Erratum to: Quadratic mixed finite element approximations of the Monge-Ampère equation in 2D

1 Erratum to: Calcolo (2015) 52:503–518 DOI 10.1007/s10092-014-0127-7

Abstract The proof of Lemma 10 in [Awanou, G.: Quadratic mixed finite element approximations of the Monge-Ampère equation in 2D. Calcolo 52(4), 503–518 (2015)] is not correct. The purpose of this erratum is to give a correct proof of the main result therein under the assumption of elliptic regularity.

2 Introduction

In [1, Lemma 10], we claimed a strict contraction property of a mapping $T_1$ in the $H^1$ seminorm. Unfortunately there was a mistake at the end of the proof of the lemma. It was stated that ”Since $\gamma < 1$, and $\alpha =h^{k+2}$, for h sufficiently small, $C h + C \alpha h ||{\text {cof}}\,Q||_{H^{k+1}(\mathcal {T}_h)} +C \alpha < 1-\gamma$”. However $\gamma$ also depends on h, see [1, p. 6]. Moreover $1 -\gamma \rightarrow 0$ at a rate higher than $h^{}$, and thus the argument as stated is not correct. As a consequence, the strategy which consists in rescaling the equation does not work.

In this erratum, using the same notation as in [1], we give a proof of the main result therein under the assumption of $W^{2,p}$ elliptic regularity. Our approach consists in adapting the proof in [3]. The main ingredient is a $W^{2,p}$ discrete elliptic regularity proved in [13].

The elliptic regularity assumption is known to hold if the domain is smooth. We refer to [14, Remark 3.2] and [13] for the formulation of the method with the weak imposition of the Dirichlet boundary condition using Nitsche’s method and the use of curvilinear coordinates near the boundary. The arguments given here can be extended to that setting.

On the other hand, $W^{2,p}$ elliptic regularity holds for the Poisson equation on a cube [17, Remark 9.1.1]. It is therefore reasonable to expect that one can prove a $W^{2,p}$ elliptic regularity result on cubes for second order equations in divergence form with smooth coefficients using an antisymmetric extension as in the proof of [17, Proposition 9.1.2]. We wish to address this issue, following the $W^{2,p}$ elliptic regularity approach in [11], in a separate work.

3 Preliminaries

We use the standard notation $W^{k,p}(\Omega )$ for the Sobolev spaces and the notation $| . |_{W^{k,p}}$ for its semi norm. We recall that $W_0^{1,p}(\Omega )$ is the subset of $W^{1,p}(\Omega )$ of elements with vanishing trace on $\partial \Omega$. We will need the following mesh dependent norm on $V_h$

$$\begin{aligned} ||v||_{\widetilde{W}^{2,p}(\mathcal {T}_h)}^p = || v||_{W^{2,p}(\mathcal {T}_h)}^p + h^{1-p} \sum _{K \in \mathcal {T}_h } || D v ||_{L^p(\partial K)}^p,\quad p \ge 2. \end{aligned}$$

We have by scaling

$$\begin{aligned} ||v||_{\widetilde{W}^{2,p}(\mathcal {T}_h)}&\le C || v||_{W^{2,p}(\mathcal {T}_h)},\quad \forall v \in V_h \end{aligned}$$

(2.1)

$$\begin{aligned} ||v||_{\widetilde{W}^{2,p}(\mathcal {T}_h)}&\le C h^{-1} ||v||_{W^{1,p}},\quad \forall v \in V_h . \end{aligned}$$

(2.2)

Moreover, there exists an interpolation operator $\tilde{I}_h$ such that for $m \in W^{k+1,p}(\Omega ) \cap W_0^{1,p}(\Omega )$, $\tilde{I}_h m \in V_h \cap W_0^{1,p}(\Omega )$ and

$$\begin{aligned} \Vert m-\tilde{I}_h m\Vert _{\widetilde{W}^{2,p}(\mathcal {T}_h) }&\le C h^{k-1} |m|_{W^{k+1,p}} \end{aligned}$$

(2.3)

$$\begin{aligned} \Vert m-\tilde{I}_h m\Vert _{W^{1,p}}&\le C h^{k} |m|_{W^{k+1,p}}. \end{aligned}$$

(2.4)

The proofs are essentially the same as the ones given for [4, Lemma 1], [4, Lemma 2] and [4, Lemma 4]. It is important to note that the constant in the above inequalities are independent of p. This follows from the fact that the constant in the Bramble-Hilbert lemma [5, (4.3.9)] is independent of p.

We recall the scale-trace inequality

$$\begin{aligned} \Vert v\Vert _{L^p(\partial K)} \le C h^{-\frac{1}{p}} \Vert v\Vert _{L^p(K)} \le C h^{-\frac{1}{2}} \Vert v\Vert _{L^p(K)},\quad p \ge 2, \end{aligned}$$

(2.5)

with a constant C independent of p.

We also recall that if w is in the Sobolev space $W^{l+1,p}(\Omega ), 1 \le p \le \infty$, $0 \le l \le d$

$$\begin{aligned} || w -I_h w ||_{W^{k,p}(\mathcal {T}_h)} \le C_{} h^{l+1-k} |w|_{W^{l+1,p}}, \end{aligned}$$

for $k=0, 1, 2$. The constant C is shown to be independent of p using [5, (4.4.5)] and shape regularity.

We will often use the inverse estimates

$$\begin{aligned} ||w_h||_{t, p,\mathcal {T}_h} \le C_{} h^{s-t+\min \left( 0,\frac{n}{p}-\frac{n}{q}\right) } ||w_h||_{s,q,\mathcal {T}_h}, \end{aligned}$$

(2.6)

for $0 \le s \le t, 1 \le p,q\le \infty$ and $w_h \in V_h$. As stated in [5], the constant C in (2.6) depends on p and q because the first step of the proof is to use a norm equivalence on the reference element. However, inspection of the proof of the equivalence of norms in a finite dimensional space reveals that the constant does not depend on q. Moreover, it only depends on p through the $W^{t,p}$ norm of the basis functions of the finite dimensional space on the reference element. The latter are bounded by a scalar multiple of their $W^{t,\infty }$ norm. We conclude that the constant C in (2.6) can be chosen independent of p.

Next, let $\phi$ be the solution of

$$\begin{aligned} - {\text {div}}\,\big ( ({\text {cof}}\,D^2 u) D \phi \big ) = r \text { in } \Omega ,\quad \phi = 0 \,\, \text { on } \,\, \partial \Omega . \end{aligned}$$

(2.7)

We make the following assumption

Assumption 2.1

For $r \in L^p(\Omega ), p \ge 2$, the weak solution $\phi$ of (2.7) is in $W^{2,p}(\Omega )$ and

$$\begin{aligned} \Vert \phi \Vert _{W^{2,p}} \le C p \Vert r\Vert _{L^p}. \end{aligned}$$

(2.8)

The result is known to hold for smooth domains, c.f. [14] and the references therein. As suggested in [16, (1.7)] the linear dependence in p of the constant in (2.8) follows by tracing constants in the proof given in [11]. Once can trace constants in the proof of [11, Theorem 9.14] and use the maximum principle [11, Theorem 9.1]. See also [7].

