Residual-based a posteriori error estimates for a conforming mixed finite element discretization of the Monge-Ampère equation

1. General introduction

The adaptive techniques have become indispensable tools and unavoidable in the field of study behavior of the error committed during solving partial differential equations (PDE). A posteriori error estimators are computable quantities, expressed in terms of the discrete solution and of the data that measure the actual discrete errors without the knowledge of the exact solution. They are essential to design adaptive mesh refinement algorithms which equi-distribute the computational effort and optimize the approximation efficiency. Since the pioneering work of Babuska and Rheinboldt

[8, 7], adaptive finite element methods based on a posteriori error estimates have been extensively investigated. Several a-posteriori error analysis methods for PDE have been developped in the last five decades [24, 23, 1, 3, 5, 6, 9, 10, 11, 28, 16, 19, 2, 21].

We consider the Monge-Ampère equation on a convex domain of $R^{2}$ with a smooth solution and our approach utilizes a slight variant of the mixed discretization proposed by Gerard Awanou and Hengguang Li in [4]. The purpose of these work is to determine to which extend the general framework for adaptivity for nonlinear problems of Gatica and his collaborators in [15, 19] can be applied to the Monge-Ampère equation. More precisely, we attempt to determine to which extent one can prove results analogous to the ones of Houédanou, Adetola and Ahounou [20]. Ideed, in [4] Gerard Awanou and Li study the mixed method for this equation and they gave a priori error estimator under the assumption of regularity for the solution of continuous problem. Omar Lakkis in his presentation of July 15, 2014, presents a family of reliable error indicators for a primal formulation [22]. However, to our best knowledge, they din’t talk about adaptative method for this mixed formulation. In this case we have for main to give a posteriori error analysis by constucting reliable and efficiency indicator error.

In [4] Gerard and Li have introduce a mixed finite element method formulation for the elliptic Monge-Ampère equation by puting $σ = D^{2} u$ . The news unknowns in the formulation are $u$ and $σ$ which been approached respectively by the discrete polynomials spaces of Lagrange. Finally, they give a result of error priori analysis with some numerical tests confirming the convergence rates. In this paper, we have got a new family of a local indicator error $Θ_{K}$ (see Definition 3.1, eq. 3.1) and global $Θ$ (eq. 26) efficiency and reliability for the mixed method of Monge-Ampère model. We prove that our indicators error are efficiency and reliability, and then, are optimal. The global inf-sup condition is the main tool yielding the reliability. In turn, The local efficiency result is derived using the technique of bubble function introduced by R. Verfürth [25] and used in similar context by C. Carstensen [13, 12].

The paper is organized as follows. Some preliminaries and notation are given in section 2. In section 3, the a posteriori error estimates are derived. We offer our conclusion and the further works in Section 4.

2. Preliminaries and Notation

2.1. Model

Let $Ω$ be a convex polygonal domain of $R^{2}$ with boundary $\partial Ω$ . We consider the following problem : find the unique strictly convex $C^{3} (¯ ¯¯ ¯ Ω)$ solution $u$ (when it exists) of

\begin{matrix} det (D^{2} u) & = f in Ω u & = g on \partial Ω . \end{matrix}

(1)

The given function $f \in C^{1} (¯ ¯¯ ¯ Ω)$ is assumed to satisfy $f > 0$ and the function $g \in C (\partial Ω)$ is also given and assumed to extend to a $C^{3} (¯ ¯¯ ¯ Ω)$ function. Here $det (D^{2} u)$ denotes the determinant of the Hessian matrix $D^{2} u = (\partial^{2} u / (\partial x_{i} \partial x_{j}))_{i, j = 1, 2}$ .

2.2. Modified continuous mixed weak formulation

We begin this subsection by introducing some useful notations. If $W$ is a bounded domain of $R^{2}$ and $m$ is a non negative integer, the Sobolev space $H^{m} (W) = W^{m, 2} (W)$ is defined in the usual way with the usual norm $∥ \cdot ∥_{m, W}$ and semi-norm $| . |_{m, W}$ . In particular, $H^{0} (W) = L^{2} (W)$ and we write $∥ \cdot ∥_{W}$ for $∥ \cdot ∥_{0, W}$ . Similarly we denote by $(\cdot, \cdot)_{W}$ the $L^{2} (W)$ , $[L^{2} (W)]^{2}$ or $[L^{2} (W)]^{2 \times 2}$ inner product. Now, we recall the continuous mixed weak formulation introduce by Gérard et al. [4]. The mixed weak formulation of (1) is : find $(σ, u) \in H^{1} (Ω)^{2 \times 2} \times H^{2} (Ω)$ such that

