection of the error. To this aim, the next result will be useful, and can be seen as a kind of a quasi-Helmholtz decomposition of functions in H(div; Ω).

Lemma 3.3.For each$\mathit{\tau }\in H\left(div;\Omega \right)$, there exist$\chi \in H^1\left(\Omega \right)$and$\mathit{\Phi }\in {\left[H_0^1\left(\Omega \right)\right]}^2$, such that

$\mathit{\tau }=\mathbf{curl}\left(\chi \right)+\mathit{\Phi }+\frac{d}{2}\left(\begin{array}{c}{x}_1-a\\ x_2-b\end{array}\right),$ (3.10)

where (a, b)^t is any fixed point belonging to Ω, and$d:=\frac{1}{\left|\Omega \right|}{\int }_{\Omega }div\left(\mathit{\tau }\right)$. In addition, there exists C > 0, such that

${\left|\chi \right|}_{H^1\left(\Omega \right)}+{||\mathit{\Phi }||}_{{\left[H^1\left(\Omega \right)\right]}^2}\le C{||\mathit{\tau }||}_{H\left(div;\Omega \right)}.$ (3.11)

Proof. We first introduce the space $M:=\left\{\mathit{\zeta }\in H\left(div;\Omega \right):{\int }_{\Omega }div\left(\mathit{\zeta }\right)=0\right\}$. Next, for each $\mathit{\tau }\in H\left(div;\Omega \right)$, we decompose $div\left(\mathit{\tau }\right)=div\left(\tilde{\mathit{\tau }}\right)+d$, where $\tilde{\mathit{\tau }}:=\mathit{\tau }-\frac{d}{2}\left(\begin{array}{c}{x}_1-a\\ x_2-b\end{array}\right)\in M$.

We remark that ${||div\left(\mathit{\tau }\right)||}_{0,\Omega }^2={||div\left(\tilde{\mathit{\tau }}\right)||}_{0,\Omega }^2+d^2\left|\Omega \right|$. Then, since $div\left(\tilde{\mathit{\tau }}\right)\in L_0^2\left(\Omega \right)$, and invoking Corollary I.2.4 in ¹⁸, there exists $\mathit{\Phi }\in {\left[H_0^1\left(\Omega \right)\right]}^2$ such that $div\left(\mathit{\Phi }\right)=div\left(\tilde{\mathit{\tau }}\right)$ in Ω and ${||\mathit{\Phi }||}_{1,\Omega }\le c{||div\left(\mathit{\tau }\right)||}_{0,\Omega }$. This implies that

where (a, b)^t is a fixed point belonging to Ω. Hence, by Theorem I.3.1 in ¹⁸, there exists a stream function $\chi \in H^1\left(\Omega \right)$ such that $\mathit{\tau }-\mathit{\Phi }-\frac{d}{2}{\left(x_1-a,x_2-b\right)}^t=\mathbf{curl}\left(\chi \right)$ in Ω. In addition, we have

$$\begin{gathered} {\left|\chi \right|}_{H^1\left(\Omega \right)}^2={||\mathbf{curl}\left(\chi \right)||}_{L^2\left(\Omega \right)}^2={||\mathit{\tau }-\mathit{\Phi }-\frac{d}{2}{\left(x_1-a,x_2-b\right)}^t||}_{L^2\left(\Omega \right)}^2 \\ \le 2\left({||\mathit{\tau }||}_{{\left[L^2\left(\Omega \right)\right]}^2}^2+{||\mathit{\Phi }||}_{{\left[L^2\left(\Omega \right)\right]}^2}^2+\frac{d^2}{4}{||\left(x_1-a,x_2-b\right)||}_{{\left[L^2\left(\Omega \right)\right]}^2}^2\right) \\ \le 2\left({||\mathit{\tau }||}_{{\left[L^2\left(\Omega \right)\right]}^2}^2+{||\mathit{\Phi }||}_{{\left[L^2\left(\Omega \right)\right]}^2}^2+\frac{d^2}{4}{\left(diam\left(\Omega \right)\right)}^2\left|\Omega \right|\right) \\ \le 2max\left\{1,c^2+\frac{1}{4}{\left(diam\left(\Omega \right)\right)}^2\right\}{||\tau ||}_{H\left(div;\Omega \right)}^2. \end{gathered}$$

As a result, we establish (3.11), and we end the proof. □

Now, considering χ and Φ as the ones provided by Lemma 3.3 for a given $\mathit{\tau }\in H\left(div;\Omega \right)$, we introduce ${\chi }_h:=I_h\left(\chi \right)$, and define

${\mathit{\tau }}_h:=\mathbf{curl}\left({\chi }_h\right)+{\pi }_h^r\left(\mathit{\Phi }\right)+\frac{d}{2}\left(\begin{array}{c}{x}_1-a\\ x_2-b\end{array}\right)\in H_h^{\mathit{\sigma }},$ (3.12)

which is referred as a discrete quasi-Helmholtz decomposition of τ _h . Therefore, we can write

$\mathit{\tau }-{\mathit{\tau }}_h=\mathbf{curl}\left(\chi -{\chi }_h\right)+\mathit{\Phi }-{\pi }_h^r\left(\mathit{\Phi }\right),$ (3.13)

that verifies

$div\left(\mathit{\tau }-{\mathit{\tau }}_h\right)=div\left(\mathit{\Phi }-{\pi }_h^r\left(\mathit{\Phi }\right)\right)$ (3.14)

On the other hand, it is not difficult to check the following orthogonality relation

$A\left(\left(\mathit{\sigma }-{\mathit{\sigma }}_h,u-u_h\right),\left({\mathit{\zeta }}_h,v_h\right)\right)=0,\forall \left({\mathit{\zeta }}_h,v_h\right)\in {\mathit{\Sigma }}_h:=H_h^{\mathit{\sigma }}\times H_h^u.$ (3.15)

