1 Introduction↩︎

Stability of the \(L^2\)-projection in norms other than \(\norm{\cdot}_{L^2}\) is a key element in the numerical analysis of finite element methods. For example, for the heat equation \(W^{1,2}\)-stability is equivalent to the inf-sup stability and quasi-optimality of Galerkin methods (see [3]). For nonlinear evolution problems stability properties provide an important ingredient in the proof of error estimates that do not rely on a coupling of time and space discretisation parameters [4]. Furthermore, decay properties of the \(L^2\)-projection are instrumental for results on \(hp\)-preconditioning [5].

Let \(\Pi\) denote the \(L^2\)-projection onto the Lagrange finite element space \(V_{h}\) of a certain polynomial degree \(K \geq 1\). We aim for stability in \(L^p\) and \(W^{1,p}\) for \(p \in [1,\infty]\), i.e., we want to show that \[\begin{align} \norm{\Pi u}_{p} \lesssim \norm{u}_{p} \qquad \text{ and } \qquad \norm{\nabla \Pi u}_{p} \lesssim \norm{\nabla u}_{p}, \end{align}\] holds for any \(u \in L^p(\Omega)\) or \(u \in W^{1,p}(\Omega)\) uniformly in the mesh size, respectively.

As proved in [6] for quasi-uniform meshes stability in \(W^{1,2}\) follows from inverse estimates and by using a Sobolev stable operator like the Scott–Zhang operator [7] or the Clément operator [8]. However, both for non-quasiuniform meshes and for \(p \neq 2\) establishing stability results is a highly challenging task. For the analysis of adaptive finite element methods it is important that Sobolev stability holds for graded meshes arising from adaptive mesh refinement, uniformly in the refinement level. Furthermore, stability estimates for \(p \neq 2\) are crucial in the error analysis of nonlinear evolution equations [4].

In dimension \(d = 1\) it is known that Sobolev stability cannot hold without conditions on the mesh grading [9], [10]. In fact, in [10] a counterexample to \(W^{1,2}\)-stability is presented for the lowest order case when neighbouring intervals differ sufficiently in size. In higher dimensions analogous counterexamples are unknown, but all available stability results are based on certain grading assumptions. Specifically, this means that the local mesh sizes of neighbouring elements are not allowed to differ too strongly.

For simplicial meshes in dimension \(d \geq 2\) numerous stability results of the \(L^2\)-projection are available. Beginning with the work on mildly graded triangulations in [9], [11], [12], stability results were subsequently established for low-order spaces on highly graded meshes, such as those generated by adaptive mesh refinement [13]–[17]. More recently, stability was established for a bounded range of polynomial degrees and on meshes exhibiting realistic grading [10], [18], [19] for dimension \(d = 2\) and in [10] for dimension \(d = 3\). The most recent results in [2], [20] build on the techniques developed in [10] and establish mesh grading results for BisecMT [21], [22], the generalisation of the newest vertex bisection, in any space dimension \(d \in \mathbb{N}\). Taken together, these works establish \(W^{1,2}\)-stability for triangulations generated by BisecMT in any dimension \(d \leq 6\) for any polynomial degree \(K \geq 1\). More generally, \(W^{1,p}\)- and \(L^p\)-stability are established in [2] under certain grading assumptions by means of an argument utilising maximal operators.

In addition to simplicial meshes, quadrilateral meshes are frequently employed for finite element discretisation [23], [24], see also [25]. In contrast to simplicial meshes, structured quadrilateral meshes admit a tensor product structure enabling highly efficient implementations. However, adaptive mesh refinement on purely quadrilateral meshes typically introduces hanging vertices. To address this, hybrid meshes consisting of simplices and quadrilaterals offer a flexible framework that accommodates adaptive refinement while retaining mesh conformity. Such refinement schemes have been presented in [26]–[28]. In particular, quadrilaterals are well-suited for anisotropic meshes [29] which are often employed in the presence of edge singularities, see, e.g., [30]. This work, however, focuses on isotropic shape-regular hybrid meshes.

To date, the only stability results for the \(L^2\)-projection onto Lagrange elements on hybrid meshes are due to Ali, Funken, and Schmidt [1]. In their work, the authors consider two-dimensional hybrid meshes generated via adaptive Q-RG [26] and Q-RB [27] refinement starting from initial meshes composed exclusively of parallelograms. The resulting hybrid meshes consist of triangles and trapezoids. For these meshes, they obtain mesh grading results by applying the approach from [17], [18]. Furthermore, by employing the technique from [15], [16] and numerically solving certain local generalised eigenvalue problems, they show \(W^{1,2}\)-stability for polynomial degree \(K \in \{2, \ldots, 9\}\).

This work extends previous stability results for the \(L^2\)-projection onto Lagrange finite element spaces on two-dimensional meshes containing simplices and quadrilaterals in the following directions:

• Our results apply to general hybrid meshes consisting of triangles and general convex quadrilaterals that satisfy certain grading and shape-regularity properties. This includes, in particular, meshes generated by the Q-RG and Q-RB refinement scheme starting from an initial mesh of general quadrilaterals; see Remark [rem:general-grading] below. This generalises the results in [1] for initial meshes consisting exclusively of parallelograms to those consisting of general quadrilaterals.

• Our stability results hold under general mesh grading assumptions and can be readily applied to other refinement schemes that yield hybrid meshes satisfying grading and shape-regularity properties.