As pointed out in the introduction, it is reasonable to expect that the result also holds for cubes.

We will refer to the result of the following lemma as discrete elliptic regularity. The result is given as [13, Lemma 4.1]. For the convenience of the reader, we give the proof.

Let $P_h: W_0^{1,p}(\Omega ) \rightarrow V_h \cap W_0^{1,p}(\Omega )$ be the projection defined by

$$\begin{aligned} \int _{\Omega } [({\text {cof}}\,D^2 u) D P_h v ] \cdot D w \,dx = \int _{\Omega } [({\text {cof}}\,D^2 u) D v ] \cdot D w \,dx,\quad \forall w \in V_h \cap W_0^{1,p}(\Omega ). \end{aligned}$$

We have the approximation property

$$\begin{aligned} || w -P_h w ||_{ W^{1,p} } \le C_{} h^{k} |w|_{W^{k+1,p}}. \end{aligned}$$

(2.9)

The result is a consequence of the stability of the Ritz projection, [15] and [12, Corollary 5.6]. Since $\Vert P_h w\Vert _{W^{1,p}} \le C \Vert w\Vert _{W^{1,p}}$ for $w \in W_0^{1,p}(\Omega )$, using $P_h I_h w=I_h w$, we obtain

$$\begin{aligned} || w -P_h w ||_{ W^{1,p} }&\le || w -I_h w ||_{ W^{1,p} } + || I_h w - P_h w ||_{ W^{1,p} } \\&= || w -I_h w ||_{ W^{1,p} } + || P_h I_h w - P_h w ||_{ W^{1,p} } \\&\le C_{} h^{k} |w|_{W^{k+1,p}} + || I_h w - w ||_{ W^{1,p} } \\&\le C_{} h^{k} |w|_{W^{k+1,p}}, \end{aligned}$$

which proves (2.9).

The independence of the constant C in p may be traced through the proof given in [12]. Alternatively, the independence of the constant C in p can be obtained through an interpolation argument we outline.

Let $1<p<\infty$ and $1\le q \le \infty$. Using a notation similar to the one used in [9], we denote by $W^{k,p,q}(\Omega )$ the interpolation space $(W^{k,1}(\Omega ),W^{k,\infty }(\Omega ))_{1-1/p,q,K}$ between $W^{k,1}(\Omega )$ and $W^{k,\infty }(\Omega )$ as defined in [6, Definition 3.2.4]. The letter K, and also $K'$ to be used below, refers to the function norm [6, (3.2.9)]. We note that it is assumed in [9] that the domain $\Omega$ is a minimally smooth domain, also known as a Lipschitz domain.

By [6, Theorem 3.2.23], $W^{k,p,q}(\Omega )$ is an exact interpolation space of order $\theta =1-1/p$ as defined in [6, Definition 3.2.22].

Moreover, by [6, Corollary 3.2.13 (a)], $W^{k,2,2}(\Omega ) \subset W^{k,2,\infty }(\Omega )$. Thus since $W^{k,2,2}(\Omega )$ is of order $\theta _1=1/2$ and $W^{k,\infty ,\infty }(\Omega )$ is of order $\theta _2=1$, by [6, Proposition 3.2.16 (a)] and the reiteration theorem [6, Theorem 3.2.20],

$$\begin{aligned} W^{k,p,q}(\Omega ) = (W^{k,2,2}(\Omega ), W^{k,\infty ,\infty }(\Omega ))_{1-2/p,q,K'},\quad 1 \le q \le \infty . \end{aligned}$$

On the other hand, it is shown in [9, p. 595] that $W^{k,p}(\Omega ) = W^{k,p,p}(\Omega )$ with equivalent norms. It can be seen from [9, Theorem 1] that the constants in the norm equivalence are independent of p. We conclude that $W^{k,p/2}(\Omega )$ is an exact interpolation space of order $1-2/p$ between $W^{k,2}(\Omega )$ and $W^{k,\infty }(\Omega )$. By [6, Definition 3.2.22], this means that since $P_h$ is a bounded linear map from $W^{k,2}(\Omega )$ into itself with norm $M_1$, and also a bounded linear map from $W^{k,\infty }(\Omega )$ into itself with norm $M_2$, then $P_h$ is a bounded linear map from $W^{k,p/2}(\Omega )$ into itself and its norm is bounded by $M_1^{1-2/p} M_2^{2/p}$ ,which is easily seen to be bounded above by a constant independent of p.

Lemma 2.2

Assume that Assumption 2.1 of elliptic regularity holds. Let $r \in L^p(\Omega ), p \ge 2$ and let $v \in V_h \cap H_0^1(\Omega )$ solve

$$\begin{aligned} \int _{\Omega } [({\text {cof}}\,D^2 u) D v] \cdot D w \,dx = \int _{\Omega } r w \,dx, w \in V_h \cap H_0^1(\Omega ). \end{aligned}$$

(2.10)

Then

$$\begin{aligned} ||v||_{\widetilde{W}^{2,p}(\mathcal {T}_h)} \le C p || r ||_{L^p}. \end{aligned}$$

(2.11)

Proof

With these notation the solution v of (2.10) is given by $v=P_h \phi$. Let $w \in V_h \cap H_0^1(\Omega )$. We have by (2.2) and (2.3)

$$\begin{aligned} ||v||_{\widetilde{W}^{2,p}(\mathcal {T}_h)}&= ||P_h \phi ||_{\widetilde{W}^{2,p}(\mathcal {T}_h)} \le ||P_h \phi - \phi ||_{\widetilde{W}^{2,p}(\mathcal {T}_h)} + || \phi ||_{\widetilde{W}^{2,p}(\mathcal {T}_h)} \\&\le || \phi - \tilde{I}_h \phi ||_{\widetilde{W}^{2,p}(\mathcal {T}_h)} + || \tilde{I}_h \phi - P_h \phi ||_{\widetilde{W}^{2,p}(\mathcal {T}_h)} + || \phi ||_{\widetilde{W}^{2,p}(\mathcal {T}_h)} \\&\le C \Vert \phi \Vert _{W^{2,p}}+ C h^{-1} || \tilde{I}_h \phi - P_h \phi ||_{W^{1,p}} + || \phi ||_{\widetilde{W}^{2,p}(\mathcal {T}_h)}. \end{aligned}$$

By (2.9) and (2.4)

$$\begin{aligned} || \tilde{I}_h \phi - P_h \phi ||_{W^{1,p}} \le || \tilde{I}_h \phi - \phi ||_{W^{1,p}} + || \phi - P_h \phi ||_{W^{1,p}} \le C h \Vert \phi \Vert _{W^{2,p}}. \end{aligned}$$

Using (2.1) we conclude by elliptic regularity that

$$\begin{aligned} ||v||_{\widetilde{W}^{2,p}(\mathcal {T}_h)} \le C \Vert \phi \Vert _{W^{2,p}} \le C p || r ||_{L^p}. \end{aligned}$$

