In the previous section, we obtained explicit bounds for the interpolation constant for triangles of general shape. Basically, such bounds from theoretical analysis only provide a raw bound for the objective constant. In this section, we propose a numerical algorithm to obtain the optimal estimation of the constant \(C^{L}(K)\) for specific triangles.
Let us define the space \(V^{L}(K):= \{u \in H^{2}(K) \mid u(p_{i}) = 0 \ (i = 1,2,3) \}\). Let \(\mathcal{T}^{h}\) be a triangulation of the domain K and define the space
$$\begin{aligned} V^{{F M}}_{h}(K):={}& \biggl\{ v \mid v \vert _{K_{h}} \in P_{2}(K_{h}), \forall {K_{h}} \in \mathcal{T}^{h}; v(p_{i}) = 0\ (i = 1,2,3); v \mbox{ is continuous} \\ &\mbox{at the nodes; } { \int _{e} \biggl(\frac{\partial v}{\partial \overrightarrow{n}} \bigg|_{K_{h}} - \frac{\partial v}{\partial \overrightarrow{n}} \bigg|_{K_{h}'} \biggr) \,\mathrm{d}s = 0 \mbox{ for each } e = K_{h} \cap K_{h'} } \biggr\} . \end{aligned}$$
For \(u_{h}, v_{h} \in V^{{F M}}_{h}(K)\), define the discretized \(H^{2}\)-inner product and seminorm by
$$\begin{aligned} \langle u_{h}, v_{h} \rangle _{h}:= \sum _{{K_{h}} \in \mathcal{T}^{h}} \int _{{K_{h}}} D^{2}{u_{h}|_{K_{h}}} \cdot D^{2} {v_{h}|_{K_{h}}} \,\mathrm{d} {K_{h}},\quad \lvert u_{h} \rvert _{2,K}:= \sqrt{ \langle u_{h}, u_{h} \rangle _{h}} . \end{aligned}$$
Let us define the two quantities over the triangle K:
$$\begin{aligned} \lambda (K):= \inf_{u \in V^{L}(K)} \frac{ \lvert u \rvert _{2,K}^{2}}{ \lVert u \rVert _{\infty,K}^{2}},\qquad \lambda _{h}(K):= \min_{u_{h} \in V_{h}^{{F M}}(K)} \frac{ \lvert u_{h} \rvert _{2,K}^{2}}{ \lVert u_{h} \rVert _{\infty,K}^{2}}. \end{aligned}$$
(8)
Note that \(C^{L}(K) = \sqrt{\lambda (K)}^{-1}\) holds. In Theorem 3.1, we describe the algorithm to bound λ by using \(\lambda _{h}\).
Given \(u \in H^{2}(K)\), the Fujino–Morley interpolation \(\Pi ^{{F M}}_{h} u\) is a function satisfying
$$\begin{aligned} \Pi ^{{F M}}_{h} u \in V^{{F M}}_{h}(K);\qquad \Pi ^{{F M}}_{h} u |_{{K_{h}}} \in P_{2}(K_{h}),\quad \forall {K_{h}} \in \mathcal{T}^{h}, \end{aligned}$$
and at the vertices \(p_{i}\) and edges \(e_{i}\) of K,
$$\begin{aligned} \bigl(u - \Pi ^{{F M}}_{h} u\bigr) (p_{i}) = 0,\qquad \int _{e_{i}} \frac{\partial}{\partial n}\bigl(u - \Pi ^{{F M}}_{h} u\bigr) \,\mathrm{d}s = 0\quad (i = 1,2,3). \end{aligned}$$
The Fujino–Morley interpolation has the property that (see, e.g., [4, 5])
$$\begin{aligned} \bigl\langle u - \Pi ^{{F M}}_{h} u, v_{h} \bigr\rangle _{h} = 0, \quad\forall v_{h} \in V^{{F M}}_{h}(K). \end{aligned}$$
(9)
Let \(V(h):= \{u + u_{h} \mid u \in V^{L}(K), u_{h} \in V^{ {F M}}_{h}(K) \}\). Thus, it is easy to see that the Fujino–Morley interpolation is just the projection \(P_{h}: V(h)\to V^{{F M}}_{h}(K)\) with respect to the inner product \(\langle \cdot, \cdot \rangle _{h}\).