• We extend the valid range of polynomial degrees from \(K \in \{2, \ldots, 9\}\) to all \(K \geq 2\). Since the decay properties of \(\Pi\) improve for higher polynomial degrees, this extension provides a foundation for future investigation of \(hp\)-methods.

• We establish \(W^{1,p}\)- and \(L^p\)-stability by following the framework in [2], which relies solely on weighted \(L^2\)-estimates and the stability of a quasi-interpolation operator, such as the Scott–Zhang operator. Consequently, any future improvements in the weighted \(L^2\)-estimates will immediately transfer into enhanced \(W^{1,p}\) and \(L^p\)-stability results.

Strategy and outline. In Section 2, we introduce finite element spaces on general hybrid meshes and review some adaptive mesh refinement schemes. Furthermore, we collect available grading and shape-regularity results. To establish the stability results in Section 3, we follow the general strategy in [2], which improves upon the approach in [10]. The core idea is to use a local approximation operator, see Section 3.1, to establish decay estimates. A (local) decomposition operator, see Sections 3.2 and 3.3, is key to verifying estimates for the approximation operator. To extend the results from simplicial meshes to general quadrilaterals, we utilise the tensor-product structure of the local spaces, but the fact that the transformations are in general non-affine poses extra challenges. While some of these difficulties can be avoided by restricting the initial meshes to parallelograms as in [1], we address the more general case of arbitrary convex quadrilaterals. The remaining steps – deducing a decay estimate in Section 3.4, weighted \(L^2\)-stability and finally \(L^p\)- and \(W^{1,p}\)-stability in Section 3.5 – proceed analogously to the simplicial case. They rely on results analogous to those in [2].

Extensions and limitations. In general dimension \(d \geq 3\), the only available grading results are those in [20] for simplicial meshes and BisecMT refinement (the generalised newest vertex bisection). The proof of this result is considerably more challenging than its two-dimensional counterparts, because a direct enumeration and investigation of worst-case scenarios is not possible. Furthermore, refinement schemes and corresponding grading results for hybrid meshes are currently unavailable in higher dimensions. Simply relying on the tensor-product structure in combination with the techniques from [2] fails to yield suitable estimates for dimensions \(d\geq 3\), see Remark [rmk:general-dim] below. For this reason, establishing stability results on hybrid meshes in higher dimensions necessitates alternative strategies.

2 Meshes and function spaces↩︎

In this work we consider two-dimensional hybrid meshes consisting of triangles and (strictly) convex quadrilaterals. In this section we introduce the precise setup for the meshes and the finite dimensional function spaces that we use.

2.1 Hybrid meshes↩︎

Let \(\Omega \subset \mathbb{R}^2\) be a bounded, polyhedral domain. We write \(a \lesssim b\), if there is a constant \(c>0\) only depending on specified parameters such that \(a \leq c b\). We write \(a \eqsim b\), if \(a \lesssim b\) and \(b \lesssim a\).

We decompose \(\Omega\) into a mesh \(\mathcal{T}\) consisting of (closed) triangles and (strictly) convex quadrilaterals, where element refers to both triangles and quadrilaterals. The latter are called quadrilaterals of type (1) in [25].

We suppose that the mesh \(\mathcal{T}\) is regular, in the following sense:

1. The mesh covers \(\overline{\Omega}\), i.e., \(\overline{\Omega}=\bigcup_{T\in\mathcal{T}} T\).

2. If \(T_1,T_2 \in \mathcal{T}\) overlap, then \(T_1 \cap T_2\) is a joint edge or vertex in \(\mathcal{T}\). In particular, there are no hanging vertices.

Let \(\mathcal{V} = \mathcal{V}(\mathcal{T})\) denote the set of vertices of \(\mathcal{T}\) and let \(\omega_i \mathrel{\vcenter{:}}= \{T \in \mathcal{T} \colon i \in T\}\) denote the vertex patch of a vertex \(i \in \mathcal{V}\). By \(\Omega_i \mathrel{\vcenter{:}}= (\cup_{T \in \omega_i} T)^\circ\) we denote the vertex patch domain.

In the following we refer to a hybrid mesh \(\mathcal{T}\) as

• simplicial, if \(\mathcal{T}\) contains only triangles,

• quadrilateral, if \(\mathcal{T}\) contains only (strictly) convex quadrilaterals,

• P-quadrilateral, if \(\mathcal{T}\) contains only parallelograms,

• SP-hybrid, if \(\mathcal{T}\) contains only triangles and parallelograms, and

• general hybrid, if \(\mathcal{T}\) contains triangles and general convex quadrilaterals.

We also use the concept of shape-regularity which measures the extent of deformation of elements. We employ the standard approach which applies both for triangles and quadrilaterals, see [25].

If \(T \in \mathcal{T}\) is a triangle, then we define \(\widehat{T}\) as the equilateral triangle with vertices \((0,0)^\top, (1,0)^\top, (\frac{1}{2},\frac{\sqrt{3}}{2})^\top\) and diameter \(\diameter(\widehat{T}) = 1\). If \(T\) is a quadrilateral, then \(\widehat{T}\) denotes the unit square \([0,1]^2\). For triangles there exists an affine linear isomorphism \(B_T\colon \widehat{T} \to T\) and for quadrilaterals there exists an affine bilinear isomorphism \(B_T\colon \widehat{T} \to T\), see Figure 1. By \(J_T\) we denote the Jacobian determinant of \(B_T\). Since \(T\) is strictly convex, one can choose \(B_T\) such that \(J_T\) is positive on \(\widehat{T}\).