This proves (2.11). $\square$

Lemma 2.3

Let $r \in V_h$. Then for $p \ge 2$

$$\begin{aligned} \Vert r\Vert _{L^p} \le C {\mathop {\mathop {\sup }\limits _{z \ne 0}}\limits _{z \in V_h}} \frac{|(r,z)|}{\Vert z\Vert _{L^q}} \qquad \frac{1}{p} + \frac{1}{q}=1. \end{aligned}$$

Proof

We have

$$\begin{aligned} \Vert r\Vert _{L^p} = \sup \limits _{\begin{array}{c} w \ne 0 \\ w \in L^q \end{array}} \frac{|(r,w)|}{\Vert w\Vert _{L^q}}. \end{aligned}$$

Let $P_{V_h}$ be the $L^2$ projection into $V_h$. The projection is known to be stable in $L^q$ [10], i.e. for $w \in L^q(\Omega )$

$$\begin{aligned} \Vert P_{V_h} w\Vert _{L^q} \le C^{\theta } \Vert w\Vert _{L^q}, \theta =\bigg | 1-\frac{2}{q}\bigg |. \end{aligned}$$

Since $p \ge 2$, $-1<1-2/q\le 2$ and hence the constant $C^{\theta }$ is bounded uniformly in q. Since $r \in V_h, (r,w) = (r, P_{V_h} w)$ and therefore

$$\begin{aligned} \frac{|(r,w)|}{\Vert w\Vert _{L^q}} \le C \frac{| (r, P_{V_h} w) |}{\Vert P_{V_h} w\Vert _{L^q} } \le C \sup \limits _{\begin{array}{c} z \ne 0 \\ z \in V_h \end{array}} \frac{|(r,z)|}{\Vert z\Vert _{L^q}}. \end{aligned}$$

This concludes the proof. $\square$

4 Error analysis of the mixed method with the elliptic regularity assumption

For this erratum the mapping $T: V_h\times \Sigma _h\rightarrow V_h\times \Sigma _h$ is defined by

$$\begin{aligned} T(w_h, \eta _h)=(T_1(w_h, \eta _h), T_2(w_h, \eta _h)), \end{aligned}$$

where $T_1(w_h, \eta _h)$ and $T_2(w_h, \eta _h)$ satisfy

$$\begin{aligned}&(\eta _h-T_2(w_h, \eta _h), \tau ) +({\text {div}}\,\tau , D(w_h-T_1(w_h, \eta _h)))\nonumber \\&\quad \qquad \qquad \qquad -\langle D(w_h-T_1(w_h, \eta _h)), \tau n\rangle =(\eta _h, \tau ) \nonumber \\&\quad \qquad \qquad \qquad +({\text {div}}\,\tau , Dw_h)-\langle Dw_h, \tau n\rangle , \quad \forall \ \tau \in \Sigma _h \end{aligned}$$

(3.1)

$$\begin{aligned}&(({\text {cof}}\,D^2u)D(w_h-T_1(w_h, \eta _h)) , D v) =(f, v)-(\det \eta _h, v), \ \forall \ v\in V_h\cap H^1_0(\Omega ) \end{aligned}$$

(3.2)

$$\begin{aligned}&w_h-T_1(w_h, \eta _h)=0 \quad \text {on}\quad \partial \Omega . \end{aligned}$$

(3.3)

It is shown in [3, Lemma 3.4] that a fixed point of (3.1)–(3.3) with $w_h=g_h$ on $\partial \Omega$ solves the nonlinear problem [1, (3)].

For this erratum we define

$$\begin{aligned} \bar{B}_h(\rho )=\{(w_h, \eta _h) \in V_h\times \Sigma _h,\ \Vert w_h-I_hu\Vert _{W^{2,\infty }(\mathcal {T}_h) }\le \rho ,\ \Vert \eta _h-I_h\sigma \Vert _{L^{\infty }}\le \rho \}. \end{aligned}$$

Recall that $B_h(\rho ) = \bar{B}_h(\rho ) \cap Z_h$ with $Z_h$ defined on [1, p. 7].

Lemma 3.1

For a positive constant $C_0$ and $\rho = C_0 h^{k-1}$, we have $B_h(\rho ) \ne \emptyset$.

Proof

It is shown in [3, Lemma 3.5] that there exists $\eta _h \in \Sigma _h$ such that $(I_h u,\eta _h) \in Z_h$. We estimate $\Vert \eta _h-I_h\sigma \Vert _{L^{\infty }}$. We have

$$\begin{aligned} (\eta _h-I_h\sigma , \tau )=(\sigma -I_h\sigma , \tau )-({\text {div}}\,\tau , D(I_hu-u))+\langle D(I_hu-u), \tau n\rangle . \end{aligned}$$

Let $p >1$ and q such that $1/p + 1/q=1$. We have by Lemma 2.3

$$\begin{aligned} \Vert \eta _h-I_h\sigma \Vert _{L^p}&\le C \sup \limits _{\begin{array}{c} \tau \ne 0 \\ \tau \in \Sigma _h \end{array}} \frac{|(\eta _h-I_h\sigma , \tau )|}{\Vert \tau \Vert _{L^q}}. \end{aligned}$$

By Cauchy-Schwarz inequality, the scale-trace inequality (2.5), inverse estimates and approximation properties of $I_h$

$$\begin{aligned} |(\eta _h-I_h\sigma , \tau )|&\le \Vert \sigma -I_h\sigma \Vert _{L^p} \Vert \tau \Vert _{L^q} + \Vert D(I_hu-u) \Vert _{L^p} \Vert {\text {div}}\,\tau \Vert _{L^q} \\&\quad + C \Vert D(I_hu-u) \Vert _{L^p(\partial \Omega )} \Vert \tau \Vert _{L^q(\partial \Omega )} \\&\le \Vert \sigma -I_h\sigma \Vert _{L^{\infty }} \Vert \tau \Vert _{L^q} + C h^{-1} \Vert D(I_hu-u) \Vert _{L^{\infty }} \Vert \tau \Vert _{L^q} \\&\quad + C h^{-\frac{1}{2}}\Vert D(I_hu-u) \Vert _{L^{\infty }} \Vert \tau \Vert _{ L^{q} } \\&\le (C h^{k+1} + C h^{k-1}) \Vert \tau \Vert _{L^q} \le C h^{k-1} \Vert \tau \Vert _{L^q}. \end{aligned}$$

We conclude that $\Vert \eta _h-I_h\sigma \Vert _{L^p} \le C h^{k-1}$. By an inverse estimate

$$\begin{aligned} \Vert \eta _h-I_h\sigma \Vert _{L^{\infty }} \le C h^{-\frac{2}{p}} \Vert \eta _h-I_h\sigma \Vert _{L^p} \le C h^{-\frac{2}{p}} h^{k-1}. \end{aligned}$$

Choosing p such that $| \ln h| \le p \le 2 |\ln h|$, we obtain $\Vert \eta _h-I_h\sigma \Vert _{L^{\infty }} \le C h^{k-1}$. This concludes the proof. $\square$

Lemma 3.2

The mapping T does not move the center ($I_hu, I_h\sigma$) of the ball $\bar{B}_h(\rho )$ too far, i.e. for h sufficiently small

$$\begin{aligned} \Vert I_h u - T_1(I_h u, I_h \sigma )\Vert _{W^{2,\infty }(\mathcal {T}_h) }&\le C_1 h^{k} \end{aligned}$$