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (σ, μ)_{Ω} + (\nabla \cdot μ, D u)_{Ω} - < D u, μ n >_{\partial Ω} & = & 0, \forall μ \in H^{1} (Ω)^{2 \times 2} (det σ, v) & = & (f, v)_{Ω}, \forall v \in H_{0}^{1} (Ω) u & = & g on \partial Ω . \end{matrix}

(2)

Let introduce Lagrange multiplier $λ = u |_{\partial Ω}$ , the modified mixed weak formulation of (1) is : find $(σ, u, λ) \in H^{1} (Ω)^{2 \times 2} \times H^{2} (Ω) \times H^{1 / 2} (\partial Ω)$ such that

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (σ, μ)_{Ω} + (\nabla \cdot μ, D u)_{Ω} - {⟨ D u, μ n ⟩}_{\partial Ω} & = & 0, \forall μ \in H^{1} (Ω)^{2 \times 2} (det σ, v)_{Ω} & = & (f, v)_{Ω}, \forall v \in H_{0}^{1} (Ω) (λ, μ)_{\partial Ω} & = & (g, μ)_{\partial Ω} \forall μ \in H^{- 1 / 2} (\partial Ω) . \end{matrix}

(3)

Lemma 2.1.

(ref. [4]) The problem (2) is well defined, and if $u$ is a smooth solution of (1), then $(u, D^{2} u)$ solves (2).

Remark 2.1.

In this formulation the boundary condition $u = g$ on $\partial Ω$ viewed as constraints and imposed via Lagrange multiplier.

We end this section with some notation. Let $P_{k}$ be the space of polynomials of total degree not larger than $k$ . In order to avoid excessive use of constants, the abbreviations $x ≲ y$ and $x \sim y$ stand for $x \leq c y$ and $c_{1} x \leq y \leq c_{2} x$ , respectively, with positive constants independent of $x$ , $y$ or $T_{h}$ (meshes).

2.3. Modified discrete formulation

Let $Ω$ be an open convex bounded subset of $R^{2}$ with boundary $\partial Ω$ and let $T_{h}$ denote a triangulation of $Ω$ into simplices $K$ . We denote by $h_{K}$ the diameter of the element $K$ and $h = max K \in T_{h} h_{K}$ . We make the assumption that the triangulation is conforming and satisfies the usual shape regularity condition, i.e. there exists a constant $σ > 0$ suth that: $\frac{h_{K}}{ρ_{K}} \leq σ,$ for all $K \in T_{h}$ where $ρ_{K}$ denotes the radius of the largest ball inside $K$ . (See Figs. 3, 3,3).

Figure 1. Isotropic element

K

R^{2}

Figure 2. Example of conforming mesh in

R^{2}

Figure 3. Example of nonconforming mesh in

R^{2}

To be able to use global inverse estimates, c.f. (2.2) and $(2.3)$ of [4], we require the triangulation to be also quasi-uniform, i.e. there is a constant $C > 0$ such that $h \leq C h_{K}$ for all $K \in T_{h}$ .

For any $K \in T_{h}$ , we denote by $E_{h} (K)$ (resp. $N_{h} (K))$ the set of its edges (resp. vertices) and set $E_{h} = ⋃ K \in T_{h} E (K)$ , $N_{h} = ⋃ K \in T_{h} N (K)$ . For $A \subset ¯ ¯¯ ¯ Ω$ we define:

E_{h} (A) = {E \in E_{h} : E \subset A} .

Let $V_{h}$ denote the standard Lagrange finite element space of degree $k \geq 3$ and $Σ_{h} = V_{h}^{2 \times 2}$ ; that is we consider the following discret spaces :

V_{h} := {v_{h} \in C^{0} (¯ ¯¯ ¯ Ω) : v_{| K} \in P_{k} (K) \forall K \in T_{h}, k \geq 3},

(4)

(5)

The discrete formulation of (2) is given by : find $(u_{h}, σ_{h}) \in V_{h} \times Σ_{h}$ such that

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (σ_{h}, τ_{h}) + (\nabla \cdot τ_{h}, D u_{h}) - ⟨ D u_{h}, τ_{h} n ⟩ & = & 0 \forall τ_{h} \in Σ_{h} (d e t σ_{h}, v_{h}) & = & (f, v_{h}) \forall v_{h} \in V_{h} u_{h} & = & g_{h} on \partial Ω, \end{matrix}

(6)