The shape-regularity constant of an element \(T \in \mathcal{T}\) is defined as \[\begin{align} \label{def:chi-T} \chi(T) \mathrel{\vcenter{:}}= \max_{\widehat{x}\in \widehat{T}} \norm{\nabla B_T(\widehat{x})} \max_{x \in T} \norm{\nabla B_T^{-1}(x)}, \end{align}\tag{1}\] where \(\norm{\cdot}\) is the spectral norm which is induced by the Euclidean vector norm. Note that \(\chi(T) \geq 1\), and that \(\chi(T)\) increases with the deformation of \(T\).

The mapping \(B_T\) is not unique, since the vertices can be rotated. However, due to the symmetry of \(\widehat{T}\), the definition of \(\chi(T)\) in 1 is independent of the specific choice of \(B_T\). This independence was also observed by Knupp [31], where he discusses several mesh-quality metrics. Moreover, by using the equilateral triangle and the unit square as reference elements, we obtain that \(\chi(\widehat{T})=1\).

By standard arguments, one can show that for \(T\in \mathcal{T}\) one has \[\begin{align} {2} \tag{2} \frac{1}{\max_T \norm{\nabla B_T^{-1}}} &\leq \frac{\diameter(T)}{\diameter(\widehat{T})} &&\leq \max_T \norm{\nabla B_T}, \\ \tag{3} \frac{1}{\max_T \abs{J_T^{-1}}} &\leq \quad\;\frac{\abs{T}}{\abs{\widehat{T}}} && \leq \max_T \abs{J_T} \end{align}\] Thus, the estimates \(\abs{J_T}^{\frac{1}{2}} \leq \norm{\nabla B_T}\) and \(\abs{J_T^{-1}}^{\frac{1}{2}} \leq \norm{\nabla B_T^{-1}}\) imply that \[\begin{align} \label{eq:diam-vs-volume} \frac{1}{\chi(T)} \frac{\diameter(\widehat{T})}{\abs{\widehat{T}}^{\frac{1}{2}}} \leq \frac{\diameter(T)}{\abs{T}^{\frac{1}{2}}} \leq \chi(T) \frac{\diameter(\widehat{T})}{\abs{\widehat{T}}^{\frac{1}{2}}} . \end{align}\tag{4}\] We define \(\chi(\mathcal{T}) \mathrel{\vcenter{:}}= \max_{T \in \mathcal{T}} \chi(T)\) as the shape-regularity constant of a hybrid mesh \(\mathcal{T}\). If \(\chi(\mathcal{T}_j)\) is bounded for a family of meshes \((\mathcal{T}_j)_{j \in \mathbb{N}}\), then, in particular, we have \[\begin{align} \label{eq:shape-regular} \diameter(T) \eqsim \abs{T}^{\frac{1}{2}} \qquad \text{for any T \in \mathcal{T}_j,} \;\; \text{for any j \in \mathbb{N},} \end{align}\tag{5}\] i.e., uniformly for the family of meshes.

2.2 Grading↩︎

Let us collect the notations for distance, graded weight functions and mesh functions as in [2] adapted to the setting of hybrid meshes.

We start by defining a metric \(d\) on \(\mathcal{T}\). For \(T \in \mathcal{T}\) we define \(d(T,T) \mathrel{\vcenter{:}}= 0\). Moreover, if \(T,T' \in \mathcal{T}\) with \(T \neq T'\) are neighbours, i.e., \(T \cap T' \not= \emptyset\), then we set \(d(T,T') \mathrel{\vcenter{:}}=1\). For all other \(T,T' \in \mathcal{T}\) we define \(d(T,T')\) as the length of the shortest chain \(T_0,\dots, T_m \in \mathcal{T}\) with \(T_0 =T\) and \(T_m=T'\) and \(d(T_j,T_{j+1})=1\). For subsets \(L, L' \subset \mathcal{T}\) we define \(d(L,L') \mathrel{\vcenter{:}}= \min \set{d(T,T')\colon T \in L, \, T' \in L'}\).

By \(\mathcal{L}^0_0(\mathcal{T})\) we denote the set of \(L^1(\Omega)\) functions that are piecewise constant on \(\mathcal{T}\). A weight \(\rho\) with grading \(\gamma_\rho\geq 1\) is a positive function \(\rho \in \mathcal{L}^0_0(\mathcal{T})\) with \[\begin{align} \label{eq:grading} \rho|_T \leq \gamma_\rho \rho|_{T'} \qquad \text{for all T,T'\in\mathcal{T} with d(T,T')=1}. \end{align}\tag{8}\] Note that this is equivalent to \[\begin{align} \label{eq:grading2} \rho|_T \leq \gamma_\rho^{d(T,T')} \rho|_{T'} \qquad \text{for all T,T' \in \mathcal{T}}. \end{align}\tag{9}\]

We call \(h\in \mathcal{L}^0_0(\mathcal{T})\) a (regularised) mesh size function of \(\mathcal{T}\), if \[\begin{align} \label{eq:hT} h|_T &\eqsim \diameter(T) \qquad \text{ for all } T \in \mathcal{T}, \end{align}\tag{10}\] where the implicit constant does not depend on \(T \in \mathcal{T}\).