From now on, given $\left(\mathit{\tau },v\right)\in \mathit{\Sigma }$, we associate it with the discrete pair $\left({\mathit{\tau }}_h,0\right)\in {\mathit{\Sigma }}_h$, where τ _h is defined as in (3.12). Hence, considering (3.15) with $\left({\mathit{\zeta }}_h,v_h\right):=\left({\mathit{\tau }}_h,0\right)$, and knowing that (σ, u ) is the unique solution of problem (2.1), we obtain

Equivalently,

$$\begin{gathered} {\left(\overline{\mathbf{\sigma }},\mathit{\tau }\right)}_{H\left(div;\Omega \right)}=F_1\left(\mathit{\tau }-{\mathit{\tau }}_h\right)\forall \mathit{\tau }\in H\left(div;\Omega \right), \\ {\left(\overline{u},v\right)}_{L^2\left(\Omega \right)}=F_2\left(v\right)\forall v\in L^2\left(\Omega \right), \end{gathered}$$

where $F_1:H\left(div;\Omega \right)\to \mathrm{ℝ}$ and $F_2:L^2\left(\Omega \right)\to \mathrm{ℝ}$ are the bounded linear functionals defined by

Hence, taking into account (3.13) and (3.14), and the fact that ${\pi }_h^k\left(\mathit{\Phi }\right)·\mathit{n}=0$ on Γ, we can rewrite F ₁(τ − τ _h ) as follows

$F_1\left(\mathit{\tau }-{\mathit{\tau }}_h\right)=R_1\left(\mathit{\Phi }\right)+R_2\left(\chi \right),$

where

Our aim now is to obtain upper bounds for each one of the terms F ₂(v), R ₁(Φ) and R ₂(χ).

Lemma 3.4. For any $v\in L^2\left(\Omega \right)$ there holds

$\left|F_2\left(v\right)\right|\le {\left(\sum _{T\in {\mathcal{T}}_h}{||f-div\left({\mathit{\sigma }}_h\right)||}_{L^2\left(T\right)}^2\right)}^{1/2}{||v||}_{L^2\left(\Omega \right)}.$

Proof. The proof follows from a straightforward application of Cauchy-Schwarz inequality. □

Lemma 3.5.There exists C > 0, independent of h, such that

$\left|R_1\left(\mathit{\Phi }\right)\right|\le C{\left(\sum _{e\in E_I}{h}_e{||\left[\left[u_h\right]\right]||}_{{\left[L^2\left(e\right)\right]}^2}^2+\sum _{T\in {\mathcal{T}}_h}{h}_T^2{||\nabla u_h+{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}^2\right)}^{1/2}{||\mathit{\tau }||}_{H\left(div;\Omega \right)}.$

Proof. It is a slight modification of Lemma 3.5 in ⁵. We omit further details. □

Lemma 3.6.Under the assumption that$g\in H^1\left(\Gamma \right)$, there exists C > 0, independent of h, such that

$$\begin{gathered} \left|R_2\left(\chi \right)\right|\le C{(\sum _{T\in {\mathcal{T}}_h}{h}_T^2{||rot\left({\mathit{\sigma }}_h\right)||}_{L^2\left(T\right)}^2 \\ +\sum _{e\in E\left(T\right)}{h}_e\left({||\left[\left[{\mathit{\sigma }}_h·\mathit{t}\right]\right]||}_{L^2\left(e\cap E_I\right)}^2+{||{\mathit{\sigma }}_h·\mathit{t}+\frac{dg}{d\mathit{t}}||}_{L^2\left(e\cap E_{\Gamma }\right)}^2\right))}^{1/2}{‖\mathit{\tau }‖}_{H\left(div;\Omega \right)}. \end{gathered}$$

Proof. Knowing that $\mathbf{curl}\left(\chi -{\chi }_h\right)·\mathit{n}=\frac{d}{d\mathit{t}}\left(\chi -{\chi }_h\right)$ on Γ, and after integrating by parts, we deduce

Therefore, the proof is completed invoking Lemma 3.1, the Cauchy-Schwarz inequality, the regularity of the mesh and (3.11). □

The previous results suggest the definition of the following residual estimator

$\eta :={\left(\sum _{T\in {\mathcal{T}}_h}{\eta }_T^2\right)}^{1/2},$ (3.16)

where

$$\begin{gathered} {\eta }_T^2:={||f-div\left({\mathit{\sigma }}_h\right)||}_{L^2\left(T\right)}^2+h_T^2{||{\mathit{\sigma }}_h+\nabla u_h||}_{{\left[L^2\left(T\right)\right]}^2}^2+h_T^2{||rot\left({\mathit{\sigma }}_h\right)||}_{L^2\left(T\right)}^2 \\ +\sum _{e\in E\left(T\right)}{h}_e\left({||\left[\left[u_h\right]\right]||}_{L^2\left(e\cap E_I\right)}^2+{||\left[\left[{\mathit{\sigma }}_h·\mathit{t}\right]\right]||}_{L^2\left(e\cap E_I\right)}^2+{||{\mathit{\sigma }}_h·\mathit{t}+\frac{dg}{d\mathit{t}}||}_{L^2\left(e\cap E_{\Gamma }\right)}^2\right). \end{gathered}$$

An upper bound for ${||\left(\overline{\mathit{\sigma }},\overline{u}\right)||}_{\mathit{\Sigma }}$ is established in the next lemma, in terms of (3.16).