We recall that $H_{0}^{1} (Ω)$ is the subset of $H^{1} (Ω)$ of elements with vanishing trace on $\partial Ω$ . Let $I_{h}$

denote the standard Lagrangian interpolation operator from

C^{0} (\partial Ω)

into the space

L_{h} := {{v_{h}}_{| \partial Ω} : v_{h} \in V_{h}}

. The modified discrete formulation of (3) is given by find : For

ρ > 0

, we difine by :

(σ_{h}, u_{h}, λ_{h}) \in V_{h} \times Σ_{h} \times L_{h}

such that

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (σ_{h}, τ_{h})_{Ω} + (\nabla \cdot τ_{h}, D u_{h})_{Ω} - ⟨ D u_{h}, τ_{h} n ⟩_{\partial Ω} & = & 0 \forall τ_{h} \in Σ_{h} (d e t σ_{h}, v_{h})_{Ω} & = & (f, v_{h})_{Ω} \forall v_{h} \in V_{h} ⟨ λ_{h}, μ ⟩_{\partial Ω} & = & ⟨ g_{h}, μ ⟩_{\partial Ω} \forall μ \in L_{h}, \end{matrix}

(7)

with

g_{h} = I_{h} g .

Now, we define

{¯ B}_{h} (ρ) = {(η_{h}, w_{h}) \in Σ_{h} \times V_{h}, ∥ w_{h} - I_{h} u ∥_{H^{1}} \leq ρ, ∥ η_{h} - I_{h} σ ∥_{L^{2}} \leq h^{- 1} ρ}

and

B_{h} (ρ) = {¯ B}_{h} (ρ) \cap Z_{h},

where

	$Z_{h} := {(w_{h}, η_{h}) \in H_{h} \times Q_{h}, w_{h} = g_{h} on \partial Ω,$
	$(η_{h}, τ_{h}) + (\nabla \cdot τ_{h}, D w_{h}) - ⟨ D w_{h}, τ_{h} n ⟩ = 0 \forall τ_{h} \in Q_{h}} .$

By simple calcultions, the problem (7) is logically equivalent to (6) and we have the result:

Lemma 2.2.

(cf. [4]) The problem (7) as unique solution in $B_{h} (ρ)$ .

Theorem 2.1.

[4] Let $(u, σ) \in H^{k + 1} (Ω) \times H^{k} (Ω)^{2 \times 2}$ be the convex solution of non linear problem (2) with $k \geq 3$ . The discrete non linear problem (6) has a unique solution $(u_{h}, σ_{h})$ in $B_{h} (ρ) \subset V_{h} \times Σ_{h}$ . Moreover the following estimate holds.

∥ u - u_{h} ∥_{H^{1}} \leq C h^{k} .

(8)

∥ σ - σ_{h} ∥_{L^{2}} \leq C h^{k - 1} .

(9)

3. A-posteriori error analysis

In order to solve Monge-Ampère problem by efficient adaptive finite element methods, reliable and efficient a posteriori error analysis is important to provide appropriated indicators. In this section, we first define the local and global indicators and then the lower and upper error bounds are derived.

3.1. Residual Error Estimators

The general philosophy of residual error estimators is to estimate an appropriate norm of the correct residual by terms that can be evaluated easier, and that involve the data at hand.

Let $U = (σ, u, λ)$ and $W = (τ, v, μ)$ . We define the operator B by

⟨ B (U), W ⟩ := (σ, τ)_{Ω} + (\nabla \cdot τ, D u)_{Ω} - < D u, τ n >_{\partial Ω} - (det σ, v)_{Ω} + ⟨ λ, μ ⟩_{\partial Ω},

and

⟨ F, W ⟩ := (- f, v)_{Ω} + (g, μ)_{\partial Ω} .

We also define by $H = H^{1} (Ω)^{2 \times 2} \times H^{2} (Ω) \times H^{1 / 2} (\partial Ω)$ and $M = H^{1} (Ω)^{2 \times 2} \times H^{1} (Ω) \times H^{1 / 2} (\partial Ω)$ . Then the continuous problem (3) is equivalent to : find $U \in H$ , such that

⟨ B (U), W ⟩ = ⟨ F, W ⟩, \forall W \in M

(10)

We define the discrete version by the same way.

Then let, $U_{h} = (σ_{h}, u_{h}, λ_{h})$ and $W = (τ, v, μ)$ . We define by:

	$⟨ B (U_{h}), W ⟩$	$=$	$(σ_{h}, μ)_{Ω} + (\nabla \cdot μ, D u_{h})_{Ω} - < D u_{h}, μ n >_{\partial Ω}$		(11)
		$+$	$(det σ_{h}, v) + ⟨ λ_{h}, μ ⟩_{\partial Ω}$		(11)

and

⟨ F, W ⟩ := (f, v)_{Ω} + (g_{h}, μ)_{\partial Ω} .