We say that the mesh \(\mathcal{T}\) has grading \(\gamma_{\mathcal{T}} \geq 1\), if there exists a mesh size function \(h \in \mathcal{L}^0_0(\mathcal{T})\) with grading \(\gamma_{\mathcal{T}}\), i.e., \[\begin{align} \label{def:grading} \frac{h|_{T}}{h|_{T'}} \leq \gamma_{\mathcal{T}} \quad \text{ for all } T,T' \in \mathcal{T} \text{ with } T \cap T' \neq \emptyset. \end{align}\tag{11}\] For a family of adaptively refined meshes, later a uniform maximal grading constant \(\gamma_{\mathcal{T}} \leq \gamma\) will be important.

2.3 Function spaces↩︎

By \(L^p(\Omega)\) and \(W^{1,p}(\Omega)\) we denote the usual Lebesgue and Sobolev spaces with norms \(\norm{\cdot}_p\) and \(\norm{\cdot}_{1,p}\), respectively, for \(p \in [1,\infty]\). For \(K \in \setN\) by \(\mathcal{P}_K(U)\) we denote the set of polynomials on a set \(U\) of total degree \(K\) (or less). Moreover, we denote the polynomials of maximal degree \(K\) on \([0,1]^{2}\) by \[\label{def:QK} \begin{align} \mathcal{Q}_K([0,1]^2) & \mathrel{\vcenter{:}}= \mathcal{P}_K([0,1]) \otimes \mathcal{P}_K([0,1]) \\ & \mathrel{\vcenter{:}}= \linearspan \set{ (x_1,x_2)^\top \mapsto p_1(x_1)p_2(x_2)\colon p_1,p_2 \in \mathcal{P}_K([0,1])}. \end{align}\tag{12}\]

Let us define the local spaces on triangles and on (strictly) convex quadrilaterals. As in Section 2.1 let \(B_T \colon \widehat{T} \to T\) denote the isomorphism from the reference element \(\widehat{T}\) to the element \(T\). For a triangle \(T \in \mathcal{T}\) we define \[\begin{align} \label{def:LK-tria} \mathcal{L}_K(T) &\mathrel{\vcenter{:}}= \set{ p \circ B_T^{-1}\colon p \in \mathcal{P}_K(\widehat{T})} = \mathcal{P}_K(T), \end{align}\tag{13}\] and for a quadrilateral \(T \in \mathcal{T}\) we define \[\begin{align} \label{def:LK-quad} \mathcal{L}_K(T) &\mathrel{\vcenter{:}}= \set{ p \circ B_T^{-1}\colon p \in \mathcal{Q}_K([0,1]^2)}, \end{align}\tag{14}\] which is in general not a space of polynomials. Note that the definition of \(\mathcal{L}_K(T)\) is independent of the choice of \(B_T\).

We define the global space of discrete functions over \(\Omega\) as \[\begin{align} \label{def:L1K} \mathcal{L}^1_K(\mathcal{T}) &\mathrel{\vcenter{:}}= \set{ v \in W^{1,\infty}(\Omega) \colon v|_T \in \mathcal{L}_K(T) \text{ for all } T \in \mathcal{T}}. \end{align}\tag{15}\] This is the special case of isoparametric elements for bilinear transformations. We refer to \(\mathcal{L}^1_K(\mathcal{T})\) as (bilinearly) mapped Lagrange elements. These spaces have approximation order \(K+1\) in \(L^2\) and order \(K\) in \(W^{1,2}\) in terms of \(h\), see [24].

Function space decomposition↩︎

Let us denote by \((\phi_i)_{i\in \mathcal{V}}\) the nodal basis of \(\mathcal{L}^1_1(\mathcal{T})\), which forms a partition of unity. Note that on triangles the \(\phi_i\) are the barycentric coordinates and thus affine linear. On quadrilaterals the \(\phi_i\) are the pullback of the tensor products of the one-dimensional barycentric coordinates of the standard square. If \(T\) is a parallelogram, then \(B_T\) is affine and \(\phi_i\) is affine bilinear. In the other cases \(\phi_i\) is a non-singular rational function.

Using the partition of unity \((\phi_i)_{i\in \mathcal{V}}\) we have the following decomposition \[\begin{align} \label{eq:space-decomposition} \mathcal{L}^1_K(\mathcal{T})=\sum_{i\in\mathcal{V}}\phi_{i}\mathcal{L}_{K-1}^{1}(\omega_{i}). \end{align}\tag{16}\]

2.4 Adaptive mesh refinement↩︎

\(W^{1,2}\)-stability of the \(L^2\)-projection mapping to Lagrange finite element spaces requires mesh conditions, specifically conditions on the mesh grading as in 11 . In dimension \(d = 1\) , there are counterexamples [10] showing that for a mesh with sufficiently strong grading, this stability cannot hold. In particular, the authors construct a sequence of one-dimensional meshes with large grading such that \(W^{1,2}\)-stability fails for \(\mathcal{L}^1_1(\mathcal{T})\). A finer analysis allows to quantify the grading threshold of this example as \[\begin{align} \label{est:gamma-threshold} \gamma \geq 1/(1+\sqrt{3}-\sqrt{3+2\sqrt{3}}) \approx 5.274510564406. \end{align}\tag{17}\]

In [1] adaptive quadrilateral meshes refined by local red, green and blue refinements are analysed. The local refinements of \([0,1]^2\) are depicted in the first row of Figure 2, and for general convex quadrilaterals in the second row. In the latter case the bilinear mapping \(B_T \colon [0,1]^2 \to T\) is used to determine the new vertices in \(T\). Then, the corresponding vertices are connected by straight lines in \(T\), see Figure 2 second line. Note that straight lines in \([0,1]^2\) are mapped to straight lines in \(T\) only if they are axis-parallel. Hence, mapping the new edges in \([0,1]^2\) does not lead to quadrilaterals with straight edges after mapping them.