(3.4)

$$\begin{aligned} \Vert I_h\sigma -T_2(I_hu, I_h\sigma ) \Vert _{L^{\infty }}&\le C_2 h^{k-1}. \end{aligned}$$

(3.5)

Proof

By [1, Lemma 2.1], on each element K

$$\begin{aligned} \Vert \det (I_h\sigma )-\det \sigma \Vert _{L^{\infty }(K)} \le C \Vert \frac{1}{2} I_h\sigma +\frac{1}{2} \sigma \Vert _{L^{\infty }(K)} \Vert I_h\sigma -\sigma \Vert _{L^{\infty }(K)}. \end{aligned}$$

By approximation properties $\Vert I_h\sigma -\sigma \Vert _{L^{\infty }(K)} \le C h^{k+1}$, so $\Vert I_h\sigma \Vert _{L^{\infty }}\le C\Vert \sigma \Vert _{L^{\infty }}$, and

$$\begin{aligned} \Vert \det (I_h\sigma )-\det \sigma \Vert _{L^{\infty }(K)} \le C \Vert I_h\sigma -\sigma \Vert _{L^{\infty }(K)} \le C h^{k+1}. \end{aligned}$$

(3.6)

By (3.2), (3.3), discrete elliptic regularity and (3.6)

$$\begin{aligned} \Vert I_h u - T_1(I_h u, I_h \sigma )\Vert _{\widetilde{W}^{2,p}(\mathcal {T}_h) }&\le C p \Vert \det I_h \sigma -f \Vert _{L^p} = C \Vert \det I_h \sigma - \det D^2 u \Vert _{L^p} \\&\le C p \Vert \det I_h \sigma - \det D^2 u \Vert _{L^{\infty }} \le C p \, h^{k+1}. \end{aligned}$$

Choosing p such that $| \ln h| \le p \le 2 |\ln h|$, we obtain by an inverse estimate

$$\begin{aligned} \Vert I_h u - T_1(I_h u, I_h \sigma )\Vert _{W^{2,\infty }(\mathcal {T}_h) }&\le C h^{-\frac{2}{p}} \Vert I_h u - T_1(I_h u, I_h \sigma )\Vert _{W^{2,p}(\mathcal {T}_h) } \\&\le C h^{-\frac{2}{p}} \Vert I_h u - T_1(I_h u, I_h \sigma )\Vert _{\widetilde{W}^{2,p}(\mathcal {T}_h) } \\&\le C\Vert I_h u - T_1(I_h u, I_h \sigma )\Vert _{\widetilde{W}^{2,p}(\mathcal {T}_h) } \le C h^{k+1} |\ln h|. \end{aligned}$$

We conclude that (3.4) holds.

Let $p >1$ and q such that $1/p + 1/q=1$. We have by Lemma 2.3

$$\begin{aligned} \Vert I_h\sigma -T_2(I_hu, I_h\sigma ) \Vert _{L^{p}} \le C \sup \limits _{\begin{array}{c} \tau \ne 0 \\ \tau \in \Sigma _h \end{array}} |( I_h\sigma -T_2(I_hu, I_h\sigma ),\tau )| / \Vert \tau \Vert _{L^q}. \end{aligned}$$

(3.7)

Moreover by (3.1) and using

$$\begin{aligned} (\sigma , \tau )+({\text {div}}\,\tau , Du)-\langle Du, \tau n\rangle =0, \quad \forall \ \tau \in H^1(\Omega ), \end{aligned}$$

we get

$$\begin{aligned} ( I_h\sigma -T_2(I_hu, I_h\sigma ),\tau )&= -({\text {div}}\,\tau , D(I_h u-T_1(I_h u, I_h\sigma )))\nonumber \\&\quad + \langle D(I_h u-T_1(I_h u, I_h\sigma )), \tau n\rangle \nonumber \\&\quad + (I_h \sigma -\sigma , \tau ) +({\text {div}}\,\tau , D(I_h u - u))\nonumber \\&\quad -\langle D (I_h u - u), \tau n\rangle . \end{aligned}$$

(3.8)

By Cauchy-Schwarz inequality, an inverse estimate, the trace-inverse inequality and approximation properties, we have

$$\begin{aligned}&\big | (I_h \sigma -\sigma , \tau ) +({\text {div}}\,\tau , D(I_h u - u)) -\langle D (I_h u - u), \tau n\rangle \big | \le (C h^{k+1} \Vert \sigma \Vert _{W^{k+1,\infty }} \nonumber \\&\quad + C h^{k-1} \Vert u \Vert _{W^{k+1,\infty }} + C h^{k-1} \Vert u \Vert _{W^{k+1,\infty }} ) ||\tau ||_{L^q} \le C h^{k-1} \Vert u \Vert _{W^{k+3,\infty }} ||\tau ||_{L^q}. \nonumber \\ \end{aligned}$$

(3.9)

Moreover

$$\begin{aligned}&-({\text {div}}\,\tau , D(I_h u-T_1(I_h u, I_h\sigma ))) + \langle D(I_h u-T_1(I_h u, I_h\sigma )), \tau n\rangle \\&\qquad = \sum _{K \in \mathcal {T}_h } (\tau , D^2(I_h u-T_1(I_h u, I_h\sigma )))_K - \sum _{K \in \mathcal {T}_h } \langle D(I_h u-T_1(I_h u, I_h\sigma )), \tau n\rangle _{\partial K} \\&\qquad \quad + \langle D(I_h u-T_1(I_h u, I_h\sigma )), \tau n\rangle . \end{aligned}$$

But by Cauchy-Schwarz inequality and the trace-inverse inequality

$$\begin{aligned}&\big | - \sum _{K \in \mathcal {T}_h } \langle D(I_h u-T_1(I_h u, I_h\sigma )), \tau n\rangle _{\partial K} + \langle D(I_h u-T_1(I_h u, I_h\sigma )), \tau n\rangle \big | \\&\quad \le \sum _{K \in \mathcal {T}_h } \big | \langle D(I_h u-T_1(I_h u, I_h\sigma )), \tau n\rangle _{\partial K} \big | \\&\quad \le \sum _{K \in \mathcal {T}_h } \Vert h^{-\frac{1}{q}}D(I_h u-T_1(I_h u, I_h\sigma ))\Vert _{L^p(\partial K)} \Vert h^{\frac{1}{q}} \tau n\Vert _{L^q(\partial K)} \\&\quad \le C \bigg ( \sum _{K \in \mathcal {T}_h } h^{-\frac{p}{q}} \Vert D(I_h u-T_1(I_h u, I_h\sigma )) \Vert ^p_{L^p(\partial K)} \bigg )^{\frac{1}{p}} ||\tau ||_{L^q}. \end{aligned}$$