Lemma 3.7.Assuming that$g\in H^1\left(\Gamma \right)$, there exists a constant C > 0, independent of h, such that

${||\left(\overline{\mathit{\sigma }},\overline{u}\right)||}_{\mathit{\Sigma }}\le C\eta .$ (3.17)

Proof. Invoking Lemmas 3.5 and 3.6, we deduce that there exists C > 0, independent of h, such that

$$\begin{gathered} \left|F_1\left(\mathit{\tau }-{\mathit{\tau }}_h\right)\right|\le C(\sum _{T\in {\mathcal{T}}_h}{h}_T^2{||{\mathit{\sigma }}_h+\nabla u_h||}_{{\left[L^2\left(T\right)\right]}^2}^2+\sum _{e\in E_h}{h}_e{({||\left[\left[{\mathit{\sigma }}_h·\mathit{t}\right]\right]||}_{L^2\left(e\cap E_I\right)}^2 \\ +{||{\mathit{\sigma }}_h·\mathit{t}+\frac{dg}{d\mathit{t}}||}_{L^2\left(e\cap E_{\Gamma }\right)}^2+{||\left[\left[u_h\right]\right]||}_{L^2\left(e\cap E_I\right)}^2))}^{1/2}{‖\mathit{\tau }‖}_{H\left(div;\Omega \right)}. \end{gathered}$$

Hence, (3.17) follows from the above bound, Lemma 3.4 and a discrete Cauchy-Schwarz inequality. □

The following theorem establishes the main result of this section, which is the reliability and efficiency of the estimator η.

Theorem 3.2.There exists a positive constant C_rel, independent of h, such that

${||\left(\mathit{\sigma }-{\mathit{\sigma }}_h,u-u_h\right)||}_{\mathit{\Sigma }}\le C_{rel}\eta .$ (3.18)

Additionally, there exists C_eff > 0, independent of h, such that

${\eta }_T^2\le C_{eff}{||\mathit{\sigma }-{\mathit{\sigma }}_h,u-u_h||}_{\mathit{\Sigma }\left(T\right)},$ (3.19)

where $\forall T\in {\mathcal{T}}_h:{||\left(\mathit{\tau },\mathit{v}\right)||}_{\Sigma \left(T\right)}^2:={||\mathit{\tau }||}_{H\left(div;T\right)}^2+{||\mathit{v}||}_{L^2\left(T\right)}^2$. Proof. The reliability of η, (3.18), follows from (3.9) and Lemma 3.7. The efficiency of η, (3.19), is treated in the next subsection. We omit further details. □

3.3 Efficiency of the estimator

In this subsection we prove the local efficiency of the estimator η (cf. (3.19)). We begin by introducing some notations and preliminary results. Given $T\in {\mathcal{T}}_h$ and $e\in E\left(T\right)$, we let ψ _T and ψ _e be the standard triangle-bubble and edge-bubble functions, respectively. In particular, ψ _T satisfies ${\psi }_T\in {\mathcal{P}}_3\left(T\right),supp\left({\psi }_T\right)\subseteq T,{\psi }_T=0$ on ∂T, and $0\le {\psi }_T\le 1$ in T. Similarly, ${\psi }_e{|}_T\in {\mathcal{P}}_2\left(T\right),supp\left({\psi }_e\right)\subseteq {\omega }_e:=\cup \left\{T\text{'}\in {\mathcal{T}}_h:e\in E\left(T\text{'}\right)\right\},{\psi }_e=0$ on ∂ω _e , and $0\le {\psi }_e\le 1$ in ω _e . We also recall from ²¹ that, given $k\in \mathrm{\mathbb{N}}\cup \left\{0\right\}$, there exists an extension operator $L:C\left(e\right)\to C\left(T\right)$ that satisfies $L\left(p\right)\in {\mathcal{P}}_k\left(T\right)$ and $L\left(p\right){|}_e=p\forall p\in {\mathcal{P}}_k\left(e\right)$. Additional properties of ψ _T , ψ _e , and L are collected in the following lemma.

Lemma 3.8. For any triangle T there exist positive constants c ₁ , c ₂ , c ₃ and c ₄ , depending only on k and the shape of T, such that for all $q\in {\mathcal{P}}_k\left(T\right)$ and $p\in {\mathcal{P}}_k\left(e\right)$ , there hold

${||{\psi }_{T}q||}_{L^2\left(T\right)}^2\le {||q||}_{L^2\left(T\right)}^2\le c_1{||{\psi }_T^{1/2}q||}_{L^2\left(T\right)}^2,$ (3.20)

${||{\psi }_{e}p||}_{L^2\left(e\right)}^2\le {||p||}_{L^2\left(e\right)}^2\le c_2{||{\psi }_e^{1/2}p||}_{L^2\left(e\right)}^2,$ (3.21)

$c_4{h}_e{||p||}_{L^2\left(e\right)}^2\le {||{\psi }_e^{1/2}L\left(p\right)||}_{L^2\left(T\right)}^2\le c_3{h}_e{||p||}_{L^2\left(e\right)}^2,$ (3.22)

Proof. See Lemma 4.1 in ²¹. □

The following inverse estimate will also be useful.

Lemma 3.9.Let$\mathcal{l},m\in \mathrm{\mathbb{N}}\cup \left\{0\right\}$such that ℓ ≤ m. Then, for any triangle T , there exists c > 0, depending only on k, ℓ, m and the shape of T, such that

${\left|q\right|}_{H^m\left(T\right)}\le c{h}_T^{\mathcal{l}-m}{\left|q\right|}_{H^{\mathcal{l}}\left(T\right)}\forall q\in {\mathcal{P}}_k\left(T\right).$ (3.23)

Proof. See Theorem 3.2.6 in ¹⁴. □

Since f = div(σ ) in Ω, we have that

${||f-div\left({\mathit{\sigma }}_h\right)||}_{L^2\left(T\right)}={||div\left(\mathit{\sigma }-{\mathit{\sigma }}_h\right)||}_{L^2\left(T\right)}.$

Lemma 3.10.There exists C₁> 0, independent of the meshsize, such that for any$T\in {\mathcal{T}}_h$