The global adaptive refinement routine uses red refinements, and temporary green refinements (Q-RG) or temporary blue refinements (Q-RB) to remove hanging nodes. The Q-RG and Q-RB refinement routines were originally introduced in [26] and in [27], respectively. Grading properties are established in [1] and the proof uses a regularised mesh size function introduced in [16].

Ali, Funken, and Schmidt showed the following grading estimate for P-quadrilateral initial meshes in [1].

In the following we shall assume that we have an initial mesh \(\mathcal{T}_0\) and an adaptive refinement scheme which generates a family of hybrid meshes \((\mathcal{T}_j)_{j \in \mathbb{N}_0}\) as in Subsection 2.1 and a grading constant \(\gamma \geq 1\) such that \[\begin{align} \label{est:grading-family} \gamma_{\mathcal{T}_j} \leq \gamma \qquad \text{ for any } j \in \mathbb{N}_0. \end{align}\tag{18}\] By Lemma 3 and Remark [rem:general-grading] for Q-RB and Q-RG refinements the grading constant is \[\begin{align} \label{eq:grading-Q-ex} \gamma_{\mathrm{Q-RB}} = \gamma_{\mathrm{Q-RG}} = 2 \end{align}\tag{19}\] for quadrilateral initial meshes.

In [1] the authors use this grading estimate for Q-RB and Q-RG applied to P-quadrilateral initial meshes to show \(W^{1,2}\)-stability for \(K=2,\dots,9\) in [1]. The special case of P-quadrilateral initial meshes leads to families of meshes consisting of parallelograms, trapezoids and triangles but not general convex quadrilaterals. This restriction simplifies the analysis. We will extend the results in [1] in several directions: We allow general hybrid meshes with grading, which includes in particular Q-RB and Q-RG starting from quadrilateral initial meshes. Moreover, we extend the range of polynomials to \(K \geq 2\). Finally, we also present \(L^p\)- and \(W^{1,p}\)-stability results, see Theorem 1 and Corollary 1 below.

3 Stability of the \(L^2\)-projection↩︎

In this section we investigate the stability of the \(L^2\)-projection onto \(\mathcal{L}^1_K(\mathcal{T})\) in Sobolev and Lebesgue norms.

Let \(\mathcal{T}\) be a hybrid mesh of a polyhedral domain \(\Omega \subset \RR^2\) as described in Section 2 and let \(K \in \setN\). Let \(\Pi \colon L^2(\Omega) \to \mathcal{L}^1_K(\mathcal{T})\) denote the \(L^2\)-projection onto \(\mathcal{L}^1_K(\mathcal{T})\), i.e., it satisfies \[\begin{align} \label{eq:defPi} \skp{\Pi u}{v_h} = \skp{u}{v_h} \qquad \text{for all v_h \in \mathcal{L}^1_K(\mathcal{T})} \end{align}\tag{20}\] for any \(u \in L^2(\Omega)\). Since \(\mathcal{L}^1_K(\mathcal{T}) \subset L^\infty(\Omega)\), the \(L^2\)-projection is also well-defined on \(L^1(\Omega)\). Naturally, \(\Pi\) is stable on \(L^2(\Omega)\) meaning that it satisfies \(\norm{\Pi u}_2 \leq \norm{u}_2\). Stability in other norms such as \(W^{1,2}(\Omega)\), or \(L^p(\Omega)\) and \(W^{1,p}(\Omega)\) for \(p \neq 2\) requires conditions on the mesh, see [10].

Stability in Lebesgue and Sobolev spaces can be obtained from certain weighted \(L^2\)-estimates of the form \[\begin{align} \label{eq:weighted} \norm{h^{-\alpha} \Pi u}_2 \lesssim \norm{h^{-\alpha} u}_2 \end{align}\tag{21}\] for some \(\alpha>0\), with \(h\) a mesh size function as in 10 . In particular, the case \(\alpha=1\) implies \(W^{1,2}\)-stability, see [2].

The proof of weighted estimates as in 21 relies on decay properties of \(\Pi\). Those can be proven by using a local operator with which one can approximate \(\Pi\). This approach has been introduced in [10] and was generalised in [2], [20]. In the following we extend this approach to hybrid meshes.

3.1 Approximating operator↩︎

In this section we construct a local, self-adjoint operator \(C\colon L^2(\Omega) \to \mathcal{L}^1_K(\mathcal{T})\) that approximates the \(L^2\)-projection \(\Pi\). The definition of \(C\) is analogous to the one in [2] for simplicial meshes.