(12)

We also define by $H_{h} = V_{h} \times Σ_{h} \times L_{h}$ . The discrete problem (7) is equivalent
to : find $U_{h} \in H_{h}$ , such that,

⟨ B (U_{h}), W ⟩ = ⟨ F, % W ⟩, \forall W \in H_{h} .

(13)

We recall the following Lemma

Lemma 3.1.

(cf. [4, Section 3.1]) : Fréchet derivate of the determinant. For $F (v) = det D^{2} v$ , we have $F^{'} (v) (u) = (c o f D^{2} v) : D^{2} u$ .

By using the Lemma 3.1, we have the operator $B$ is differentiable and for all $U = (σ, u, λ) \in H$ , $W = (τ, v, μ) \in M$ and $V = (τ^{'}, v^{'}, μ^{'}) \in M$ , its differential at $V = (τ^{'}, v^{'}, μ^{'})$ is given by:

	$⟨ D_{V} B (U), W ⟩$	$=$	$(σ, τ)_{Ω} + (\nabla \cdot τ, D u)_{Ω} - < D u, τ n >_{\partial Ω}$		(14)
		$+$	$(c o f D^{2} v^{'} : D^{2} u, v) + ⟨ λ, μ ⟩_{\partial Ω} .$		(14)

We deduce the existence of a positive constant $C_{m}$ , independent of $ξ \in H$ and the continuous and discrete solutions, such that the following global inf-sup condition holds:

C_{m} ∥ ξ ∥_{H}

\leq

sup W \in M - {0} \frac{| ⟨ D B_{V} (ξ), W ⟩ |}{∥ W ∥_{M}}

(15)

We have the following lemma:

Lemma 3.2.

There holds

∥ U - U_{h} ∥_{H}

\leq

C ∥ R ∥_{H^{'}} + ∥ f - det σ_{h} - A : D^{2} u_{h} ∥_{L^{2} (Ω)} .

(16)

Where $R$ is the residual functional define by $R (W) = [F - B (U_{h}), W]$ $\forall W \in M$ , which satisfies:

R (W_{h}) = 0 \forall W_{h} \in H_{h} .

Proof.

Let $U = (σ, u, λ)$ , $W = (τ, v, μ)$ and $V = (τ^{'}, v^{'}, μ^{'})$ , we have

	$⟨ D_{V} B (U), W ⟩$	$=$	$(σ, τ)_{Ω} + (\nabla \cdot τ, D u)_{Ω}$
		$-$	$< D u, τ n >_{\partial Ω} - (c o f D^{2} v^{'} : D^{2} u, v) + ⟨ λ, μ ⟩_{\partial Ω}$

and

$⟨ D_{V} B (U - U_{h}), W ⟩$	$=$	$(σ, τ)_{Ω} + (\nabla \cdot τ, D u)_{Ω}$
	$-$	$< D u, τ n >_{\partial Ω} - (c o f D^{2} v^{'} : D^{2} u, v)$
	$+$	$⟨ λ, μ ⟩_{\partial Ω}$
	$-$	$[(σ_{h}, τ)_{Ω} + (\nabla \cdot τ, D u_{h})_{Ω}$
	$-$	$< D u_{h}, τ n >_{\partial Ω} + (c o f D^{2} v^{'} : D^{2} u_{h}, v)$
	$+$	$⟨ λ_{h}, μ ⟩_{\partial Ω}]$

Particulary for $V = U$ , we have

$⟨ D_{V} B (U - U_{h}), W ⟩$	$=$	$⟨ F, W ⟩ - [(σ_{h}, τ)_{Ω}$
	$+$	$(\nabla \cdot τ, D u_{h})_{Ω} - < D u_{h}, τ n >_{\partial Ω} + (A : D^{2} u_{h}, v)$
	$+$	$(det σ_{h}, v) - (det σ_{h}, v) + (det σ, v) + ⟨ λ_{h}, μ ⟩_{\partial Ω}],$

where $A := c o f D^{2} u$ . Therefore,

$⟨ D_{V} B (U - U_{h}), W ⟩$	$=$	$⟨ F - B (U_{h}), W ⟩ + (f - det σ_{h}, v) - (A : D^{2} u_{h}, v)$	(17)
	$=$	$⟨ F - B (U_{h}), W ⟩ + (f - det σ_{h} - A : D^{2} u_{h}, v)$
	$=$	$R (W) + (f - det σ_{h} - A : D^{2} u_{h}, v)$