Therefore by Cauchy-Schwarz inequality

$$\begin{aligned}&\big | -({\text {div}}\,\tau , D(I_h u-T_1(I_h u, I_h\sigma ))) + \langle D(I_h u-T_1(I_h u, I_h\sigma )), \tau n\rangle \big | \nonumber \\&\quad \le C \left( \Vert I_h u-T_1(I_h u, I_h\sigma )\Vert _{W^{2,p}(\mathcal {T}_h )}\right. \nonumber \\&\qquad \left. + \left( \sum _{K \in \mathcal {T}_h } h^{1-p} \Vert D(I_h u-T_1(I_h u, I_h\sigma )) \Vert ^p_{L^p(\partial K)} \right) ^{\frac{1}{p}} \right) ||\tau ||_{L^q}. \end{aligned}$$

(3.10)

We conclude from (3.7), (3.8), (3.9) and (3.10) that

$$\begin{aligned} | ( I_h\sigma -T_2(I_hu, I_h\sigma ),\tau )| \le (C h^{k-1} + C \Vert I_h u - T_1(I_h u, I_h \sigma )\Vert _{\widetilde{W}^{2,p}(\mathcal {T}_h) }) \Vert \tau \Vert _{L^q}. \end{aligned}$$

Thus using (2.1)

$$\begin{aligned} | ( I_h\sigma -T_2(I_hu, I_h\sigma ),\tau )|&\le (C h^{k-1} + C \Vert I_h u - T_1(I_h u, I_h \sigma )\Vert _{W^{2,p}(\mathcal {T}_h) }) \Vert \tau \Vert _{L^q} \\&\le (C h^{k-1} + C \Vert I_h u - T_1(I_h u, I_h \sigma )\Vert _{W^{2,\infty }(\mathcal {T}_h) }) \Vert \tau \Vert _{L^q}. \end{aligned}$$

Choosing p such that $| \ln h| \le p \le 2 |\ln h|$ and using (3.7), we obtain

$$\begin{aligned} \Vert I_h\sigma \!-\!T_2(I_hu, I_h\sigma ) \Vert _{L^{\infty }}&\!\le \! C h^{-\frac{2}{p}} \Vert I_h\sigma \!-\!T_2(I_hu, I_h\sigma ) \Vert _{L^{p}} \le C h^{-\frac{2}{p}} h^{k-1} \le C h^{k-1}. \end{aligned}$$

This concludes the proof. $\square$

Lemma 3.3

Let $\rho >0$ and $(w_1, \eta _1)$ and $(w_2, \eta _2)$ in $B_h(\rho )$. We have

$$\begin{aligned} ||T_2(w_1, \eta _1)-T_2(w_2, \eta _2) ||_{L^{\infty }} \le C_3 || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{W^{2,\infty }(\mathcal {T}_h )}, \end{aligned}$$

(3.11)

for a constant $C_3 \ge 1$.

Proof

For $(w_1, \eta _1)$ and $(w_2, \eta _2)$ in $B_h(\rho )$. We have using (3.1)

$$\begin{aligned} (T_2(w_1, \eta _1)-T_2(w_2, \eta _2),\tau )&=- ({\text {div}}\,\tau , D(T_1(w_1, \eta _1)-T_1(w_2, \eta _2))) \\&\quad + \langle D( T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ),\tau n\rangle \\&= \sum _{K \in \mathcal {T}_h} (\tau , D^2(T_1(w_1, \eta _1)-T_1(w_2, \eta _2) )_K \\&\quad - \sum _{K \in \mathcal {T}_h} \langle \tau n, D(T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ) \rangle _{\partial K} \\&\quad + \langle D( T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ),\tau n\rangle . \end{aligned}$$

Let $p \ge 2$ and q such that $1/p + 1/q=1$. We have

$$\begin{aligned} \bigg | \sum _{K \in \mathcal {T}_h} (\tau , D^2(T_1(w_1, \eta _1)-T_1(w_2, \eta _2) )_K \bigg |&\le C || T_1(w_1, \eta _1)\\&\quad -T_1(w_2, \eta _2) ||_{W^{2,p}(\mathcal {T}_h )} \Vert \tau \Vert _{L^q} \\&\le C || T_1(w_1, \eta _1)\\&\quad -T_1(w_2, \eta _2) ||_{W^{2,\infty }(\mathcal {T}_h )} \Vert \tau \Vert _{L^q}. \end{aligned}$$

Moreover by Cauchy-Schwarz and the scale-trace inequality (2.5)

$$\begin{aligned}&\bigg | - \sum _{K \in \mathcal {T}_h} \langle \tau n, D(T_1(w_1, \eta _1)\!-\!T_1(w_2, \eta _2) ) \rangle _{\partial K} \!+\! \langle D( T_1(w_1, \eta _1)\!-\!T_1(w_2, \eta _2) ),\tau n\rangle \bigg | \\&\quad \le \sum _{K \in \mathcal {T}_h} | \langle \tau n, D(T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ) \rangle _{\partial K}| \\&\quad = \sum _{K \in \mathcal {T}_h} | \langle h^{\frac{1}{q}} \tau n, h^{-\frac{1}{q}} D(T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ) \rangle _{\partial K}| \le C \Vert \tau \Vert _{L^q} \\&\qquad \times \bigg ( \sum _{K \in \mathcal {T}_h } h^{-\frac{p}{q}} \Vert D(T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ) \Vert ^p_{L^2(\partial K)} \bigg )^{\frac{1}{p}}. \end{aligned}$$

Since $-p/q = 1-p$ we obtain

$$\begin{aligned} | (T_2(w_1, \eta _1)-T_2(w_2, \eta _2),\tau ) | \le C || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{\widetilde{W}^{2,p}(\mathcal {T}_h )} \Vert \tau \Vert _{L^q}. \end{aligned}$$

And thus using (2.1)

$$\begin{aligned} | (T_2(w_1, \eta _1)-T_2(w_2, \eta _2),\tau ) |&\le C || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{W^{2,p}(\mathcal {T}_h )} \Vert \tau \Vert _{L^q} \\&\le C || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{W^{2,\infty }(\mathcal {T}_h )} \Vert \tau \Vert _{L^q}. \end{aligned}$$

We conclude that

$$\begin{aligned} \Vert I_h\sigma -T_2(I_hu, I_h\sigma ) \Vert _{L^{\infty }}&\le C h^{-\frac{2}{p}} \Vert I_h\sigma -T_2(I_hu, I_h\sigma ) \Vert _{L^{p}} \\&\le C h^{-\frac{2}{p}} \sup \limits _{\begin{array}{c} \tau \ne 0 \\ \tau \in \Sigma _h \end{array}} |( I_h\sigma -T_2(I_hu, I_h\sigma ),\tau )| / \Vert \tau \Vert _{L^q} \\&\le C h^{-\frac{2}{p}} || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{W^{2,\infty }(\mathcal {T}_h )} \\&\le C || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{W^{2,\infty }(\mathcal {T}_h )}, \end{aligned}$$

where we used Lemma 2.3 and choose p such that $| \ln h| \le p \le 2 |\ln h|$. This concludes the proof. $\square$

For $(w_h, \eta _h) \in Z_h$ we define

$$\begin{aligned} \Gamma = (({\text {cof}}\,D^2 u): \eta _h,v) + (({\text {cof}}\,D^2 u) D w_h , D v ). \end{aligned}$$

(3.12)

We have the following analogue of [3, Lemma 3.7]

Lemma 3.4

Let $(w_h, \eta _h) \in Z_h$. Then

$$\begin{aligned} \begin{aligned} | (({\text {cof}}\,D^2 u): \eta _h,v) + (({\text {cof}}\,D^2 u) D w_h , D v ) |&\le C h^{\frac{1}{q}} ||w_h||_{\widetilde{W}^{2,p}(\mathcal {T}_h )} ||v||_{L^q}, \end{aligned} \end{aligned}$$

(3.13)

for all $v \in V_h \cap H_0^1(\Omega )$ and $p \ge 2, 1/p+1/q=1$.