Below, let us introduce the theorem that provides an explicit lower bound of λ. Such a result is inspired by the idea of [17] for the lower bounds of eigenvalue problems.
Let \(C_{h}^{{F M}}\) be a quantity that makes the following inequality hold.
$$\begin{aligned} \bigl\lVert u - \Pi ^{{\scriptsize F M}}_{h} u \bigr\rVert _{ \infty,K} \le C^{{F M}}_{h} \bigl\lvert u - \Pi ^{{\scriptsize F M}}_{h} u \bigr\rvert _{2,K},\quad \forall u \in V^{L}(K). \end{aligned}$$
(10)
The existence of \(C_{h}^{{F M}}\) is confirmed by the argument in Sect. 3.1.
Theorem 3.1
With the quantity \(C_{h}^{{F M}}\), we have a lower bound of \(\lambda (K)\) as follows:
$$\begin{aligned} \lambda (K) \ge \frac{\lambda _{h}}{1+(C_{h}^{{F M}})^{2}\lambda _{h}} . \end{aligned}$$
(11)
Proof
For any \(u \in V^{L}(K)\), noting that \(\lvert \Pi ^{{\scriptsize F M}}_{h} u \rvert _{2,K} \ge \sqrt{\lambda _{h}} \lVert \Pi ^{{\scriptsize F M}}_{h} u \rVert _{\infty,K}\) and applying the inequality (10), we have
$$\begin{aligned} \lVert u \rVert _{\infty,K}& = \bigl\lVert \Pi _{h}^{{ \scriptsize F M}}u + u-\Pi _{h}^{{\scriptsize F M}}u \bigr\rVert _{ \infty,K} \\ & \le \bigl\lVert \Pi _{h}^{{\scriptsize F M}}u \bigr\rVert _{ \infty,K} + \bigl\lVert u-\Pi _{h}^{{\scriptsize F M}}u \bigr\rVert _{\infty,K} \\ & \le \frac{ \lvert \Pi _{h}^{{\scriptsize F M}}u \rvert _{2,K}}{\sqrt{\lambda _{h}}} + C_{h}^{{F M}} \bigl\lvert u-\Pi _{h}^{{\scriptsize F M}}u \bigr\rvert _{2,K} \\ & \le \sqrt{\frac{1}{\lambda _{h}} + \bigl(C_{h}^{{F M}} \bigr)^{2}} \sqrt{ \bigl\lvert \Pi _{h}^{{\scriptsize F M}}u \bigr\rvert ^{2}_{2,K} + \bigl\lvert u-\Pi _{h}^{{\scriptsize F M}}u \bigr\rvert ^{2}_{2,K}} . \end{aligned}$$
From the orthogonality in (9), we have
$$\begin{aligned} \bigl\lvert \Pi _{h}^{{\scriptsize F M}}u \bigr\rvert ^{2}_{2,K} + \bigl\lvert u-\Pi _{h}^{{\scriptsize F M}}u \bigr\rvert ^{2}_{2,K} = \lvert u \rvert _{2,K}^{2}. \end{aligned}$$
Thus,
$$\begin{aligned} \lVert u \rVert _{\infty,K} \le \sqrt{ \frac{1+(C_{h}^{{F M}})^{2}\lambda _{h}}{\lambda _{h}}} \lvert u \rvert _{2,K}, \quad \forall u \in V^{L}(K). \end{aligned}$$
From the definition of λ in (8), we draw the conclusion. □
To apply Theorem 3.1 for bounding λ, an explicit value of \(C_{h}^{{F M}}\) is needed. Below, let us describe the way to obtain this explicit value by utilizing the raw bound of \(C^{L}(\alpha,\theta )\).