$h_T{||{\mathit{\sigma }}_h+\nabla u_h||}_{{\left[L^2\left(T\right)\right]}^2}\le C_1\left({||u-u_h||}_{L^2\left(T\right)}+h_T{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}\right).$

Proof. We introduce ${\mathit{\rho }}_h:={\mathit{\sigma }}_h+\nabla u_h$ in T. Then, taking into account the property (3.20) and integrating by parts, we have

$$\begin{gathered} c_1^{-1}{||{\mathit{\rho }}_h||}_{{\left[L^2\left(T\right)\right]}^2}^2\le {||{\psi }_T^{1/2}{\mathit{\rho }}_h||}_{{\left[L^2\left(T\right)\right]}^2}^2={\int }_T\left({\mathit{\sigma }}_h+\nabla u_h\right)·{\psi }_T{\mathit{\rho }}_h \\ ={\int }_T{\mathit{\sigma }}_h·{\psi }_T{\mathit{\rho }}_h+{\int }_T\nabla u_h·{\psi }_T{\mathit{\rho }}_h={\int }_T{\mathit{\sigma }}_h·{\psi }_T{\mathit{\rho }}_h-{\int }_T{u}_{h}div\left({\psi }_T{\mathit{\rho }}_h\right) \\ ={\int }_T\left({\mathit{\sigma }}_h-\mathit{\sigma }\right)·{\psi }_T{\mathit{\rho }}_h+{\int }_T\left(u-u_h\right)div\left({\psi }_T{\mathit{\rho }}_h\right). \end{gathered}$$

Now, applying Cauchy-Schwarz inequality as well as inverse inequality (3.23) and property 0 ≤ ψ _T ≤ 1, we derive

$$\begin{gathered} c_1^{-1}{||{\mathit{\rho }}_h||}_{{\left[L^2\left(T\right)\right]}^2}^2\le \{{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}{||{\psi }_T^{1/2}{\mathit{\rho }}_h||}_{{\left[L^2\left(T\right)\right]}^2}^2 \\ +{||u-u_h||}_{L^2\left(T\right)}{||div\left({\psi }_T{\mathit{\rho }}_h\right)||}_{{\left[L^2\left(T\right)\right]}^2}\} \\ \le {||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}{||{\mathit{\rho }}_h||}_{{\left[L^2\left(T\right)\right]}^2}+\sqrt{2}{||u-u_h||}_{L^2\left(T\right)}{||\nabla \left({\psi }_T{\mathit{\rho }}_h\right)||}_{{\left[L^2\left(T\right)\right]}^{2\times 2}} \\ \le C\left\{{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}+\sqrt{2}{h}_T^{-1}{||u-u_h||}_{L^2\left(T\right)}\right\}{||{\mathit{\rho }}_h||}_{{\left[L^2\left(T\right)\right]}^2}. \end{gathered}$$

Hence, simplifying ${||{\mathit{\rho }}_h||}_{{\left[L^2\left(T\right)\right]}^2}$ and multiplying by the factor h _T , we complete the proof of the lemma. □

In the following lemma, we bound the jump of u _h ,

Lemma 3.11.There exists C₂> 0, independent of the mesh size, such that for any$e\in E_I$

$h_e{||\left[\left[u_h\right]\right]||}_{L^2\left(e\right)}^2\le C_2\left\{{||u-u_h||}_{L^2\left({\omega }_e\right)}^2+{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{L^2{\left({\omega }_e\right)}^2}^2\right\}.$ (3.24)

Proof. First, given $e\in E_I$ we set ${\omega }_e:=T\cup T\text{'}$, with $T,T\text{'}\in {\mathcal{T}}_h$ such that $e=\partial T\cap \partial T\text{'}$. Next, we introduce $w_h:=\left[\left[u_h\right]\right]$ on e and ${\mathit{\rho }}_e:={\psi }_{e}L\left(w_h{\mathit{n}}_{T,e}\right)$ in ω _e , which belongs to H(div, ω _e ). Taking into account (3.21), knowing that [[u]]= 0 on E _I , and integrating by parts, we derive

$$\begin{gathered} c_2^{-1}{||w_h||}_{L^2\left(e\right)}^2\le {||{\psi }_e^{1/2}{w}_h||}_{L^2\left(e\right)}^2={\int }_e{\psi }_{e}L\left(w_h\right)\left[\left[u_h-u\right]\right]={\int }_e\left[\left[u_h-u\right]\right]{\mathit{\rho }}_e·{\mathit{n}}_T \\ ={\int }_{{\omega }_e}\left(u_h-u\right)div\left({\mathit{\rho }}_e\right)+{\int }_{{\omega }_e}{\nabla }_h\left(u_h-u\right)·\left({\mathit{\rho }}_e\right) \\ ={\int }_{{\omega }_e}\left(u_h-u\right)div\left({\mathit{\rho }}_e\right)+{\int }_{{\omega }_e}\left({\mathit{\sigma }}_h+{\nabla }_h{u}_h\right)·{\mathit{\rho }}_e+{\int }_{{\omega }_e}\left(\mathit{\sigma }-{\mathit{\sigma }}_h\right)·{\mathit{\rho }}_e\mathbf{.} \end{gathered}$$