Proof

Denote by $P_{\Sigma _h}$ the $L^2$ projection into the space $\Sigma _h$. Put $A={\text {cof}}\,D^2 u$. It is proven in the proof of [3, Lemma 3.7] that for $v \in V_h \cap H_0^1(\Omega )$

$$\begin{aligned} \Gamma = - \sum _{K \in \mathcal {T}_h} ( {\text {div}}\,( P_{\Sigma _h} (v A) - vA ), D w_h)_K + \langle (P_{\Sigma _h} (v A) - vA) n , D w_h\rangle _{\partial \Omega }. \end{aligned}$$

We have

$$\begin{aligned} \Gamma&= \sum _{K \in \mathcal {T}_h} ( P_{\Sigma _h} (v A) - vA , D^2 w_h)_K - \sum _{K \in \mathcal {T}_h} \langle (P_{\Sigma _h} (v A) - vA) n , D w_h\rangle _{\partial K} \nonumber \\&\quad + \langle (P_{\Sigma _h} (v A) - vA) n , D w_h\rangle _{\partial \Omega }. \end{aligned}$$

(3.14)

By Cauchy-Schwarz inequality

$$\begin{aligned} \bigg | \sum _{K \in \mathcal {T}_h} ( P_{\Sigma _h} (v A) - vA , D^2 w_h)_K \bigg | \le \Vert P_{\Sigma _h} (v A) - vA \Vert _{L^q} \Vert w_h \Vert _{W^{2,p}(\mathcal {T}_h)}, \end{aligned}$$

(3.15)

and by Cauchy-Schwarz and the trace inequalities

$$\begin{aligned}&\bigg | - \sum _{K \in \mathcal {T}_h} \langle (P_{\Sigma _h} (v A) - vA) n , D w_h\rangle _{\partial K} + \langle (P_{\Sigma _h} (v A) - vA) n , D w_h\rangle _{\partial \Omega } \bigg | \nonumber \\&\quad \le \sum _{K \in \mathcal {T}_h} | \langle (P_{\Sigma _h} (v A) - vA) n , D w_h\rangle _{\partial K} | \nonumber \\&\quad = \sum _{K \in \mathcal {T}_h} | \langle h^{\frac{1}{q}} (P_{\Sigma _h} (v A) - vA) n , h^{-\frac{1}{q}} D w_h\rangle _{\partial K} | \nonumber \\&\quad \le C \sum _{K \in \mathcal {T}_h} \Vert h^{\frac{1}{q}} (P_{\Sigma _h} (v A) - vA) \Vert _{L^q(\partial K)} \Vert h^{-\frac{1}{q}} D w_h \Vert _{L^p(\partial K)} \nonumber \\&\quad \le C h^{\frac{1}{q} } \Vert P_{\Sigma _h} (v A) - vA \Vert _{W^{1,q}} \bigg ( \sum _{K \in \mathcal {T}_h } h^{1-p} \Vert D w_h \Vert ^p_{L^p(\partial K)} \bigg )^{\frac{1}{p}}. \end{aligned}$$

(3.16)

Arguing as in the proof of [14, Lemma 4.4] we have for $m=0,1$

$$\begin{aligned} ||P_{\Sigma _h} (v A) - vA||_{W^{m,q}}&\le C h^{1-m} ||v||_{L^q}. \end{aligned}$$

(3.17)

This follows from the stability in $L^q$ and $W^{1,q}$ of the $L^2$ projection [8], i.e. for $v \in W^{m,q}(\Omega )$, $v=0$ on $\partial \Omega$, $\Vert P_{\Sigma _h} (v A) \Vert _{W^{m,q}} \le C \Vert v A \Vert _{W^{m,q}}$.

As in the proof of Lemma 2.3, the constant in the $L^q$ stability of the $L^2$ projection is independent of q. For the $W^{1,q}$ stability, the independence in q of the constant is obtained by tracing constants in the proof of [8, Theorem 4 and Theorem 3]. More precisely, constants in the interpolation estimates and inverse estimates used therein are independent of q, c.f. Sect. 2. In addition, the constant $\alpha$ in [8] is equal to 1 for quasi uniform triangulations, making the constants in the estimates independent of q.

Since $P_{\Sigma _h} I_h (v A) = I_h (v A)$,

$$\begin{aligned} ||P_{\Sigma _h} (v A) - vA||_{W^{m,q}}&\le ||P_{\Sigma _h} (v A) - I_h (vA) ||_{W^{m,q}} + || I_h (vA) - v A ||_{W^{m,q}} \\&= ||P_{\Sigma _h} (v A) - P_{\Sigma _h} I_h (vA) ||_{W^{m,q}} + || I_h (vA) - v A ||_{W^{m,q}} \\&\le C \Vert I_h (vA) - v A \Vert _{W^{m,q}} \le C h^{k+1-m} ||v||_{W^{k+1,q}} \\&= C h^{k+1-m} ||v||_{W^{k,q}} \le C h^{1-m} ||v||_{ L^2 }, \end{aligned}$$

where in the last steps, we note that v is a piecewise polynomial of degree k and use an inverse estimate. It therefore follows from (3.14)–(3.16) that (3.13) holds. $\square$

The mapping $T_1$ has a fixed contraction property, i.e.

Lemma 3.5

For h sufficiently small, we have for $(w_1, \eta _1)$ and $(w_2, \eta _2)$ in $B_h(\rho )$

$$\begin{aligned} || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{W^{2,\infty }(\mathcal {T}_h )}&\le \frac{1}{4 C_3} || w_1-w_2 ||_{W^{2,\infty }(\mathcal {T}_h )} \nonumber \\&\quad + \left( \frac{1}{4 C_3} + C |\ln h| \rho \right) \Vert \eta _1-\eta _2 \Vert _{L^{\infty }}.\nonumber \\ \end{aligned}$$

(3.18)

Proof

The proof is a variant of [3, Lemma 3.10] and [3, Lemma 3.11]. Using (3.2) we have

$$\begin{aligned}&(({\text {cof}}\,D^2u) D (T_1(w_1, \eta _1)-T_1(w_2, \eta _2)),D v) = (({\text {cof}}\,D^2u)D (w_1-w_2), D v) \\&\quad + (\det \eta _1 - \det \eta _2,v) \!+\! (({\text {cof}}\,D^2u):(\eta _1-\eta _2),v) \! -\! (({\text {cof}}\,D^2u):(\eta _1-\eta _2),v), \end{aligned}$$

for all $v \in V_h$. Using the definition of $\Gamma$, (3.12) with $w_h=w_1-w_2, \eta _h=\eta _1-\eta _2$, and Lemma 3.4, we have

$$\begin{aligned} \begin{aligned} (({\text {cof}}\,D^2u) D (T_1(w_1, \eta _1)-T_1(w_2, \eta _2)),D v)&= - (({\text {cof}}\,D^2u):(\eta _1-\eta _2),v) \\&\quad + (\det \eta _1 - \det \eta _2,v) + \Gamma , \end{aligned} \end{aligned}$$

(3.19)

for all $v \in V_h$ with

$$\begin{aligned} | \Gamma | \le h^{\frac{1}{q}} ||w_h||_{\widetilde{W}^{2,p}(\mathcal {T}_h )} ||v||_{L^q}, \end{aligned}$$

(3.20)

with $p \ge 2, 1/p+1/q=1$.

