In this section, we study the hp finite element method and the backward Euler discretization approximation of convex optimal control problems governed by nonlinear parabolic equations. We shall take the state space \(W=L^{2}(0,T;Y)\) with \(Y=H^{1}_{0}(\Omega)\), the control space \(X=L^{2}(0,T;U)\) with \(U=L^{2}(\Omega_{U})\) and \(H=L^{2}(\Omega)\) to fix the idea. Let B be a linear continuous operator from X to \(L^{2}(0,T;Y)\). We are interested in the following nonlinear parabolic optimal control problems:
$$\begin{aligned}& \min_{u\in K} \biggl\{ \frac{1}{2} \int_{0}^{T}\bigl(\Vert y-y_{d}\Vert ^{2}_{L^{2}(\Omega)}+\Vert u\Vert ^{2}_{L^{2}(\Omega_{U})} \bigr)\,dt \biggr\} , \end{aligned}$$
(2.1)
$$\begin{aligned}& y_{t}-\operatorname {div}(A\nabla y)+ \phi(y)=f+Bu,\quad x\in\Omega, t \in(0,T], \end{aligned}$$
(2.2)
$$\begin{aligned}& y(x,t)=0,\quad x\in \partial\Omega, t\in(0,T], \end{aligned}$$
(2.3)
$$\begin{aligned}& y(x,0)=y_{0}(x), \quad x\in\Omega, \end{aligned}$$
(2.4)
where Ω and \(\Omega_{U}\) are bounded open sets in \(\mathbb{R}^{2}\) with a Lipschitz boundary ∂Ω and \(\partial\Omega_{U}\), K is a set defined by \(K= \{v\in X:\int_{0}^{T}\int_{\Omega_{U}}v \,dx \,dt\geq0 \}\), and \(f, y_{d}\in L^{2}(0,T;H)\), \(y_{0}(x)\in V=H_{0}^{1}(\Omega)\), and \(A(\cdot)=(a_{i,j}(\cdot))_{2\times 2}\in(C^{\infty}(\overline{\Omega}))^{2\times2}\), such that there is a constant \(c>0\) satisfying \(\xi^{t}A\xi\geq c\Vert \xi \Vert ^{2}\), \(\xi\in\mathbb{R}^{2}\). The function \(\phi(\cdot)\in W^{2,\infty}(-R,R)\) for any \(R>0\), \(\phi^{\prime}(y)\in L^{2}(\Omega)\) for any \(y\in H^{1}(\Omega)\), and \(\phi^{\prime}(y)\geq0\).
Let \(a(v,w)=\int_{\Omega }(A\nabla v)\cdot\nabla w \,dx\), \(\forall v,w\in V\), \((f_{1},f_{2})=\int_{\Omega}f_{1}f_{2}\,dx \), \(\forall f_{1},f_{2}\in H\), \((v,w)_{U}= \int_{\Omega_{U}}vw \,dx\), \(\forall v,w\in U\). It follows from the assumptions on A that there are constants c and \(C>0\) such that
$$\begin{aligned} a(v,v)\geq c\Vert v\Vert _{H^{1}(\Omega)}^{2}, \qquad \bigl\vert a(v,w) \bigr\vert \leq C\vert v\vert _{H^{1}(\Omega)}^{2}\vert w\vert _{H^{1}(\Omega)}^{2}, \quad \forall v,w\in Y. \end{aligned}$$
Then a weak formula of the convex nonlinear parabolic optimal control problems (2.1)-(2.4) reads
$$\begin{aligned} &\min_{u(t)\in K} \biggl\{ \frac{1}{2} \int_{0}^{T} \bigl(\Vert y-y_{d} \Vert ^{2}_{L^{2}(\Omega )}+\Vert u\Vert ^{2}_{L^{2}(\Omega_{U})} \bigr)\,dt \biggr\} , \end{aligned}$$
(2.5)
where \(y\in W\), \(u\in X\), \(u(t)\in K\), subject to
$$\begin{aligned}& (y_{t},w)+a(y,w)+\bigl(\phi(y),w\bigr)=(f+Bu,w),\quad \forall w\in Y, t \in(0,T], \end{aligned}$$
(2.6)
$$\begin{aligned}& y(x,0)=y_{0}(x), \quad x\in\Omega. \end{aligned}$$
(2.7)
It is well known (see, e.g., [21]) that the optimal control problem (2.5)-(2.7) has at least a solution \((y,u)\), and if that a pair \((y,u)\) is the solution of (2.5)-(2.7), then there is a co-state \(p\in W\) such that the triplet \((y,p,u)\) satisfies the following optimality conditions:
$$\begin{aligned}& (y_{t},w)+a(y,w)+\bigl(\phi(y),w\bigr)=(f+Bu,w),\quad \forall w\in Y, y(0)=y_{0}(x), \end{aligned}$$
(2.8)
$$\begin{aligned}& -(p_{t}, w)+a(q,p)+\bigl(\phi^{\prime}(y)p,q \bigr)=(y-y_{d},q),\quad \forall q\in Y, p(T)=0, \end{aligned}$$
(2.9)
$$\begin{aligned}& \int_{0}^{T}\bigl(u+B^{*}p,v-u \bigr)_{U}\,dt\geq0,\quad \forall v\in K, \end{aligned}$$
(2.10)
where \(B^{*}\) is the adjoint operators of B, and \((\cdot,\cdot)_{U}\) is the inner product of U, which will be simply written as \((\cdot,\cdot)\) in the rest of the paper when no confusion is caused.
Due to the special structure of the control constraint set K, we can derive a relationship between the control variable and the co-state variable of (2.8)-(2.10) in the following lemma.
Lemma 2.1
Let
\((y,p,u)\)
be the solution of (2.8)-(2.10). Then we have
$$\begin{aligned} u=\max \bigl\{ 0,\overline{B^{*}p} \bigr\} -B^{*}p, \end{aligned}$$
where
\(\overline{B^{*}p}=\frac{\int_{0}^{T}\int_{\Omega_{U}}B^{*}p \,dx \,dt}{\int_{0}^{T}\int_{\Omega_{U}}1 \,dx \,dt}\)
denotes the integral average on
\(\Omega_{U}\times[0,T]\)
of the function
\(B^{*}p\).
Now, we consider the hp finite element approximation for the nonlinear parabolic optimal control problems. We assume that Ω and \(\Omega_{U}\) are polygonal. We consider the triangulation \(\mathcal{T}\) of the set Ω⊂ \(\mathbb{R}^{2}\), which is a collection of elements \(\tau\in \mathcal{T}\) associated with each element τ, and an affine element map \(F_{\tau}:\widehat{\tau}\rightarrow\tau\), where the reference element τ̂ is the reference triangle
$$\bigl\{ (x,y)\in\mathbb{R}^{2} : 0< x< 1,0< y< \min(x,1-x) \bigr\} . $$
We consider the triangulation \(\mathcal{T}\) which satisfies the standard conditions defined in [16] and write \(h_{\tau}=\operatorname {diam}\tau\). Additionally we assume that triangulation \(\mathcal{T}\) is γ-shape regular, i.e.,
$$\begin{aligned} h_{\tau}^{-1}\bigl\Vert F_{\tau}^{\prime} \bigr\Vert _{L^{\infty }(\widehat{\tau})}+h_{\tau}\bigl\Vert \bigl(F_{\tau}^{\prime} \bigr)^{-1}\bigr\Vert _{L^{\infty }(\widehat{\tau})}\leq \gamma. \end{aligned}$$
(2.11)
This implies that there exists a constant \(C>0\) that depends solely on γ such that
$$\begin{aligned} C^{-1}h_{\tau }\leq h_{\tau^{\prime}}\leq Ch_{\tau}, \quad \tau,\tau^{\prime}\in\mathcal {T} \mbox{ with } \overline{\tau}\cap\overline{\tau}^{\prime}\neq\emptyset, \end{aligned}$$
(2.12)
and there exists a constant \(M\in\mathbb{N}\) that depends solely on γ such that no more than M elements share a common vertex. We assume that the triangulation \(\mathcal{T}_{U}\) of \(\Omega_{U}\) which is a collection of elements \(\tau_{U}\in\mathcal{T}_{U}\), is γ-shape regular which satisfies the standard conditions as \(\mathcal{T}\). Associated with each element \(\tau_{U}\) is an affine element map \(F_{\tau_{U}}:\widehat{\tau}\rightarrow\tau_{U}\). We further assume the triangulation \(\mathcal{T}\) satisfies the relation between the patch and the reference patch in [16].