Using the fact that ${\int }_{{\omega }_e}={\int }_T+{\int }_{T\text{'}}$, and applying Cauchy-Schwarz inequality, we deduce

$$\begin{gathered} c_2^{-1}{||w_h||}_{L^2\left(e\right)}^2 \\ \le {||u-u_h||}_{L^2\left(T\right)}{||div\left({\mathit{\rho }}_e\right)||}_{L^2\left(T\right)}+{||u-u_h||}_{L^2\left(T\text{'}\right)}{||div\left({\mathit{\rho }}_e\right)||}_{L^2\left(T\text{'}\right)} \\ +{||{\mathit{\sigma }}_h+\nabla u_h||}_{{\left[L^2\left(T\right)\right]}^2}{||{\mathit{\rho }}_e||}_{{\left[L^2\left(T\right)\right]}^2}+{||{\mathit{\sigma }}_h\nabla u_h||}_{{\left[L^2\left(T\text{'}\right)\right]}^2}{||{\mathit{\rho }}_e||}_{{\left[L^2\left(T\text{'}\right)\right]}^2} \\ +{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}{||{\mathit{\rho }}_e||}_{{\left[L^2\left(T\right)\right]}^2}+{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\text{'}\right)\right]}^2}{||{\mathit{\rho }}_e||}_{{\left[L^2\left(T\text{'}\right)\right]}^2}. \end{gathered}$$ (3.25)

Now, invoking the inverse inequality (3.23) and knowing that 0 ≤ ψ _e ≤ 1 in ω _e together with (3.22), we arrive for each $T\in {\omega }_e$

$$\begin{gathered} {||div\left({\mathit{\rho }}_e\right)||}_{L^2\left(T\right)}\le \sqrt{2}{||\nabla {\mathit{\rho }}_e||}_{{\left[L^2\left(T\right)\right]}^{2\times 2}}\le c\sqrt{2}{h}_T^{-1}{||{\mathit{\rho }}_e||}_{{\left[L^2\left(T\right)\right]}^2} \\ =c\sqrt{2}{h}_T^{-1}{||{\psi }_e^{1/2}L\left(w_h\right)||}_{L^2\left(T\right)}\le c{c}_3\sqrt{2}{h}_T^{-1/2}{||w_h||}_{L^2\left(e\right)}. \end{gathered}$$

This inequality, together with (3.22), allow us to rewrite (3.25) as follows: There exists c > 0 independent of mesh size, such that

$$\begin{gathered} c{||w_h||}_{L^2\left(e\right)}^2\le \{h_T^{-1/2}{||u-u_h||}_{L^2\left(T\right)}+h_{T\text{'}}^{-1/2}{||u-u_h||}_{L^2\left(T\text{'}\right)} \\ +h_T{||{\mathit{\sigma }}_T+\nabla u_h||}_{{\left[L^2\left(T\right)\right]}^2}+h_{T\text{'}}{||{\mathit{\sigma }}_T+\nabla u_h||}_{{\left[L^2\left(T\text{'}\right)\right]}^2} \\ +h_T{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}+h_{T\text{'}}{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\text{'}\right)\right]}^2}\}{||w_h||}_{L^2\left(e\right)}. \end{gathered}$$

Then the proof follows after multiplying by h _e , and applying Lemma 3.10. □

Lemma 3.12.There exists C₃ > 0, independent of the meshsize, such that for any$T\in {\mathcal{T}}_h$

$h_T{||rot\left({\mathit{\sigma }}_h\right)||}_{L^2\left(T\right)}\le C_3{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}.$

Proof. We introduce ${\rho }_h:=rot\left({\mathit{\sigma }}_h\right)$ in T. Then, invoking the property (3.20), rot(σ) = 0 in T, and integrating by parts, we have

$$\begin{gathered} c_1^{-1}{||{\rho }_h||}_{L^2\left(T\right)}^2\le {||{\psi }_T^{1/2}{\rho }_h||}_{{\left[L^2\left(T\right)\right]}^2}^2={\int }_{T}rot\left({\mathit{\sigma }}_h\right){\psi }_T{\rho }_h \\ ={\int }_{T}rot\left({\mathit{\sigma }}_h-\mathit{\sigma }\right){\psi }_T{\rho }_h={\int }_T\left({\mathit{\sigma }}_h-\mathit{\sigma }\right)·\mathbf{curl}\left({\psi }_T{\rho }_h\right). \end{gathered}$$

Now, applying Cauchy-Schwarz inequality, as well as inverse inequality (3.23) and the fact that 0 ≤ ψ_T ≤ 1 in T, we derive

$$\begin{gathered} c_1^{-1}{||{\rho }_h||}_{{\left[L^2\left(T\right)\right]}^2}^2\le {||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}{||\mathbf{curl}\left({\psi }_T{\rho }_h\right)||}_{{\left[L^2\left(T\right)\right]}^2} \\ ={||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}{||\nabla \left({\psi }_T{\rho }_h\right)||}_{{\left[L^2\left(T\right)\right]}^2} \\ \le C{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}{h}_T^{-1}{||{\psi }_T{\rho }_h||}_{L^2\left(T\right)} \\ \le C{h}_T^{-1}{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}{||{\rho }_h||}_{L^2\left(T\right)}. \end{gathered}$$

Hence, simplifying ${||{\rho }_h||}_{{\left[L^2\left(T\right)\right]}^2}$ and multiplying by the factor h _T , we complete the proof of the lemma. □

The tangential component jump of σ _h is treated in the next lemma.

Lemma 3.13.There exists C₄ > 0, independent of the mesh size, such that for any$e\in {\mathcal{E}}_I$

$h_e{||\left[\left[{\mathit{\sigma }}_h·\mathit{t}\right]\right]||}_{L^2\left(e\right)}^2\le C_4{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L\left({\omega }_e\right)\right]}^2}^2.$ (3.26)