By [1, Lemma 1], on each element K we have

$$\begin{aligned} \det \eta _1 - \det \eta _2 = {\text {cof}}\,\left( \frac{1}{2} \eta _1 +\frac{1}{2} \eta _2\right) :(\eta _1-\eta _2). \end{aligned}$$

Therefore on each element K

$$\begin{aligned} \begin{aligned}&({\text {cof}}\,D^2u):(\eta _1-\eta _2) - (\det \eta _1 - \det \eta _2) \\&\quad = \big ( ({\text {cof}}\,D^2u) - {\text {cof}}\,\big ( \frac{1}{2} \eta _1 +\frac{1}{2} \eta _2 \big ):(\eta _1-\eta _2) \\&\quad = {\text {cof}}\,\big ( D^2 u - \frac{1}{2} \eta _1 -\frac{1}{2} \eta _2 \big ):(\eta _1-\eta _2). \end{aligned} \end{aligned}$$

(3.21)

Let us define

$$\begin{aligned} A = \big ({\text {cof}}\,\sigma - \frac{1}{2} \eta _1 -\frac{1}{2} \eta _2 \big ):(\eta _1-\eta _2). \end{aligned}$$

We have

$$\begin{aligned} \sigma - \left( \frac{1}{2} \eta _1 +\frac{1}{2} \eta _2\right)&=\sigma - I_h \sigma + \frac{1}{2} I_h \sigma + \frac{1}{2} I_h \sigma - \left( \frac{1}{2} \eta _1+\frac{1}{2} \eta _2\right) \\&= \sigma - I_h \sigma + \frac{1}{2} (I_h \sigma - \eta _1) + \frac{1}{2} (I_h \sigma - \eta _2). \end{aligned}$$

We conclude that

$$\begin{aligned} \begin{aligned} ||\sigma - \left( \frac{1}{2} \eta _1 +\frac{1}{2} \eta _2\right) ||_{L^{\infty }(K)}&\le ||\sigma - I_h \sigma ||_{L^{\infty }(K)}\\&+ \frac{1}{2} ||I_h \sigma - \eta _1||_{L^{\infty }(K)}+\frac{1}{2} ||I_h \sigma - \eta _2||_{L^{\infty }(K)}\\&\le C h^{k+1} + C\rho . \end{aligned} \end{aligned}$$

It follows from (3.21) that

$$\begin{aligned} \Vert ({\text {cof}}\,D^2u):(\eta _1-\eta _2) - (\det \eta _1 - \det \eta _2) \Vert _{L^p} \le \big ( C h^{k+1} + C \rho \big ) \Vert \eta _1-\eta _2 \Vert _{L^{\infty }}. \end{aligned}$$

(3.22)

Let us define the linear form L on $V_h$ by

$$\begin{aligned} L(v) = (({\text {cof}}\,D^2u) D (T_1(w_1, \eta _1)-T_1(w_2, \eta _2)),D v). \end{aligned}$$

By the Riesz representation theorem, there exists $r \in V_h$ with $L(v) = (r,v)$ for all $v \in V_h$. Moreover by Lemma 2.3 $\Vert r\Vert _{L^p} \le C \sup \limits _{\begin{array}{c} v \ne 0 \\ v \in V_h \end{array}} |L(v)|/\Vert v\Vert _{L^q}$. We conclude from (3.19), (3.20) and (3.22) that

$$\begin{aligned} \Vert r\Vert _{L^p} \le C h^{\frac{1}{q}} || w_1-w_2 ||_{\widetilde{W}^{2,p}(\mathcal {T}_h )} + \big ( C h^{k+1} + C \rho \big ) \Vert \eta _1-\eta _2 \Vert _{L^{\infty }}. \end{aligned}$$

By discrete elliptic regularity and (2.1)

$$\begin{aligned} || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{\widetilde{W}^{2,p}(\mathcal {T}_h )}&\le C p h^{\frac{1}{q}} || w_1-w_2 ||_{\widetilde{W}^{2,p}(\mathcal {T}_h )} \\&\quad + \big ( C h^{k+1} + C \rho \big ) p \Vert \eta _1-\eta _2 \Vert _{L^{\infty }}\\&\le C p h^{\frac{1}{q}} || w_1-w_2 ||_{W^{2,\infty }(\mathcal {T}_h )} \\&\quad + \big ( C h^{k+1} + C \rho \big ) p \Vert \eta _1-\eta _2 \Vert _{L^{\infty }}. \end{aligned}$$

Since $p \ge 2$ and $0<h \le 1$, we have $h^{1/q} \le h^{1/2}$. Choosing p such that $| \ln h| \le p \le 2 |\ln h|$ we have $p h^{1/2} \le C |\ln h| h^{1/2} \le 1/(4 C_3)$ for h sufficiently small. Similarly $C h^{k+1} |\ln h| \le 1/(4 C_3)$ for h sufficiently small. We conclude using an inverse estimate that

$$\begin{aligned} || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{W^{2,\infty }(\mathcal {T}_h )}&\le C h^{-\frac{2}{p}} || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{W^{2,p}(\mathcal {T}_h )} \\&\le C h^{-\frac{2}{p}} || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{\widetilde{W}^{2,p}(\mathcal {T}_h )} \\&\le C || T_1(w_1, \eta _1)-T_1(w_2, \eta _2) ||_{\widetilde{W}^{2,p}(\mathcal {T}_h )} \\&\le \frac{1}{4 C_3} || w_1-w_2 ||_{W^{2,\infty }(\mathcal {T}_h )} \\&\quad + \left( \frac{1}{4 C_3} + C |\ln h| \rho \right) \Vert \eta _1-\eta _2 \Vert _{L^{\infty }}. \end{aligned}$$

This completes the proof. $\square$

Lemma 3.6

Let $\rho (h)=2C_4 h^{k-1}$ where $C_4=\max (C_0,C_1,2 C_2)$ with $C_0$ the constant in Lemma 3.1 and $C_1,C_2$ the constants from Lemma 3.2. Then T maps $B_h(\rho )$ into itself for h sufficiently small.