Our construction is based on the decomposition 16 . Let \((\phi_i)_{i\in \mathcal{V}}\) denote the nodal basis of \(\mathcal{L}^1_1(\mathcal{T})\), which is a partition of unity. For each \(i \in \mathcal{V}\), \(\phi_i\) is a weight on \(\Omega_i = \support \phi_i\). This allows to define the weighted scalar products \[\begin{align} \label{eq:scalarproduct-phii} \skp{u}{v}_{\phi_i} \mathrel{\vcenter{:}}= \int_{\Omega_i} u v \, \phi_i\, \mathrm{d}x. \end{align}\tag{22}\] Let \(C_i \colon L^{2}(\Omega)\to\mathcal{L}^1_{K-1}(\omega_{i})\) denote the orthogonal projection with respect to \(\skp{\cdot}{\cdot}_{\phi_{i}}\) onto \(\mathcal{L}^1_{K-1}(\omega_i)\), i.e., \[\begin{align} \label{eq:Ci} \langle C_{i}u,v_{K-1}\rangle_{\phi_{i}}=\langle u,v_{K-1}\rangle_{\phi_{i}}\quad \text{for all } v_{K-1}\in\mathcal{L}_{K-1}^{1}(\omega_i), \end{align}\tag{23}\] and for any \(u \in L^2(\Omega)\). Note that \(\Pi\) also satisfies 23 in the sense that \[\begin{align} \label{eq:Ci-Pi} \langle \Pi u,v_{K-1}\rangle_{\phi_{i}}=\langle u,v_{K-1}\rangle_{\phi_{i}}\quad \text{for all } v_{K-1}\in\mathcal{L}_{K-1}^{1}(\omega_i), \end{align}\tag{24}\] and for any \(u \in L^2(\Omega)\). This property of \(\Pi\) is the essential feature for our definition of the operators \(C_i\).

We define the global operator \(C\colon L^{2}(\Omega)\rightarrow\mathcal{L}^1_K(\mathcal{T})\) as the (weighted) sum of the local operators \(C_{i}\) \[\begin{align} \label{eq:defC} C \mathrel{\vcenter{:}}= \sum_{i\in\mathcal{V}}\phi_{i}C_{i}, \end{align}\tag{25}\] which resembles 16 . Exactly as in [2] we obtain the following basic properties of \(C\).

To obtain decay estimates for \(\Pi\) as in [2], [10] it remains to estimate from above the condition number\[\operatorname{cond}_2(C|_{\mathcal{L}^1_K(\mathcal{T})}) = \frac{\lambda_{\max}(C|_{\mathcal{L}^1_K(\mathcal{T})})}{\lambda_{\min}(C|_{\mathcal{L}^1_K(\mathcal{T})})}.\] By Proposition [pro:C] we know that \(\lambda_{\max}(C|_{\mathcal{L}^1_K(\mathcal{T})})\leq 1\), so we only need an estimate for the smallest eigenvalue of \(C|_{\mathcal{L}^1_K(\mathcal{T})}\). This is achieved by means of a decomposition operator in the next subsection.

3.2 Decomposition operator↩︎

In this subsection we introduce a decomposition operator of functions in \(\mathcal{L}^1_K(\mathcal{T})\). The general idea goes back to [10]. Here, we use a decomposition operator as in [2], which is based on the function space decomposition 16 .

In the following we construct decomposition operators \(\frD_i \colon \mathcal{L}^1_K(\mathcal{T}) \to \mathcal{L}^1_{K-1}(\omega_i)\) such that \[\begin{align} \label{def:decomp} v_K = \sum_{i \in \mathcal{V}} \phi_i \frD_i v_K \qquad \text{for all v_K \in \mathcal{L}^1_K(\mathcal{T})}. \end{align}\tag{26}\] Exactly as in [2] we have the following statement.

Let us now construct the decomposition operators \(\frD_i\) for general hybrid meshes. These operators are defined locally on each element \(T \in \mathcal{T}\). Denoting \(\frD_i^T \mathrel{\vcenter{:}}= \frD_i|_{T}\) we aim for an estimate of the form \[\begin{align} \label{est:DiT-local} \sum_{i \in \mathcal{V}(T)}\int_T \phi_i \abs{\frD_i^T v_K}^2 \, \mathrm{d}x \leq K_1 \int_{T} \abs{v_K}^2 \, \mathrm{d}x \qquad \text{ for all } v_K \in \mathcal{P}_K(T), \end{align}\tag{28}\] and any \(T \in \mathcal{T}\). We will show that \(\frD_i v_K \in \mathcal{L}^1_{K-1}(\omega_i)\), i.e., that the local definitions are continuous across edges, and that 28 implies the global estimate 27 . In fact, we only need to extend the construction in [2] for triangles to general quadrilaterals.