3.1 Explicit estimation of \(C^{{F M}}_{h}\)
To have an explicit value of \(C^{{F M}}_{h}\), we first define the quantity \(C^{{F M}}_{\mathrm{res}}({K_{h}})\) for each element \({K_{h}}\) in the triangulation \(\mathcal{T}^{h}\):
$$\begin{aligned} C^{{F M}}_{\mathrm{res}}({K_{h}}):= \sup _{u \in H^{2}( {K_{h}})} \frac{ \lVert u-\Pi _{h}^{{\scriptsize F M}}u \rVert _{\infty, {K_{h}}}}{ \lvert u-\Pi _{h}^{{\scriptsize F M}}u \rvert _{2,{K_{h}}}} = \sup_{w \in W_{1}} \frac{ \lVert w \rVert _{\infty, {K_{h}}}}{ \lvert w \rvert _{2,{K_{h}}}}. \end{aligned}$$
Here, \(W_{1}:= \{w \in H^{2}({K_{h}}) \mid w(p_{i}) = 0, \int _{e_{i}} \frac{\partial w}{\partial n }\,\mathrm{d}s = 0 \ (i = 1,2,3) \}\). Noting that \(W_{1} \subseteq W_{2}\) for \(W_{2}:= \{ w \in H^{2}({K_{h}}) \mid w(p_{i}) = 0 \ (i=1,2,3) \}\), from the definition of \(C^{L}\) in (3), we have
$$\begin{aligned} C^{{F M}}_{\mathrm{res}}({K_{h}}) \le \sup _{w \in W_{2}} \frac{ \lVert w \rVert _{\infty, {K_{h}}}}{ \lvert w \rvert _{2,{K_{h}}}} = C^{L}( {K_{h}}). \end{aligned}$$
Then, the following \(C^{{F M}}_{h}\) with an upper bound makes certain (10) holds:
$$\begin{aligned} C^{{F M}}_{h}:= \max_{{K_{h}} \in \mathcal{T}^{h}} C^{ {F M}}_{\mathrm{res}}({K_{h}}) \Bigl( \le \max _{{K_{h}} \in \mathcal{T}^{h}} C^{L}({K_{h}}) \Bigr). \end{aligned}$$
(12)
Remark 3.1
Let \(\mathcal{T}^{h}\) be a uniform triangulation of a right isosceles triangle; see a sample mesh in Fig. 8. We choose an explicit upper bound of \(C^{{F M}}_{h} \) as \(C^{{F M}}_{h} \le 1.3712h\), since for each \({K_{h}} \in \mathcal{T}^{h}\), \(C^{{F M}}_{\mathrm{res}} \le C^{L}({K_{h}}) \le 1.3712h\), where h is the leg length of each right triangle element.
3.2 Estimation of \(\lambda _{h}\) by solving the finite-dimensional optimization problem
In this subsection, we present a method to estimate \(\lambda _{h}\), which is required in Theorem 3.1 for bounding λ. Let \(M:= \operatorname{Dim}(V^{{F M}}_{h})\). The estimation of \(\lambda _{h}\) is equivalent to finding the solution to the optimization problem
$$\begin{aligned} \lambda _{h} =\operatorname{min} \mathbf{x}^{T} \mathbf{A} \mathbf{x}, \quad\mbox{subject to}\quad \Biggl\lVert \sum _{i=1}^{M} \mathbf{x}_{i} \phi _{i} \Biggr\rVert _{\infty,K} \ge 1 , \end{aligned}$$
(13)
where the components \(a_{ij}\) of A are given by \(a_{ij} = \langle \phi _{i}, \phi _{j} \rangle _{h}\), \(\{\phi _{i} \}_{i=1,\ldots,M}\) are the basis functions for the Fujino–Morley space \(V^{{F M}}_{h}\), and denotes the Fujino–Morley coefficient vector of \(u_{h} \in V^{{F M}}_{h}\).
To solve the optimization problem (13) is not an easy task since the \(L^{\infty}\)-norm of the function appears in the constraint. Here, we introduce the technique to apply Bernstein polynomials and their convex-hull property to solve the problem. Strictly speaking, a new optimization problem (14) utilizing the Bernstein polynomials will be formulated to provide a lower bound for the solution of (13).
As preparation, let us introduce the definition of Bernstein polynomials along with the convex-hull property; refer to, e.g., [18, 19] for detailed discussion.