Proof

Let $(w_h, \eta _h) \in B_h(\rho )$. By definition, $||w_h-I_hu||_{W^{2,\infty }(\mathcal {T}_h )} \le \rho$ and $||\eta _h - I_h \sigma ||_{L^{\infty }} \le \rho$. By (3.18) and (3.4), for h sufficiently small

$$\begin{aligned} ||T_1(w_h,\eta _h)-I_h u||_{W^{2,\infty }(\mathcal {T}_h )}&\le ||T_1(w_h,\eta _h)-T_1(I_h u, I_h \sigma )||_{W^{2,\infty }(\mathcal {T}_h )}\\&\quad + ||T_1(I_h u, I_h \sigma )-I_h u||_{W^{2,\infty }(\mathcal {T}_h )} \\&\le \left( \frac{1}{4} + C |\ln h| h^{k-1}\right) ||\eta _h-I_h \sigma ||_{L^{\infty }}\\&\quad + \frac{1}{4} ||w_h- I_h u ||_{\widetilde{H}^2(\mathcal {T}_h )} + C_1 h^{k}\\&\le \frac{3 \rho }{4} + C_1 h^{k} = \frac{3 \rho }{4} + \frac{C_1 h}{2 C_5} \rho \le \rho . \end{aligned}$$

In addition, by (3.18), (3.11) and (3.5) and a similar argument we get

$$\begin{aligned}&||T_2(w_h,\eta _h)-I_h \sigma ||_{L^{\infty }}\\&\quad \le ||T_2(w_h,\eta _h)-T_2(I_h u, I_h \sigma )||_{L^{\infty }} + ||T_2(I_h u, I_h \sigma )-I_h \sigma ||_{L^{\infty }} \\&\quad \le C_3 ||T_1(w_h,\eta _h)-T_1(I_h u, I_h \sigma )||_{W^{2,\infty }(\mathcal {T}_h )} + ||T_2(I_h u, I_h \sigma )-I_h \sigma ||_{L^{\infty }} \\&\quad \le \frac{1}{4} ||\eta _h-I_h \sigma ||_{L^{\infty }} + C |\ln h| \rho ||\eta _h-I_h \sigma ||_{L^{\infty }} \\&\qquad + \frac{1}{4} ||w_h-I_h u||_{W^{2,\infty }(\mathcal {T}_h )} + C_2 h^{k-1}\\&\quad \le \rho , \end{aligned}$$

for h sufficiently small. By (3.1) $(T_1(w_h,\eta _h), T_2(w_h,\eta _h))$ is in the space $Z_h$. This concludes the proof. $\square$

We can now claim

Theorem 3.7

Let $(u, \sigma ) \in H^{k+3}(\Omega ) \times H^{k+1}(\Omega )^{d \times d}$ denotes the unique convex solution of [1, (1)] with $k \ge 2$. Then the problem [1, (3)] has a unique solution in $B_h(\rho ) \subset V_h \times \Sigma _h$ for h sufficiently small and with $\rho (h)$ given in Lemma 3.6.

Proof

The proof follows from the Brouwer fixed point theorem. For h sufficiently small and for $(w_1,\eta _1),(w_2,\eta _2) \in B_h(\rho )$, by (3.18) and (3.11)

$$\begin{aligned}&||T_1(w_1, \eta _1)-T_1(w_2, \eta _2)||_{W^{2,\infty }(\mathcal {T}_h)} + ||T_2(w_1, \eta _1)-T_2(w_2, \eta _2)||_{L^{\infty }} \\&\quad \le C ||T_1(w_1, \eta _1)-T_1(w_2, \eta _2)||_{W^{2,\infty }(\mathcal {T}_h)} \\&\quad \le C ||w_1-w_2 ||_{W^{2,\infty }(\mathcal {T}_h)} + C||\eta _1-\eta _2 ||_{L^{\infty }}. \end{aligned}$$

Hence the mapping T is continuous in $B_h(\rho )$. Since for h sufficiently small and the choice of $\rho (h)$, the continuous mapping T maps the closed ball $B_h(\rho )$ into itself, there exists a fixed point $(u_h,\sigma _h)$ in $B_h(\rho )$.

Assume that $(w_h^1,\eta _h^1)$ and $(w_h^2,\eta _h^2)$ are two fixed points of T. Then $T_1(w_h^1,\eta _h^1)=w_h^1$ and $T_1(w_h^2,\eta _h^2)=w_h^2$. By (3.18) we have

$$\begin{aligned} ||w_h^1 - w_h^2||_{W^{2,\infty }(\mathcal {T}_h)} \le \frac{1}{2 C_3} ||\eta _h^1 - \eta _h^2||_{L^{\infty }} + \frac{1}{4} ||w_h^1 - w_h^2||_{W^{2,\infty }(\mathcal {T}_h)}, \end{aligned}$$

and so

$$\begin{aligned} ||w_h^1 - w_h^2||_{W^{2,\infty }(\mathcal {T}_h)} \le \frac{2}{3 C_3} ||\eta _h^1 - \eta _h^2||_{L^{\infty }}. \end{aligned}$$

We also have $T_2(w_h^1,\eta _h^1)=\eta _h^1$ and $T_2(w_h^2,\eta _h^2)=\eta _h^2$. By (3.11)

$$\begin{aligned} ||\eta _h^1 - \eta _h^2||_{L^{\infty }} \le C_3 ||w_h^1 - w_h^2||_{W^{2,\infty }(\mathcal {T}_h)} \le \frac{2}{3} ||\eta _h^1 - \eta _h^2||_{L^2}. \end{aligned}$$

This implies $\eta _h^1 = \eta _h^2$ and so $w_h^1 = w_h^2$. This proves uniqueness. $\square$

The following error estimates hold

Theorem 3.8

Under the assumptions of Theorem 3.7, the solution $(u_h,\sigma _h)$ of (3.1)–(3.3) satisfies

$$\begin{aligned} \Vert u - u_h\Vert _{W^{2,\infty }(\mathcal {T}_h)}&\le C h^{k-1} \end{aligned}$$

(3.23)

$$\begin{aligned} ||\sigma -\sigma _h||_{L^{\infty } }&\le C h^{k-1} . \end{aligned}$$

(3.24)

Proof

By the definition of the ball $B_h(\rho )$, the existence of the solution $(u_h,\sigma _h)$ in $B_h(\rho )$ with $\rho = O(h^{k-1})$ given in Theorem 3.7, we have

$$\begin{aligned} ||I_h u-u_h||_{W^{2,\infty }(\mathcal {T}_h)}&\le C h^{k-1} \\ || I_h \sigma -\sigma _h||_{L^{\infty }}&\le C h^{k-1}. \end{aligned}$$

The estimates (3.23) and (3.24) then follow from triangular inequalities and standard interpolation inequalities. $\square$

Remark 3.9

Since it is now known that T has a fixed point $(u_h,\sigma _h)$ with $\Vert u_h - I_h \sigma \Vert _{L^{\infty }} \le C h^{k-1}$, it should be possible to derive a O$(h^k)$ error estimate in the $H^1$ norm for $u-u_h$ by using [3, Lemma 3.10] and [3, Lemma 3.11]. Note that in the proof of [3, Lemma 3.10] an inverse estimate, used to estimate $\Vert u_h - I_h \sigma \Vert _{L^{\infty }}$ from $\Vert u_h - I_h \sigma \Vert _{L^{2}}$ can now be avoided.