3.2.1 Decomposition on triangles↩︎

We begin by reviewing the decomposition for a triangle \(T \in \mathcal{T}\) as introduced in [2]. Let \(\lambda_0,\lambda_1,\lambda_2\) denote the barycentric coordinates of \(T\). We consider the polynomials \(\lambda^\beta = \lambda_0^{\beta_0} \lambda_1^{\beta_1} \lambda_2^{\beta_2}\) for all multi-indices \(\beta=(\beta_0,\beta_1,\beta_2) \in \setN_0^3\). Let \(e_0=(1,0,0)^\top\), \(e_1=(0,1,0)^\top\) and \(e_2=(0,0,1)^\top\) be the canonical unit vectors. Let \(\{x_\alpha \colon \alpha \in \setN_0^3, \abs{\alpha}=K \}\) denote the set of Lagrange nodes of \(\mathcal{L}^1_K(T)\). We define for \(i \in \{0,1,2\}\) and \(\alpha \in \mathbb{N}^3_0\) with \(\abs{\alpha} = K\) the operator \[\begin{align} \label{eq:def-Di-triangle} \frD_i^T(\lambda^\alpha) \mathrel{\vcenter{:}}= \frac{\alpha_i}{K} \lambda^{\alpha-e_i} = \begin{cases} \lambda_i(x_\alpha) \lambda^{\alpha-e_i} &\qquad \text{if \alpha_i >0}, \\ 0 &\qquad \text{if \alpha_i=0,} \end{cases} \end{align}\tag{29}\] with the convention \(0 \cdot \lambda^{\alpha-e_i}=0\). By definition it follows that \[\begin{align} \label{eq:Di-identity} \lambda_i \frD_i^T(\lambda^\alpha) = \lambda_i(x_\alpha) \lambda^\alpha \qquad \text{for i= \in \{ 0,1,2 \}\, and \abs{\alpha}=K.} \end{align}\tag{30}\] Note that this represents a local version of the decomposition in 26 . From the proof of Prop. 3.14 in [2] it follows that for any triangle \(T \in \mathcal{T}\) one has \[\begin{align} \label{eq:Di-est-triangle} \sum_{i=0}^2\int_T \lambda_i \abs{\frD_i^T v_K}^2 \, \mathrm{d}x\leq \tfrac{2K+2}{K} \int_{T} \abs{v_K}^2 \, \mathrm{d}x, \end{align}\tag{31}\] where \(i \in \{0,1,2\}\) corresponds to the vertex of \(T\) defined by \(\lambda_i = 1\). Indeed, this verifies 28 for triangles.

3.2.2 Decomposition on a general convex quadrilateral↩︎

We continue with the case that \(T \in \mathcal{T}\) is a (strictly) convex quadrilateral. As in the definition of \(\mathcal{L}_K(T)\) we use the tensor product structure. For this purpose, we define the local decomposition first for the unit square \(\square \mathrel{\vcenter{:}}= I \times I\) for \(I\mathrel{\vcenter{:}}= [0,1]\). Recall from Section 2.3 that for \(K \in \mathbb{N}\) we have \(\mathcal{Q}_K(\square) = \mathcal{P}_K(I) \otimes \mathcal{P}_K(I)\). We denote by \(\bar{\lambda}_0,\bar{\lambda}_1\) the (one-dimensional) barycentric coordinates of \(I\). We define the one-dimensional decomposition operator \(\frD^I \colon \mathcal{P}_K(I) \to \mathcal{P}_{K-1}(I)\) by \[\begin{align} \label{def:DI-1D} \frD^I_j(\bar{\lambda}^\alpha) = \frac{\alpha_j}{K} \bar{\lambda}^{\alpha-\bar e_j}, \end{align}\tag{32}\] where \(\alpha \in \setN_0^2\) with \(\abs{\alpha}=K\) and \(\bar{e}_0=(1,0)^\top\) and \(\bar{e}_1=(0,1)^\top\). Since this is the special case of 29 for one space dimension, by 30 we have that \[\begin{align} \label{eq:Di-identity-interval} \bar{\lambda}_j \frD_j^I(\bar{\lambda}^\alpha) = \bar{\lambda}_j(x_\alpha) \bar{\lambda}^\alpha \qquad \text{for j \in \{0,1\} and \abs{\alpha}=K.} \end{align}\tag{33}\]

Let \(\mathcal{V}(\square)\) denote the four vertices of \(\square\), namely \((0,0)^\top, (0,1)^\top, (1,0)^\top, (1,1)^\top\). Then for \(i =(i_1,i_2)^\top\in \mathcal{V}(\square)\) we define \(\frD^\square_i\colon \mathcal{Q}_K(\square) \to \mathcal{Q}_{K-1}(\square)\) by \[\begin{align} \label{eq:frD-tensor} \frD^\square_i(\bar{\lambda}^\alpha \otimes \bar{\lambda}^\beta) \mathrel{\vcenter{:}}= \frD^I_{i_1}(\bar{\lambda}^\alpha) \otimes \frD^I_{i_2}(\bar{\lambda}^\beta), \end{align}\tag{34}\] where \((f \otimes g)(x_1,x_2) \mathrel{\vcenter{:}}= f (x_1)\, g(x_2)\) and \(\alpha, \beta \in \mathbb{N}_0^2\) with \(\abs{\alpha}=\abs{\beta}=K\). The decomposition property is inherited from the one-dimensional decomposition.

For general quadrilaterals \(T \in \mathcal{T}\) the local decomposition operators \(\frD_i^T\) with \(i \in \mathcal{V}(T)\) are defined as the pullback of the decompositions on \(\square\), i.e., \[\begin{align} \label{def:DT-quad-local} \frD^T_i(v_K \circ B_T^{-1}) \mathrel{\vcenter{:}}= (\frD^\square_{B_T^{-1}(i)}v_K) \circ B_T^{-1}, \end{align}\tag{35}\] where \(B_T\colon [0,1]^2 \to T\) is an affine bilinear transformation as above and \(v_K \in \mathcal{Q}_K(\square)\). Recall from Section 2.1 that \(J_T\) is the Jacobian determinant of \(B_T\), which is chosen to be positive. Hence, by transformation 28 is equivalent to \[\begin{align} \label{est:K1-gen} \sum_{i \in \mathcal{V}(\square)} \int_{\square} \phi_i \abs{\frD_{i}^{\square}(v_K)}^2 {J_T} \, \mathrm{d}x \leq K_1 \int_{\square} \abs{v_K}^2 {J_T} \, \mathrm{d}x, \end{align}\tag{36}\] for any \(v_K \in \mathcal{Q}_K(\square)\). We now proceed to determine \(K_1\) such that 36 holds.