For each element \(\tau\in\mathcal{T}\), we denote \(\mathcal {E}(\tau)\) the set of edges of τ and \(\mathcal{N}(\tau)\) the set of vertices of τ, and choose a polynomial degree \(p_{\tau}\in\mathbb{N}\) and collect these numbers in the polynomial degree vector \(\mathbf{p}_{1}=(p_{\tau})_{\tau\in\mathcal{T}}\). Similarly, for each element \(\tau_{U}\in\mathcal{T}_{U}\), we choose a polynomial degree vector \(\mathbf{p}_{2}=(p_{\tau_{U}})_{\tau_{U}\in \mathcal {T}_{U}}\) (\(p_{\tau_{U}}\in\mathbb{N}\)). \(\mathcal{N}(\mathcal {T})\) denotes the set of all vertices of \(\mathcal{T}\), \(\mathcal {E}(\mathcal{T})\) denotes the set of all edges. Additionally, we introduce the following notation (\(V\in\mathcal{N}(\mathcal{T})\), \(e\in\mathcal{E}(\mathcal{T})\)):
$$\begin{aligned}& \mathcal {N}(e)=\bigl\{ V\in\mathcal {N}(\mathcal{T}):V\in\overline{e}\bigr\} , \\& w_{V}=\bigl\{ x\in\Omega:x\in\overline{\tau}\mbox{ and } \overline{\tau}\cap\{V\}\neq\emptyset\bigr\} ^{0}, \\& w^{1}_{e}=\bigcup_{V\in\mathcal{N}(e)}w_{V},\qquad w^{1}_{\tau}=\bigcup_{V\in\mathcal{N}(\tau)}w_{V}, \\& h_{\tau_{U}}=\operatorname {diam}\tau_{U}, \qquad p_{e}=\max\bigl\{ p_{\tau}: e\in\mathcal{E}(\tau)\bigr\} , \end{aligned}$$
where \(\chi^{0}\) denotes the interior of the set χ. Finally, we denote by \(h_{e}\) the length of the edge e.
Next, we define the hp-finite element space \(S^{\mathbf{p}_{1}}(\mathcal{T})\subset H^{1}(\Omega)\) and the hp-discontinuous Galerkin finite element space \(U^{ \mathbf{p}_{2}}(\mathcal{T}_{U})\subset L^{2}(\Omega_{U})\) by
$$\begin{aligned}& S^{ \mathbf{p}_{1}}(\mathcal{T})= \bigl\{ v\in C(\Omega):v|_{\tau}\circ F_{\tau}\in\Pi_{p_{\tau}}(\widehat{\tau}) \bigr\} , \\& U^{ \mathbf{p}_{2}}(\mathcal{T}_{U})= \bigl\{ v\in L^{2}( \Omega_{U}):v|_{\tau_{U}}\circ F_{\tau_{U}}\in \Pi_{p_{\tau_{U}}}(\widehat{\tau}) \bigr\} , \end{aligned}$$
where we set
$$\Pi_{k}(\widehat{\tau})= \textstyle\begin{cases} {P}_{k}=\operatorname {span}\{x^{i}y^{j}:0\leq i+j\leq k\},& \mbox{if } \widehat{\tau}=T,\\ {Q}_{k}=\operatorname {span}\{x^{i}y^{j}:0\leq i,j\leq k\}, & \mbox{if } \widehat{\tau}=S. \end{cases} $$
We also assume that the polynomial degree vector \(\mathbf{p}_{1}\) satisfies
$$\begin{aligned} \gamma^{-1}p_{\tau}\leq p_{\tau^{\prime}}\leq \gamma p_{\tau}, \quad \tau,\tau^{\prime}\in\mathcal{T} \mbox{ with } \overline{\tau}\cap\overline{\tau}^{\prime}\neq\emptyset. \end{aligned}$$
(2.13)
Then we can introduce the finite dimensional spaces \(K_{hp}=K\cap U^{ \mathbf{p}_{2}}(\mathcal{T}_{U})\), \(V_{hp}=V\cap S^{\mathbf{p}_{1}}(\mathcal{T})\).