Convex-hull property of Bernstein polynomials
Given a triangle K, let \((u,v,w)\) be the barycentric coordinates for a point x in K. A Bernstein polynomial p of degree n over a triangle K is defined by
$$\begin{aligned} {p}:= \sum_{i+j+k = n} d_{i,j,k} J^{(n)}_{i,j,k},\qquad J^{(n)}_{i,j,k}(x) := \frac{n!}{i!j!k!} u^{i} v^{j} w^{k}. \end{aligned}$$
Here, \(J^{(n)}_{i,j,k}(x)\) are the Bernstein basis polynomials; the coefficients \(d_{i,j,k}\) are the control points of p. Noting that
$$\begin{aligned} J^{(n)}_{i,j,k}\ge 0, \qquad\sum_{i+j+k = n} J^{(n)}_{i,j,k}=1, \end{aligned}$$
we can easily obtain the following convex-hull property of Bernstein polynomials:
$$\begin{aligned} \lVert{p} \rVert _{\infty,K} \le \max \lvert d_{i,j,k} \rvert. \end{aligned}$$
Given \(u_{h} \in V^{{F M}}_{h}(K)\), for each \({K_{h}} \in \mathcal{T}^{h}\), \(u_{h}|_{{K_{h}}} \in P_{2}({K_{h}})\) can be represented by the Bernstein basis polynomials of degree two. Let B be the \(N \times M\) matrix that transforms the Fujino–Morley coefficients x to the Bernstein coefficients \(d^{B}\). Note that \(u_{h}\) is regarded as a piecewise Bernstein polynomial so that its Bernstein coefficient vector \(d^{B}\) has the dimension \(N=6\times \# \{elements\}\). The dimension of \(d^{B}\) can be further reduced considering the continuity of \(u_{h}\) at the vertices of the triangulation. However, it is difficult to utilize the constraints of \(u_{h}\) that cross the edges to reduce the dimension N. From the convex-hull property of the Bernstein polynomials, the following inequality holds:
$$\begin{aligned} 1 \le \Biggl\lVert \sum_{i=1}^{M} \mathbf{x}_{i} \phi _{i} \Biggr\rVert _{\infty,K} \le \lVert \mathbf{Bx} \rVert _{ \infty}. \end{aligned}$$
Based on this inequality, we propose a new optimization by relaxing the constraint condition of (13):
$$\begin{aligned} \lambda _{h,B} = \operatorname{min} \mathbf{x}^{T} \mathbf{Ax}, \quad\mbox{subject to}\quad \lVert \mathbf{Bx} \rVert _{\infty} \ge 1. \end{aligned}$$
(14)
The solution to problem (14) provides a lower bound for (13), i.e., \(\lambda _{h} \ge \lambda _{h,B}\).
Below, we propose an algorithm to solve the problem (14). Since A is positive-definite, let us consider the Cholesky decomposition of A: \(\mathbf{A} = \mathbf{R}^{T}\mathbf{R}\), where R is an \(M\times M\) upper triangular matrix. Then, by letting \(\mathbf{y}:= \mathbf{Rx}\) and \(\mathbf{\widehat{B}}:= \mathbf{BR}^{-1}\), problem (14) becomes
$$\begin{aligned} \lambda _{h,B}=\operatorname{min} \mathbf{y}^{T} \mathbf{y},\quad \mbox{subject to}\quad \lVert \widehat{\mathbf{B}} \mathbf{y} \rVert _{\infty} \ge 1. \end{aligned}$$
(15)
The following lemma shows the solution for problem (15).
Lemma 3.1
Let \({b^{T}_{i}}\) (\(i=1,\ldots,N\)) be the ith row of \(\widehat{\mathbf{B}}\) and \(b^{T}_{\mathrm{max}}\) be a row of \(\widehat{\mathbf{B}}\) satisfying \(\lVert b_{\mathrm{max}} \rVert _{2} = \max_{i=1,\ldots,N} \lVert b_{i} \rVert _{2}\). Then, the optimal value of problem (15) is given byFootnote 1
$$\begin{aligned} \lambda _{h,B} =\frac{1}{ \lVert b_{\mathrm{max}} \rVert _{2}^{2}}. \end{aligned}$$
Proof