Proof. Given $e\in E_I$, let $T,T\text{'}\in {\mathcal{T}}_h$ such that ${\omega }_e=T\cup T\text{'}$ and they share e, i.e. $\partial T\cap \partial T\text{'}=e$. Denoting by $w_h:=\left[\left[{\mathit{\sigma }}_h·\mathit{t}\right]\right]$ on e, and using 3.21, it follows that

$$\begin{gathered} c_2^{-1}{||w_h||}_{L^2\left(e\right)}^2\le {||{\psi }_e^{1/2}{w}_h||}_{L^2\left(e\right)}^2={\int }_e{\psi }_{e}L\left(w_h\right)\left[\left[{\mathit{\sigma }}_h·\mathit{t}\right]\right] \\ ={\int }_e{\psi }_{e}L\left(w_h\right){\mathit{\sigma }}_h·{\mathit{t}}_T+{\int }_e{\psi }_{e}L\left(w_h\right){\mathit{\sigma }}_h·{\mathit{t}}_{T\text{'}} \\ =-{\int }_{{\omega }_e}\mathbf{curl}\left({\psi }_{e}L\left(w_h\right)\right)·{\mathit{\sigma }}_h+{\int }_{{\omega }_e}{\psi }_{e}L\left(w_h\right)rot\left({\mathit{\sigma }}_h\right) \\ ={\int }_{{\omega }_e}\mathbf{curl}\left({\psi }_{e}L\left(w_h\right)\right)·\left(\mathit{\sigma }-{\mathit{\sigma }}_h\right)+{\int }_{{\omega }_e}{\psi }_{e}L\left(w_h\right)rot\left({\mathit{\sigma }}_h\right), \end{gathered}$$ (3.27)

where in the last equality we take into account

${\int }_{{\omega }_e}\mathbf{curl}\left({\psi }_{e}L\left(w_h\right)\right)·\mathit{\sigma }=-{\int }_{{\omega }_e}\mathbf{curl}\left({\psi }_{e}L\left(w_h\right)\right)·\nabla u={\int }_{\partial {\omega }_e}{\psi }_{e}L\left(w_h\right)\nabla u·\mathit{t}=0.$

In addition, realizing that ${\int }_{{\omega }_e}={\int }_T+{\int }_{T\text{'}}$ and applying Cauchy-Schwarz inequality, we deduce

$$\begin{gathered} c_2^{-1}{||w_h||}_{L^2\left(e\right)}^2\le {||\mathbf{curl}\left({\psi }_{e}L\left(w_h\right)\right)||}_{{\left[L^2\left(T\right)\right]}^2}{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2} \\ +{||{\psi }_{e}L\left(w_h\right)||}_{L^2\left(T\right)}{||rot\left({\mathit{\sigma }}_h\right)||}_{L^2\left(T\right)} \\ +{||\mathbf{curl}\left({\psi }_{e}L\left(w_h\right)\right)||}_{{\left[L^2\left(T\text{'}\right)\right]}^2}{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\text{'}\right)\right]}^2} \\ +{||{\psi }_{e}L\left(w_h\right)||}_{L^2\left(T\text{'}\right)}{||rot\left({\mathit{\sigma }}_h\right)||}_{L^2\left(T\text{'}\right)}. \end{gathered}$$ (3.28)

Now, knowing that $0\le {\psi }_e^{1/2}\le 1$, and taking into account (3.22), for each $T\in {\mathcal{T}}_h$, we deduce

${||{\psi }_{e}L\left(w_h\right)||}_{L^2\left(T\right)}\le c_3{h}_T^{1/2}{||w_h||}_{L^2\left(e\right)}.$ (3.29)

Now, the inverse inequality (3.23)), the fact that $0\le {\psi }_e^{1/2}\le 1$ in ω_e , together with (3.22)), implies for each $T\in {\omega }_e$

$$\begin{gathered} {||\mathbf{curl}\left({\psi }_{e}L\left(w_h\right)\right)||}_{{\left[L^2\left(T\right)\right]}^2}={||\nabla \left({\psi }_{e}L\left(w_h\right)\right)||}_{{\left[L^2\left(T\right)\right]}^2} \\ \le c{h}_T^{-1}{||{\psi }_{e}L\left(w_h\right)||}_{L^2\left(T\right)}\le c{h}_T^{-1}{||{\psi }_e^{1/2}L\left(w_h\right)||}_{L^2\left(T\right)} \\ \le c{c}_3{h}_T^{-1/2}{||w_h||}_{L^2\left(e\right)}. \end{gathered}$$ (3.30)

Inequalities (3.29) and (3.30) allow us to rewrite (3.28) as follows: There exists c > 0 independent of meshsize, such that

$$\begin{gathered} c{||w_h||}_{L^2\left(e\right)}^2\le \{h_T^{-1/2}{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\right)\right]}^2}+h_{T\text{'}}^{-1/2}{||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(T\text{'}\right)\right]}^2} \\ +h_T^{1/2}{||rot\left({\mathit{\sigma }}_h\right)||}_{L^2\left(T\right)}+h_{T\text{'}}^{1/2}{||rot\left({\mathit{\sigma }}_h\right)||}_{L^2\left(T\text{'}\right)}\}{||w_h||}_{L^2\left(e\right)}. \end{gathered}$$

Then, (3.26) follows after simplifying ${||w_h||}_{L^2\left(e\right)}$, multiplying by $h_e^{1/2}$ and invoking Lemma 3.12. □

Remark 3.14.The current a posteriori error analysis can be extended to three dimensions. To this aim, we consider Ω a bounded and simply connected polyhedral domain in ℝ³ . Now, given a partition 𝒯_h of$\overline{\Omega }$made of tetrahedral, we take into account similar notations as the ones introduced in Section 3, with face instead of edge. In addition, for any smooth enough vector field ρ, respectively, we set$\mathbf{curl}\left(\mathit{\rho }\right):=\nabla \times \mathit{\rho }$, while the jump of tangential trace of ρ across$e\in E_h$, by