The semidiscrete hp finite element approximation of (2.1)-(2.4) is as follows:
$$\begin{aligned}& \min_{u_{hp}\in K_{hp}} \biggl\{ \frac{1}{2} \int_{0}^{T}\bigl(\Vert y_{hp}-y_{d} \Vert ^{2}_{L^{2}(\Omega )}+\Vert u_{hp}\Vert ^{2}_{L^{2}(\Omega_{U})}\bigr)\,dt \biggr\} , \end{aligned}$$
(2.14)
$$\begin{aligned}& \biggl(\frac{\partial y_{hp}}{\partial t},w_{hp} \biggr)+a(y_{hp},w_{hp})+ \bigl(\phi(y_{hp}),w\bigr)=(f+Bu_{hp},w_{hp}),\quad \forall w_{hp}\in V_{hp}, \end{aligned}$$
(2.15)
$$\begin{aligned}& y_{hp}(x,0)=y_{0}^{hp}(x),\quad x\in \Omega, \end{aligned}$$
(2.16)
where \(y_{hp}\in H^{1}(0,T;V_{hp})\) and \(y_{0}^{hp}\in V_{hp}\) is an finite element approximation of \(y_{0}\).
It follows that the optimal control problems (2.14)-(2.16) has at least a solution \((y_{hp},u_{hp})\) and if that a pair \((y_{hp},u_{hp})\) is the solution of (2.14)-(2.16), then there is a co-state \(p_{hp}\in V_{hp}\) such that the triplet \((y_{hp},p_{hp},u_{hp})\) satisfies the following optimality conditions:
$$\begin{aligned}& \biggl(\frac{\partial y_{hp}}{\partial t},w_{hp} \biggr)+a(y_{hp},w_{hp})+ \bigl(\phi(y_{hp}),w\bigr)=(f+Bu_{hp},w_{hp}),\quad \forall w_{hp}\in V_{hp}, \end{aligned}$$
(2.17)
$$\begin{aligned}& y_{hp}(x,0)=y_{0}^{hp}(x),\quad x\in\Omega, \\& - \biggl(\frac{\partial p_{hp}}{\partial t},q_{hp} \biggr)+a(q_{hp},p_{hp})+ \bigl(\phi^{\prime }(y_{hp})p_{hp},q_{hp} \bigr)=(y_{hp}-y_{d},q_{hp}),\quad \forall w_{hp}\in V_{hp}, \end{aligned}$$
(2.18)
$$\begin{aligned}& p_{hp}(x,T)=0, \quad x\in\Omega, \\& \bigl(u_{hp}+B^{*}p_{hp},v_{hp}-u_{hp} \bigr)_{U}\geq0,\quad \forall v_{hp}\in K_{hp}. \end{aligned}$$
(2.19)
Furthermore, we consider the fully discrete finite element approximation for the above semidiscrete problems by using the backward Euler scheme. Let \(0=t_{0}< t_{1}<\cdots<t_{M-1}<t_{M}=T\), \(k_{i}=t_{i}-t_{i-1}\), \(i=1,2,\ldots,M\), \(k=\max\limits_{1\leq i\leq M}\{k_{i}\}\).