Using Cauchy-Schwarz inequality and the inequality (15), the result follows. ∎

Now we define the residual equation:

R (W) = [F - B (U_{h}), W],

(18)

hence,

$R (W) =$	$=$	$(f, v)_{Ω} + (g, μ)_{\partial Ω} - (σ_{h}, τ)_{Ω} - (\nabla τ, D u_{h})_{Ω} + ⟨ τ n, D u_{h} ⟩$
	$-$	$(A : D^{2} u_{h}, v)_{Ω} - (λ_{h}, μ)_{\partial Ω}$
	$=$	$(f - A : D^{2} u_{h}, v)_{Ω} - (σ_{h}, τ)_{Ω} - (\nabla τ, D u_{h})_{Ω}$
	$+$	$⟨ τ n, D u_{h} ⟩_{\partial Ω} + (g - λ_{h}, μ)_{\partial Ω} .$

By integrating by parts, we obtain for $W = (v, τ, μ) \in M$ , the equation:

R (W) = R_{1} (v) + R_{2} (τ) + R_{3} (μ),

(19)

where:

$R_{1} (v)$	$:=$	$(f - A : D^{2} u_{h}, v)_{Ω}, v \in H := H^{1} (Ω)$
$R_{2} (τ)$	$:=$	$(D^{2} u_{h} - σ_{h}, τ)_{Ω}, τ \in Σ := [H^{1} (Ω)]^{2 \times 2}$
$R_{3} (μ)$	$:=$	$(g - λ_{h}, μ)_{\partial Ω}, μ \in H^{1 / 2} (\partial Ω)$

In this way, it follows that:

∥ R ∥_{M^{'}} \leq C {∥ R_{1} ∥_{H^{'}} + ∥ R_{2} ∥_{Σ^{'}} + ∥ R_{3} ∥_{H^{- 1 / 2} (\partial Ω)}}

(20)

and hence our next purpose is to derive suitable upper bounds for each one of the terms on the right hand side of (20). We start with the following lemma, which is a direct consequence of the Cauchy-Schwarz inequality.

Lemma 3.3.

There exist $C_{1}$ , $C_{2}$ and $C_{3} > 0$ , independent of the meshsizes, such that:

∥ R_{1} ∥_{H^{'}} \leq C_{1} {⎧ ⎨ ⎩ \sum K \in T_{h} (∥ f_{h} - A : D^{2} u_{h} ∥_{K}^{2} + ∥ f - f_{h} ∥_{K}^{2}) ⎫ ⎬ ⎭}^{1 / 2}

(21)

and

∥ R_{2} ∥_{Σ^{'}} \leq C_{2} {⎧ ⎨ ⎩ \sum K \in T_{h} ∥ σ_{h} - D^{2} u_{h} ∥_{K}^{2} ⎫ ⎬ ⎭}^{1 / 2} .

(22)

In addition there holds

∥ R_{3} ∥_{H^{- 1 / 2} (Ω)} \leq C_{3} {⎧ ⎨ ⎩ \sum E \in E_{h} (\partial Ω) ∥ g - λ_{h} ∥_{E}^{2} ⎫ ⎬ ⎭}^{1 / 2} .

(23)

Lemma 3.4.

There exist a positive constant $C$ , such that

∥ R ∥_{M^{'}} \leq C ⎧ ⎪ ⎨ ⎪ ⎩ {⎛ ⎝ \sum K \in T_{h} ∥ f_{h} - A : D^{2} u_{h} ∥_{K}^{2} ⎞ ⎠}^{1 / 2} + {⎛ ⎝ \sum K \in T_{h} ∥ σ_{h} - D^{2} u_{h} ∥_{K}^{2} ⎞ ⎠}^{1 / 2} + {⎛ ⎝ \sum E \in E_{h} (\partial Ω) ∥ g - λ_{h} ∥_{E}^{2} ⎞ ⎠}^{1 / 2} + {⎛ ⎝ \sum K \in T_{h} ∥ f_{h} - f ∥_{K}^{2} ⎞ ⎠}^{1 / 2} ⎫ ⎪ ⎬ ⎪ ⎭ .

(24)

Proof.

By using (20),(21),(22),(23) and Cauchy-Schwarz inequality, the result follows. ∎

3.1.1. A posteriori error indicators

Now, we define the error indicators:

Definition 3.1 (A-posteriori error indicators).