$\left[\left[\mathit{n}\times \mathit{\rho }\right]\right]:=\left\{\begin{array}{l}{\mathit{n}}_{T,e}\times {\mathit{\rho }}_{T,e}+{\mathit{n}}_{T\text{'},e}\times {\mathit{\rho }}_{T\text{'},e}e\in E_I,\\ {\mathit{n}}_{T,e}\times {\mathit{\rho }}_{T,e}e\in E_{\Gamma },\end{array}\right.$

where$T,T\text{'}\in {\mathcal{T}}_h$are the pair of tetrahedral sharing the face$e\in E_I$. On the other hand, when$e\in E_{\Gamma }$, by$T\in {\mathcal{T}}_h$we refer to the unique element having e as a boundary face. Now, following the ideas given in the proof of Lemma 3.3, and applying Theorem I.3.5 in¹⁸, we can establish the 3D-version of the quasi-Helmholtz decomposition of functions belonging to H(div; Ω) presented in Lemma 3.3, which in addition is also stable (invoking Theorem 2.1 in¹³). This means that for any$\mathit{\tau }\in H\left(div;\Omega \right)$, there exist$\mathit{\chi }\in {\left[H^1\left(\Omega \right)\right]}^3$and$\mathit{\Phi }\in {\left[H_0^1\left(\Omega \right)\right]}^3$, such that

$\mathit{\tau }=\mathbf{curl}\left(\mathit{\chi }\right)+\mathit{\Phi }+\frac{d}{3}\left(\begin{array}{c}{x}_1-a\\ x_2-b\\ x_3-c\end{array}\right),$

where (a, b, c)^t is any fixed point belonging to Ω, and$d:=\frac{1}{\left|\Omega \right|}{\int }_{\Omega }div\left(\mathit{\tau }\right)$. In addition, there exists C > 0, such that

${\left|\mathit{\chi }\right|}_{{\left[H^1\left(\Omega \right)\right]}^3}^2+{||\Phi ||}_{{\left[H^1\left(\Omega \right)\right]}^3}^2\le C{||\mathit{\tau }||}_{H\left(div;\Omega \right)}^2.$

Then, proceeding in analogous way as in Section 3, we prove a similar result to Theorem 3.2, where the local a posteriori error estimator now reads as

$$\begin{gathered} {\eta }_T^2:={||f+div\left({\mathit{\sigma }}_h\right)||}_{{\left[L^2\left(T\right)\right]}^3}^2+h_T^2{||{\mathit{\sigma }}_h+\nabla {\mathit{u}}_h||}_{{\left[L^2\left(T\right)\right]}^3}^2+h_T^2{||\mathbf{curl}\left({\mathit{\sigma }}_h\right)||}_{{\left[L^2\left(T\right)\right]}^3}^2 \\ +\sum _{e\in E\left(T\right)\cap E_I}\left\{h_e{||\left[\left[{\mathit{u}}_h\right]\right]||}_{{\left[L^2\left(e\right)\right]}^3}^2+h_e{||\left[\mathit{n}\times {\mathit{\sigma }}_h\right]||}_{{\left[L^2\left(e\right)\right]}^3}^2\right\} \\ +\sum _{e\in E\left(T\right)\cap E_{\Gamma }}{h}_e{||\mathit{n}\times \left({\mathit{\sigma }}_h+\nabla \mathit{g}\right)||}_{{\left[L^2\left(e\right)\right]}^3}^2. \end{gathered}$$ (3.31)

4 NUMERICAL EXAMPLE

In this section, we present one numerical example illustrating the performance of the dual mixed method when applied to the Poisson problem, with Dirichlet condition, as well as of the corresponding adaptive procedure. We consider the lowest finite element ${\mathcal{RT}}_0\left(T\right)-{\mathcal{P}}_0\left(T\right)$ for our approximation. We remark that the computational implementation has been done using a MATLAB code.

Hereafter, the number of degrees of freedom (unknowns) is given by N: = number of edges + number of elements, induced by the triangulation. Moreover, the involved individual and total errors are defined as $e_0\left(u\right):={||u-u_h||}_{L^2\left(\Omega \right)},e\left(\mathit{\sigma }\right):={\left({||\mathit{\sigma }-{\mathit{\sigma }}_h||}_{{\left[L^2\left(\Omega \right)\right]}^2}^2+{||div\left(\mathit{\sigma }-{\mathit{\sigma }}_h\right)||}_{L^2\left(\Omega \right)}^2\right)}^{1/2}$ and $e:={\left(e_0{\left(u\right)}^2+e{\left(\sigma \right)}^2\right)}^{1/2}$, where $\left(\mathit{\sigma },u\right)\in H\left(div;\Omega \right)\times L^2\left(\Omega \right)$ and $\left({\mathit{\sigma }}_{\mathbf{h}},u_h\right)\in H_{h,r}^{\mathit{\sigma }}\times H_{h,r}^u$ are the corresponding unique solutions of the continuous (2.1) and discrete (2.2) formulations. Additionally, if e and e’ stand for the errors at two consecutive triangulations with N and N’ number of degrees of freedom, respectively, we set the experimental rate of convergence of the global error as $r:=-2\frac{\log \left(e/e\text{'}\right)}{\log \left(N/N\text{'}\right)}$. We define r0(u) and r(σ ) in analogous way.

The data f and g for our example, are chosen so that the exact solution is $u\left(x,y\right)=\frac{xy}{{\left(x+1.05\right)}^2+y^2}$ and $\Omega :={\left(-1,1\right)}^2\backslash {\left[0,1\right]}^2$. We notice that in this case u has a singularity at (−1.05, 0), which does not belong to Ω, but it is very close to ∂Ω. Then, u has a numerical singularity in a neighborhood of $\left(-1,0\right)\in \Gamma$.

Then, the purpose of this example, is to show the performance of the following adaptive algorithm (cf. ²²). Given an a posteriori error estimator$\eta :=\sum _{T\in {\mathcal{T}}_h}{\eta }_T^2$:

• 1. Start with a coarse mesh 𝒯_h .

• 2. Solve the Galerkin scheme for the current mesh 𝒯_h .

• 3. Compute ηT for each triangle T ∈ 𝒯_h .

• 4. Consider stopping criterion and decide to finish or go to the next step.