For \(i=1,2,\ldots,M\), construct the hp finite element approximation spaces \(V_{hp}^{i}\subset H^{1}_{0}(\Omega)\) (similar to \(V_{hp}\)) on the ith time step. Similarly, we construct the hp finite element approximation spaces \(K^{i}_{hp}\subset L^{2}(\Omega_{U})\) (similar to \(K_{hp}\)) on the ith time step. The fully discrete hp finite element approximation scheme (2.17)-(2.19) is to find \((y_{hp}^{i},u_{hp}^{i})\in V^{i}_{hp}\times K^{i}_{hp}\), \(i=1,2,\ldots,M\), such that
$$\begin{aligned}& \min_{u_{hp}^{i}\in K^{i}_{hp}} \Biggl\{ \sum _{i=1}^{M} \biggl(\frac{1}{2}\bigl\Vert y_{hp}^{i}-y_{d}(x,t_{i})\bigr\Vert ^{2}_{L^{2}(\Omega)}+\frac{1}{2}\bigl\Vert u_{hp}^{i} \bigr\Vert ^{2}_{L^{2}(\Omega_{U})} \biggr) \Biggr\} , \end{aligned}$$
(2.20)
$$\begin{aligned}& \biggl(\frac{y_{hp}^{i}-y_{hp}^{i-1}}{k_{i}},w_{hp} \biggr)+a \bigl(y_{hp}^{i},w_{hp}\bigr)+\bigl(\phi \bigl(y_{hp}^{i}\bigr),w_{hp}\bigr) \\& \quad = \bigl(f(x,t_{i})+Bu_{hp}^{i},w_{hp} \bigr), \quad \forall w_{hp}\in V_{hp}^{i}, \end{aligned}$$
(2.21)
$$\begin{aligned}& y_{hp}^{0}(x)=y_{0}^{hp}(x),\quad x\in\Omega. \end{aligned}$$
(2.22)
It follows that the optimal control problem (2.20)-(2.22) has at least a solution \((Y_{hp}^{i},U_{hp}^{i})\), and if a pair \((Y_{hp}^{i},U_{hp}^{i})\in V^{i}_{hp}\times K^{i}_{hp}\) is the solution of (2.20)-(2.22), then there is a co-state \(P_{hp}^{i-1}\in V^{i}_{hp}\), such that the triplet \((Y_{hp}^{i},P_{hp}^{i-1},U_{hp}^{i})\in V^{i}_{hp}\times V^{i}_{hp}\times K^{i}_{hp}\), satisfies the following optimality conditions:
$$\begin{aligned}& \biggl(\frac{Y_{hp}^{i}-Y_{hp}^{i-1}}{k_{i}},w_{hp} \biggr)+a\bigl(Y_{hp}^{i},w_{hp} \bigr)+\bigl(\phi \bigl(Y_{hp}^{i}\bigr),w_{hp} \bigr)=\bigl(f(x,t_{i})+BU_{hp}^{i},w_{hp} \bigr), \end{aligned}$$
(2.23)
$$\begin{aligned}& \forall w_{hp}\in V_{hp}^{i}\subset V=H_{0}^{1}(\Omega),\quad i=1,2,\ldots,M,\qquad Y_{hp}^{0}(x)=y_{0}^{hp}(x),\quad x\in\Omega, \\& \biggl(\frac{P_{hp}^{i-1}-P_{hp}^{i}}{k_{i}},q_{hp} \biggr)+a\bigl(q_{hp},P_{hp}^{i-1} \bigr)+\bigl(\phi^{\prime }\bigl(Y_{hp}^{i-1} \bigr){P_{hp}^{i-1}},q_{hp}\bigr)= \bigl(Y_{hp}^{i}-y_{d}(x,t_{i}),q_{hp} \bigr), \end{aligned}$$
(2.24)
$$\begin{aligned}& \forall q_{hp}\in V_{hp}^{i}\subset V=H_{0}^{1}(\Omega), \quad i=M,\ldots,2,1,\qquad P_{hp}^{M}(x)=0,\quad x\in\Omega, \\& \bigl(U_{hp}^{i}+B^{*}P_{hp}^{i-1},v_{hp}-U_{hp}^{i} \bigr)_{U}\geq0, \quad \forall v_{hp}\in K_{hp}^{i}, i=1,2,\ldots,M. \end{aligned}$$