Let $U_{h} = (σ_{h}, u_{h}, λ_{h}) \in H_{h}$ be the finite element solution. Then, the residual error estimator is locally defined by:

$Θ_{K}^{2} (U_{h})$	$:=$	$∥ f_{h} - A : D^{2} u_{h} ∥_{K}^{2} + ∥ σ_{h} - D^{2} u_{h} ∥_{K}^{2}$
	$+$	$∥ f_{h} - det σ_{h} - A : D^{2} u_{h} ∥_{K}^{2}$
	$+$	$\sum E \in E_{h} (K \cap \partial Ω) ∥ g_{h} - λ_{h} ∥_{E}^{2} .$

The global residual error estimator is given by:

Θ (U_{h})

:=

{⎛ ⎝ \sum K \in T_{h} Θ_{K}^{2} (U_{h}) ⎞ ⎠}_{h}^{1 / 2} .

(26)

Furthermore denote the local and global approximation terms by:

ζ_{K}^{2} := ∥ f - f_{h} ∥_{K}^{2} + \sum E \in E_{h} (K \cap \partial Ω) ∥ g - g_{h} ∥_{E}^{2},

and

ζ := {⎛ ⎝ \sum K \in T_{h} ζ_{K}^{2} ⎞ ⎠}^{1 / 2} .

3.1.2. Reliability of $Θ$

The first main result is given by the following theorem.

Theorem 3.1 (Reliability of the a posteriori error estimator).

Let $U = (σ, u, λ) \in H$ be the exact solution and $U_{h} = (σ_{h}, u_{h}, λ_{h}) \in H_{h}$ be the finite element solution. Then, there exist a positive constant $C$ such that:

∥ U - U_{h} ∥_{H}

\leq

C [Θ (U_{h}) + ζ] .

(27)

Proof.

By using Lemma 3.2 and the estimate (24), the result follows. ∎

3.1.3. Efficiency of $Θ$

The second main result of this paper is the efficiency of the a posteriori error estimator $Θ$ . In order to derive the local lower bounds, we proceed similarly as in [12, 20, 14] (see also [18]), by applying inverse inequalities, and the localization technique based on simplex-bubble and face-bubble functions. To this end, we recall some notation and introduce further preliminary results. Given $K \in T_{h}$ , and $E \in E_{h} (K)$ , we let $b_{K}$ and $b_{E}$ be the usual simplexe-bubble and face-bubble functions respectively (see (1.5) and (1.6) in [27]). In particular, $b_{K}$ satisfies $b_{K} \in P^{2} (K)$ , $s u p p (b_{K}) \subseteq K$ , $b_{K} = 0 on \partial K$ , and $0 \leq b_{K} \leq 1 on K$ . Similarly, $b_{E} \in P^{2} (K)$ , $s u p p (b_{E}) \subseteq ω_{E} := {K^{'} \in T_{h} : E \in E_{h} (K^{'})}$ , $b_{E} = 0 on \partial K ∖ E$ and $0 \leq b_{E} \leq 1 in ω_{E}$ . We also recall from [26] that, given $k \in N$ , there exists an extension operator $L : C (E) ⟶ C (K)$ that satisfies $L (p) \in P^{k} (K)$ and $L (p)_{| E} = p, \forall p \in P^{k} (E)$ . A corresponding vectorial version of $L$ , that is, the componentwise application of $L$ , is denoted by L. Additional properties of $b_{K}$ , $b_{E}$ and $L$ are collected in the following lemma (see [26]):

Lemma 3.5.

Given $k \in N^{*}$ , there exist positive constants depending only on $k$ and shape-regularity of the triangulations (minimum angle condition), such that for each simplexe $K$ and $E \in E_{h} (K)$ there hold

$∥ q ∥_{K}$	$≲$	$∥ q b_{K}^{1 / 2} ∥_{K} ≲∥ q ∥_{K}, \forall q \in P^{k} (K)$	(28)
$\| q b_{K} \|_{1, K}$	$≲$	$h_{K}^{- 1} ∥ q ∥_{K}, \forall q \in P^{k} (K)$	(29)
$∥ p ∥_{E}$	$≲$	$∥ b_{E}^{1 / 2} p ∥_{E} ≲∥ p ∥_{E}, \forall p \in P^{k} (E)$	(30)
$∥ L (p) ∥_{K} + h_{E} \| L (p) \|_{1, K}$	$≲$	$h_{E}^{1 / 2} ∥ p ∥_{E} \forall p \in P^{k} (E)$	(31)