For general convex quadrilaterals \(B_T\) is affine bilinear, and hence \(J_T\) is affine, i.e., of the form \(J_T(x) = c_0 + c_1 x_1 + c_2 x_2\), for \(x \in \square\) and for \(c_i \in \mathbb{R}\) with \(i \in \{0,1,2\}\). Note that for strictly convex quadrilaterals one has \(c_0 > 0\). To prove 36 it suffices to investigate the case \(c_0 = 1\), and hence we consider \[\begin{align} \label{def:JT} J_T(x) = 1 + c_1 x_1 + c_2 x_2. \end{align}\tag{37}\] for \(c_1, c_2 \in \mathbb{R}\). Since \(J_T>0\) on \(\square\) we only need to show 36 for \(J_T\) as in 37 with \[\begin{align} \label{def:C-set} (c_1,c_2)^\top \in \mathcal{C} \mathrel{\vcenter{:}}= \set{(t_1, t_2)^\top \in \RR^2 \colon t_1 \geq -1, t_2 \geq -1, t_1 + t_2 \ge -1}, \end{align}\tag{38}\] as depicted in Figure 4.

In the special case of a parallelogram \(B_T\) is affine, and hence \(J_T\) is constant, meaning that \(c_1 = c_2 = 0\) in 37 .

In the following we consider the two cases, where \(T \in \mathcal{T}\)

1. is a parallelogram, i.e., \(c_1=c_2=0\);

2. is a general convex quadrilateral, i.e., \((c_1,c_2)^\top \in \mathcal{C}\).

Let us consider another special case, which will be used in the estimate for general convex quadrilaterals below.

With this auxiliary result we are in the position to find a bound \(K_1\) for general convex quadrilaterals.

3.3 Global decomposition↩︎

Let us now define the global decomposition operators \(\frD_i\colon \mathcal{L}^1_K(\mathcal{T}) \to \mathcal{L}^1_{K-1}(\omega_i)\) for vertex \(i \in \mathcal{V}\) satisfying 26 . For this purpose, we patch together the local decomposition operators, introduced in the previous subsection. For \(T \in \mathcal{T}\) with \(i \in T\) let \(j \in \mathcal{V}(T)\) be the local index of the vertex \(i\) in \(T\). We define \(\frD_i(v_K)\) for \(v_K \in \mathcal{L}^1(\mathcal{T})\) on \(T\) by \[\begin{align} \label{def:globalD} (\frD_iv_K)|_T \mathrel{\vcenter{:}}= \frD_j^T (v_K|_T), \end{align}\tag{58}\] with \(\frD_j^T\) as defined in 29 and 35 for triangles and for general (convex) quadrilaterals, respectively. This local definition ensures that \(\frD_i\colon \mathcal{L}^1_K(\mathcal{T}) \to \mathcal{L}^0_{K-1}(\omega_i)\). The following proposition shows that \(\frD_i\) maps to \(\mathcal{L}^1_{K-1}(\omega_i)\), i.e., to continuous functions.

3.4 Decay estimates↩︎

Based on Proposition [pro:C] the technique of [10] allows to derive a decay estimate for the \(L^2\)-projection \(\Pi \colon L^2(\Omega) \to \mathcal{L}^1_K(\mathcal{T})\) with explicit decay rate. The rate is independent of the shape regularity and only depends on the polynomial degree \(K\). Decay estimates, in turn, are key to establish \(L^p\)- and \(W^{1,p}\)-stability estimates with the maximal function approach developed in [2], see Section 3.5 below.

3.5 Stability results↩︎

Based on the decay estimate in Proposition [pro:decay] we obtain the following stability estimates directly as in [2].

Recall that for any family of hybrid meshes generated by the adaptive refinement routines Q-RB and Q-RG starting from a quadrilateral mesh \(\mathcal{T}_0\) the shape-regularity constant is uniformly bounded by \(4\chi(\mathcal{T}_0)\), see Lemma 2. Moreover, by Lemma 3 and Remark [rem:general-grading] the grading constant is \(\gamma=2\), see 19 . For the convenience of the reader let us state the stability results without additional weights in this special case.

Corollary 1 extends the results by [1] on \(W^{1,2}\)-stability for the Q-RG and Q-RB refinement from \(K \in \set{2,\dots,9}\) to \(K \geq 2\). Moreover, we allow for quadrilateral initial meshes, while [1] is restricted to P-quadrilateral initial meshes. Additionally, we cover also the case of \(L^p\)- and \(W^{1,p}\)-stability.

Acknowledgements↩︎

The work by T.T. was supported by the German Research Foundation (DFG) via grant TRR 154, subproject C09, project number 239904186, and by the Graduate School CE within Computational Engineering at Technische Universität Darmstadt.

References↩︎