(2.25)
For \(i=1,2,\ldots,M\), let
$$\begin{aligned}& Y_{hp}|_{(t_{i-1},t_{i}]}= \bigl((t_{i}-t)Y_{hp}^{i-1}+(t-t_{i-1})Y_{hp}^{i} \bigr)/k_{i}, \\& P_{hp}|_{(t_{i-1},t_{i}]}= \bigl((t_{i}-t)P_{hp}^{i-1}+(t-t_{i-1})P_{hp}^{i} \bigr)/k_{i}, \\& U_{hp}|_{(t_{i-1},t_{i}]}=U_{hp}^{i}. \end{aligned}$$
For any function \(w\in C(0,T;L^{2}(\Omega))\), let \(\hat{w}(x,t)|_{t\in(t_{i-1},t_{i}]}=w(x,t_{i})\), \(\tilde{w}(x,t)|_{t\in(t_{i-1},t_{i}]}=w(x,t_{i-1})\). Then the optimality conditions (2.23)-(2.25) can be restated as
$$\begin{aligned}& \biggl(\frac{\partial Y_{hp}}{\partial t},w_{hp} \biggr)+a(\hat {Y}_{hp},w_{hp})+\bigl(\phi(\hat{Y}_{hp}),w_{hp} \bigr)=(\hat {f}+BU_{hp},w_{hp}), \end{aligned}$$
(2.26)
$$\begin{aligned}& \forall w_{hp}\in V_{hp}^{i}\subset V=H_{0}^{1}(\Omega), \quad t\in(t_{i-1},t_{i}], i=1,2,\ldots,M, \\& Y_{hp}(x,0)=y_{0}^{hp}(x),\quad x\in\Omega, \\& - \biggl(\frac{\partial P_{hp}}{\partial t},q_{hp} \biggr)+a(q_{hp}, \tilde{P}_{hp})+\bigl(\phi^{\prime}(\tilde{Y}_{hp}){ \tilde {P}_{hp}},q_{hp}\bigr)=(\hat{Y}_{hp}- \hat{y}_{d},q_{hp}), \end{aligned}$$
(2.27)
$$\begin{aligned}& \forall q_{hp}\in V_{hp}^{i}\subset V=H_{0}^{1}(\Omega),\quad t\in(t_{i-1},t_{i}], i=M,\ldots,2,1, \\& P_{hp}(x,T)=0, \quad x\in\Omega, \\& \bigl(U_{hp}+B^{*}\tilde{P}_{hp},v_{hp}-U_{hp} \bigr)_{U}\geq0, \\& \forall v_{hp}\in K_{hp}^{i},\quad t \in(t_{i-1},t_{i}], i=1,2,\ldots,M. \end{aligned}$$
(2.28)
The following lemmas [10, 16, 19] are important in deriving a posteriori error estimates of residual type.
Lemma 2.2
There exist a constant
\(C>0\)
independent of
v, \(h_{\tau_{U}}\), and
\(p_{\tau_{U}}\)
and a mapping
\(\pi_{p_{\tau_{U}}}^{h_{\tau_{U}}}:H^{1}(\tau_{U})\rightarrow \mathscr{P}_{p_{\tau_{U}}}(\tau_{U})\)
such that
\(\forall v\in H^{1}(\tau_{U})\), \(\tau_{U}\in\mathcal{T}_{U}\)
the following inequality is valid:
$$\bigl\Vert v-\pi_{p_{\tau_{U}}}^{h_{\tau_{U}}}\bigr\Vert _{L^{2}(\tau _{U})}\leq C\frac{h_{\tau_{U}}}{p_{\tau_{U}}}\Vert v\Vert _{H^{1}(\tau_{U})}, $$
where we will write
\(v\in\mathscr{P}_{p_{\tau_{U}}}(\tau_{U})\)
if the following satisfied: \(v|_{\tau_{U}}\circ F_{\tau_{U}}\in P_{p_{\tau_{U}}}(\widehat{\tau})\)
if
\(\tau_{U}\)
is a triangle; \(v|_{\tau_{U}}\circ F_{\tau_{U}}\in Q_{p_{\tau_{U}}}(\widehat{\tau})\)
if
\(\tau_{U}\)
is a parallelogram.