To this end, we recall some notation. We define the error respect to $σ$ , $u$ and $λ$ respectively by $e_{σ} = σ - σ_{h}$ , $e_{u} = u - u_{h}$ and $e_{λ} = λ - λ_{h}$ . Then we recall the global error define by $∥ U - U_{h} ∥_{H} := {(∥ e_{σ} ∥_{1, Ω}^{2} + ∥ e_{u} ∥_{2, Ω}^{2} + ∥ e_{λ} ∥_{1 / 2, \partial Ω}^{2})}_{σ}^{1 / 2}$ . To prove local efficiency for $w \subset Ω$ , let us denote by:

∥ V ∥_{h, w} := {⎡ ⎣ \sum K \subset ¯ w ⎛ ⎝ ∥ v ∥_{2, K}^{2} + ∥ τ ∥_{0, K}^{2} + \sum E \in E_{h} (¯ K) ∥ μ ∥_{0, E}^{2} ⎞ ⎠ ⎤ ⎦}^{1 / 2}, where V = (τ, v, μ) \in H .

The main result of this subsection can be given as follows:

Theorem 3.2.

Let $f \in L^{2} (Ω)$ , $g \in L^{2} (\partial Ω)$ . Let $(σ, u, λ)$ be the unique solution of the continous problem and $(σ_{h}, u_{h}, λ_{h})$ be the unique solution of discret problem. Then, the local error estimator $Θ_{K}$ satisfies :

Θ_{K} ≲ ∥ (e_{σ}, e_{u}, e_{λ}) ∥_{h, {~ w}_{K}} + \sum K^{'} \subset {~ w}_{K} ζ_{K}^{'}, \forall K \in T_{h} .

(32)

Where ${~ w}_{K}$ is a finite union of neighbouring elements of $K$ .

Proof.

To establish the lower error bound (32), we will make extensive use of the original system of equations given by (1) and (3), which is recovered from the mixed formulation (6) by choosing suitable test functions and integrating by parts backwardly the corresponding equations. Thereby, we bound each term of the residual separately.

Residual element $(f_{h} - A : D^{2} u_{h}) :$

Let us define by $v_{K} = (f_{h} - A : D^{2} u_{h}) b_{K}$ where $b_{K}$ is the bubble fonction define in the Section 3.1.3. We have

$(f_{h} - A : D^{2} u_{h}, v_{K})_{K} = \int_{K} (f_{h} - A : D^{2} u_{h}) . v_{K}$ (33)

Introduce $f$ and use the modified continuous formulation (3) to get:

$(f_{h} - A : D^{2} u_{h}, v_{K})_{K} = \int_{K} (f - f_{h}) . v_{K} - \int_{K} (A : D^{2} u_{h}) . v_{K} + \int_{K} (det σ) . v_{K}$

$= \int_{K} (f - f_{h}) . v_{K} + \int_{K} ((A : D^{2} u) - \int_{K} (A : D^{2} u_{h})) . v_{K}$

$= \int_{K} (f - f_{h}) . v_{K} + \int_{K} A : ((D^{2} u) - (D^{2} u_{h})) . v_{K}$

Using Cauchy-Schwarz inequality we obtain

$∥ (f_{h} - A : D^{2} u_{h}) v_{K}^{\frac{1}{2}} ∥ \leq C ((∥ f - f_{h} ∥_{K} ∥ v_{K} ∥ + ∥ e_{u} ∥_{2, K} ∥ v_{K} ∥_{K}) .$ (34)

Using an inverses inequalities we have

$∥ f_{h} - A : D^{2} u_{h} ∥_{K} ≲ ∥ e_{u} ∥_{K} + ζ_{K} .$ (35)

Thus,

$∥ f_{h} - A : D^{2} u_{h} ∥_{K}$ $≲$ $∥ (e_{σ}, e_{u}, e_{λ}) ∥_{h, {~ w}_{K}} + ζ_{K} .$ (36)
Residual element $(σ_{h} - D^{2} u_{h}) :$ Let $K \in T_{h}$ . We have

$σ_{h} - D^{2} u_{h}$ $=$ $(σ_{h} - σ) + (σ - D^{2} u_{h})$

$=$ $σ_{h} - σ + (D^{2} u - D^{2} u_{h})$

$=$ $e_{σ} + (D^{2} e_{u})$

The triangular inequality leads to

$∥ σ_{h} - D^{2} u_{h} ∥_{K}$ $≲$ $∥ e_{σ} ∥_{K} + ∥ e_{u} ∥_{2, K}$

Hence,

$∥ σ_{h} - D^{2} u_{h} ∥_{K}$ $≲$ $∥ (e_{σ}, e_{u}, e_{λ}) ∥_{h, {~ w}_{K}} + ζ_{K} .$ (37)
Residual element $(f_{h} + det σ_{h} - A : D^{2} u_{h})$ : From the residu equation of differential form (17), we obtain for $W = (τ, v, μ)$ :