• 5. Apply Blue-green procedure to refine each element T’ ∈ 𝒯_h such that

${\eta }_{T\text{'}}\ge \frac{1}{2}max\left\{{\eta }_T:T\in {\mathcal{T}}_h\right\}.$

• 6. Define the resulting mesh as the new 𝒯_h and go to step 2.

Table 1 reports the histories of convergence of the individual and total errors for a sequence of uniform and adaptive refinements, respectively. We notice that the adaptive refinement algorithm is able to recognize the numerical singularity, and then the induced sequence of adapted meshes let us to improve the quality of approximation, better than the corresponding when uniform refinement is performed. In addition, we observe that index of efficiency e/η remains bounded, indicating that η is reliable and efficient, despite the fact that g in this case is not piecewise polynomial. Figure 1 displays some adapted meshes, generated by the proposed adaptive algorithm, from which we observe that the numerical singularity is detected.

Table 1:
History of convergence of Example provided, considering uniform (up) and adaptive (bottom) refinements.

N	e 0 (u)	r 0 (u)	e(σ)	r(σ)	e	r	η	e/η
34	4.6846e-01	-	4.2795e+00	-	4.3051e+00	-	806.54	0.0053
128	3.8157e-01	0.3095	4.5340e+00	-	4.5500e+00	-	434.93	0.0105
496	2.3974e-01	0.6862	5.1609e+00	-	5.1665e+00	-	213.32	0.0242
1952	1.1636e-01	1.0552	4.5542e+00	0.1826	4.5557e+00	0.1837	103.90	0.0438
7744	5.4734e-02	1.0947	4.5472e+00	0.0022	4.5475e+00	0.0026	50.360	0.0903
30848	2.0412e-02	1.4272	3.4610e+00	0.3950	3.4611e+00	0.3951	23.629	0.1465
123136	5.4906e-03	1.8972	1.8229e+00	0.9264	1.8229e+00	0.9264	11.484	0.1587
34	4.6846e-01	-	4.2795e+00	-	4.3051e+00	-	806.54	0.0053
99	3.8166e-01	0.3835	4.5365e+00	-	4.5526e+00	-	434.93	0.0105
192	2.3979e-01	1.4034	5.2105e+00	-	5.2160e+00	-	213.28	0.0245
289	1.1864e-01	3.4415	4.7172e+00	0.4865	4.7187e+00	0.4901	103.90	0.0454
386	5.9916e-02	4.7210	4.8647e+00	-	4.8651e+00	-	50.740	0.0959
483	3.6723e-02	4.3672	4.1212e+00	1.4798	4.1213e+00	1.4802	25.579	0.1611
580	3.3800e-02	0.9067	3.2165e+00	2.7084	3.2167e+00	2.7083	17.125	0.1878
1021	3.2498e-02	0.1389	2.4170e+00	1.0106	2.4172e+00	1.0105	11.726	0.2061
1747	2.2408e-02	1.3843	1.8179e+00	1.0606	1.8181e+00	1.0607	8.7937	0.2067
3633	2.2308e-02	0.0122	1.2888e+00	0.9395	1.2890e+00	0.9393	6.0359	0.2136
7208	6.5209e-03	3.5904	8.8348e-01	1.1024	8.8350e-01	1.1027	4.2152	0.2096
14979	3.9386e-03	1.3786	6.3307e-01	0.9113	6.3309e-01	0.9113	2.9475	0.2148
29052	1.7638e-03	2.4257	4.4686e-01	1.0517	4.4687e-01	1.0517	2.1120	0.2116
56844	1.1159e-03	1.3639	3.2513e-01	0.9476	3.2513e-01	0.9476	1.5154	0.2146
114267	4.5185e-04	2.5896	2.2625e-01	1.0386	2.2625e-01	1.0386	1.0765	0.2102
219394	2.8879e-04	1.3725	1.6567e-01	0.9555	1.6567e-01	0.9555	0.7781	0.2129

Figure 1:
Adapted meshes corresponding (top-bottom, left-right) to 192, 386, 3633 and 29052 dofs, for Example considered, with Dirichlet boundary condition (based on η).

CONCLUDING REMARKS

In this paper, we have developed an a posteriori error analysis for a dual mixed formulation of Poisson problem in the plane, with non homogeneous Dirichlet boundary condition. By establishing a new kind of quasi-Helmholtz decomposition of functions in H(div; Ω) (cf. Lemma 3.3), we are able to obtain an a posteriori error estimator, which consists of six residual terms, and results to be reliable and locally efficient with respect to the error measured in its natural norm on $H\left(div;\Omega \right)\times L^2\left(\Omega \right)$. In this sense, we have generalized the results obtained in previous works ( ¹, ¹⁰, for example), and without invoking the so called saturation assumption.

The results of numerical experiment, included in this work, are in agreement with our theoretical analysis. Here, we notice that the estimator is able to help us to identify which part of the domain is localized the numerical singularity of the exact solution. As a consequence, the adaptive algorithm, based on this estimator, let us to improve the quality of the approximation.

Finally, since Lemma 3.3 can be proved for 3d case too, the current work can be extended to 3d, obtaining a reliable and locally efficient residual a posteriori error estimator, consisting also of six residual terms (cf. (3.31)).

Acknowledgments

This research was partially supported by ANID-Chile through the the project Centro de Modelamiento Matemático (AFB170001) of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal CONICYT-Chile and FONDECYT grant No. 1200051; by Direcciones de Investigación y de Postgrado de la Universidad Católica de la Santísima Concepción (Chile), through Incentivo Mensual and Becas de Mantención programs, and by Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción (Chile).

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Publication Dates

• Publication in this collection
  05 Sept 2022
• Date of issue
  Jul-Sep 2022

History

• Received
  27 Sept 2021
• Accepted
  24 Mar 2022