Lemma 2.3
Let
\(\mathbf{p}_{1}\)
be an arbitrary polynomial degree distribution satisfies (2.13). Then there exists a linear operator
\(E_{1}:H^{1}_{0}(\Omega)\rightarrow S^{ \mathbf{p}_{1}}(\mathcal {T})\cap H^{1}_{0}(\Omega)\), and there exists a constant
\(C>0\)
depending solely on
γ
such that for every
\(v\in H^{1}_{0}(\Omega)\)
and all elements
\(\tau\in\mathcal{T}\)
and all edges
\(e\in\mathcal{E}(\mathcal{T})\):
$$\begin{aligned}& \Vert v-E_{1}v\Vert _{L^{2}(\tau)}+\frac{h_{\tau}}{p_{\tau }}\bigl\Vert \nabla (v-E_{1}v)\bigr\Vert _{L^{2}(\tau)}\leq C \frac{h_{\tau}}{p_{\tau}} \Vert \nabla v\Vert _{L^{2}(w_{\tau }^{1})}, \\& \Vert v-E_{1}v\Vert _{L^{2}(e)}\leq C \biggl( \frac{h_{e}}{p_{e}} \biggr)^{\frac{1}{2}}\Vert \nabla v\Vert _{L^{2}(w_{e}^{1})}. \end{aligned}$$
Lemma 2.4
Let
\(\mathbf{p}_{1}\)
be an arbitrary polynomial degree distribution satisfying (2.13) and
\(p_{\tau}\geq2\), \(\forall \tau\in\mathcal{T}\). Then there exists a bounded linear operator
\(E_{2}:H^{1}_{0}(\Omega)\cap H^{2}(\Omega)\rightarrow S^{ \mathbf{p}_{1}}(\mathcal{T})\cap H^{1}_{0}(\Omega)\), and there exists a constant
\(C>0\)
that depends solely on
γ
such that for every
\(v\in H^{1}_{0}(\Omega)\cap H^{2}(\Omega)\)
and all elements
\(\tau\in\mathcal{T}\)
and all edges
\(e\in\mathcal{E}(\mathcal {T})\):
$$\begin{aligned}& \Vert v-E_{2}v\Vert _{L^{2}(\tau)}+\frac{h_{\tau}}{p_{\tau }}\bigl\Vert \nabla (v-E_{2}v)\bigr\Vert _{L^{2}(\tau)}\leq C \biggl( \frac{h_{\tau}}{p_{\tau}} \biggr)^{2}\vert v\vert _{H^{2}(w_{\tau }^{1})}, \\& \Vert v-E_{2}v\Vert _{L^{2}(e)}\leq C \biggl( \frac{h_{e}}{p_{e}} \biggr)^{\frac {3}{2}}\vert v\vert _{H^{2}(w_{e}^{1})}. \end{aligned}$$
For \(\varphi\in W_{h}\), we shall write
$$\begin{aligned} \phi(\varphi)-\phi(\rho)=-\tilde{\phi}^{\prime}(\varphi) ( \rho -\varphi )=-\phi^{\prime}(\rho) (\rho-\varphi) +\tilde{ \phi}^{\prime\prime}(\varphi) (\rho-\varphi)^{2}, \end{aligned}$$
(2.29)
where
$$\begin{aligned}& \tilde{\phi}^{\prime}(\varphi)= \int_{0}^{1}\phi^{\prime}\bigl(\varphi +s(\rho -\varphi)\bigr)\,ds, \qquad \tilde{\phi}^{\prime\prime}(\varphi)= \int_{0}^{1}(1-s)\phi^{\prime \prime }\bigl(\rho+s( \varphi-\rho)\bigr)\,ds \end{aligned}$$
are bounded functions in Ω̄ [22].