$(f - det σ_{h} - A : D^{2} u_{h}, v) = ⟨ D_{U} B (U - U_{h}), W ⟩ - R (W)$

and we deduce for $W = (0, v, 0)$ :

$(f_{h} - det σ_{h} - A : D^{2}$

Residual-based a posteriori error estimates for a conforming mixed finite element discretization of the Monge-Ampère equation

Authors

A posteriori error analysis for a new fully-mixed isotropic discretization to the stationary Stokes-Darcy coupled problem

Residual-based a posteriori error estimates of mixed methods in Biot's consolidation model

A posteriori error estimates for the Brinkman-Darcy-Forchheimer problem

Quantifying discretization errors for soft-tissue simulation in computer assisted surgery: a preliminary study

A posteriori error analysis for a Lagrange multiplier method for a Stokes/Biot fluid-poroelastic structure interaction model

Lower a posteriori error estimates on anisotropic meshes

Adaptive C^0 interior penalty methods for Hamilton-Jacobi-Bellman equations with Cordes coefficients

1. General introduction

2. Preliminaries and Notation

2.1. Model

2.2. Modified continuous mixed weak formulation

Lemma 2.1.

Remark 2.1.

2.3. Modified discrete formulation

Lemma 2.2.

Theorem 2.1.

3. A-posteriori error analysis

3.1. Residual Error Estimators

Lemma 3.1.

Lemma 3.2.

Proof.

Lemma 3.3.

Lemma 3.4.

Proof.

3.1.1. A posteriori error indicators

Definition 3.1 (A-posteriori error indicators).

3.1.2. Reliability of $Θ$

Theorem 3.1 (Reliability of the a posteriori error estimator).

Proof.

3.1.3. Efficiency of $Θ$

Lemma 3.5.

Theorem 3.2.

Proof.

	$(f_{h} - A : D^{2} u_{h}, v_{K})_{K} = \int_{K} (f - f_{h}) . v_{K} - \int_{K} (A : D^{2} u_{h}) . v_{K} + \int_{K} (det σ) . v_{K}$
	$= \int_{K} (f - f_{h}) . v_{K} + \int_{K} ((A : D^{2} u) - \int_{K} (A : D^{2} u_{h})) . v_{K}$
	$= \int_{K} (f - f_{h}) . v_{K} + \int_{K} A : ((D^{2} u) - (D^{2} u_{h})) . v_{K}$

$σ_{h} - D^{2} u_{h}$	$=$	$(σ_{h} - σ) + (σ - D^{2} u_{h})$
	$=$	$σ_{h} - σ + (D^{2} u - D^{2} u_{h})$
	$=$	$e_{σ} + (D^{2} e_{u})$

Residual-based a posteriori error estimates for a conforming mixed finite element discretization of the Monge-Ampère equation

Authors

A posteriori error analysis for a new fully-mixed isotropic discretization to the stationary Stokes-Darcy coupled problem

Residual-based a posteriori error estimates of mixed methods in Biot's consolidation model

A posteriori error estimates for the Brinkman-Darcy-Forchheimer problem

Quantifying discretization errors for soft-tissue simulation in computer assisted surgery: a preliminary study

A posteriori error analysis for a Lagrange multiplier method for a Stokes/Biot fluid-poroelastic structure interaction model

Lower a posteriori error estimates on anisotropic meshes

Adaptive C^0 interior penalty methods for Hamilton-Jacobi-Bellman equations with Cordes coefficients

This week in AI

1. General introduction

2. Preliminaries and Notation

2.1. Model

2.2. Modified continuous mixed weak formulation

Lemma 2.1.

Remark 2.1.

2.3. Modified discrete formulation

Lemma 2.2.

Theorem 2.1.

3. A-posteriori error analysis

3.1. Residual Error Estimators

Lemma 3.1.

Lemma 3.2.

Proof.

Lemma 3.3.

Lemma 3.4.

Proof.

3.1.1. A posteriori error indicators

Definition 3.1 (A-posteriori error indicators).

3.1.2. Reliability of Θ

Theorem 3.1 (Reliability of the a posteriori error estimator).

Proof.

3.1.3. Efficiency of Θ

Lemma 3.5.

Theorem 3.2.

Proof.

3.1.2. Reliability of $Θ$

3.1.3. Efficiency of $Θ$