Let \(S:= \{\mathbf{y} \mid \lVert \widehat{\mathbf{B}} \mathbf{y} \rVert _{\infty }\ge 1 \}\) and \(\bar{\mathbf{y}}:= \lVert b_{\mathrm{max}} \rVert _{2}^{-2}b_{\mathrm{max}}\). Then, we have \(\bar{\mathbf{y}}\in S\) because
$$\begin{aligned} \lVert \widehat{\mathbf{B}}\bar{\mathbf{y}} \rVert _{ \infty }= \max _{i=1,\ldots,N} \bigl\lvert {b^{T}_{i}} \bar{ \mathbf{y}} \bigr\rvert \ge \bigl\lvert {b^{T}_{\mathrm{max}}} \bar{ \mathbf{y}} \bigr\rvert = 1. \end{aligned}$$
Hence,
$$\begin{aligned} \min_{\mathbf{y}\in S} \mathbf{y}^{T}\mathbf{y} \le \bar{\mathbf{y}}^{T} \bar{\mathbf{y}} = \frac{1}{ \lVert b_{\mathrm{max}} \rVert _{2}^{2}}. \end{aligned}$$
(16)
For any \(\mathbf{y} \in S\), from the Cauchy–Schwarz inequality,
$$\begin{aligned} 1 \le \max_{i=1,\ldots,N} \bigl\lvert {b_{i}}^{T} \mathbf{y} \bigr\rvert \le \max_{i=1,\ldots,N} \lVert b_{i} \rVert _{2} \lVert \mathbf{y} \rVert _{2} = \lVert b_{\mathrm{max}} \rVert _{2} \lVert \mathbf{y} \rVert _{2}. \end{aligned}$$
Thus,
$$\begin{aligned} \frac{1}{ \lVert b_{\mathrm{max}} \rVert _{2}^{2}} \le \min_{ \mathbf{y}\in S} \mathbf{y}^{T}\mathbf{y}. \end{aligned}$$
(17)
From (16) and (17), we draw the conclusion. □
Note that the diagonal elements of \(\mathbf{B}\mathbf{A}^{-1}\mathbf{B}^{T}=\widehat{\mathbf{B}} \widehat{\mathbf{B}}^{T}\) correspond to \(\|b_{i}\|_{2}^{2}\) (\(i=1, \ldots, N\)). Therefore, we can solve problem (14) without performing the Cholesky decomposition of A, as shown by the following lemma.
Lemma 3.2
Let \(\mathbf{D}:=\mathbf{B} \mathbf{A}^{-1}\mathbf{B}^{T}\). The optimal value of (14) is given by
$$\begin{aligned} \lambda _{h,B}=\frac{1}{\max (\mathrm{diag}(\mathbf{D}))}, \end{aligned}$$
where \(\mathrm{diag}(\mathbf{D})\) is the diagonal elements of D.
Theorem 3.1 gives a lower bound for λ. Since \(C^{L}(K) = \sqrt{\lambda (K)}^{-1}\), this lower bound is used to obtain an upper bound for \(C^{L}(K)\). Below, let us summarize the procedure to obtain a lower bound for λ.
Algorithm for calculating the lower bound of
\(\lambda (K)\)
-
a.
Set up the FEM space \(V^{{F M}}_{h}(K)=\operatorname{span}\{\phi _{i}\}_{i=1}^{M}\) over a triangulation of the triangle domain K.
-
b.
Assemble the global matrix \(\mathbf{A} = ( a_{ij} )_{M \times M}\) (\(a_{ij} = \langle \phi _{i}, \phi _{j} \rangle _{h}\)) and the transformation matrix B from Fujino–Morley coefficients to Bernstein coefficients.
-
c.
Apply Lemma 2.3 to obtain a raw bound for \(C^{{F M}}_{h}\).
-
d.
Apply Lemma 3.1 or Lemma 3.2 to calculate \(\lambda _{h,B} (\le \lambda _{h})\).
-
e.
The lower bound for λ is obtained through Theorem 3.1 by using \(\lambda _{h,B}\) and the upper bound of \(C^{{F M}}_{h}\).
Using uniform triangulation of a domain K, a direct estimation of the lower bound for λ without using \(C^{{F M}}_{h}\) is available.
Corollary 3.1
For a uniform triangulation of \(K=K_{\alpha,\theta,h}\) with N subdivisions for each side, the following holds:
$$\begin{aligned} \lambda (K) \ge \lambda _{h}\bigl(1-(1/N)^{2} \bigr). \end{aligned}$$
(18)
Proof
Since \((C^{L}(K))^{2} = 1/\lambda (K)\) and each \({K_{h}} \in \mathcal{T}^{h}\) is similar to K, we have,
$$\begin{aligned} \lambda (K) \ge \frac{\lambda _{h}}{1+ (C_{h}^{{F M}})^{2} \lambda _{h} } \ge \frac{\lambda _{h}}{1+(C^{L}({K_{h}}))^{2} \lambda _{h} } = \frac{\lambda _{h}}{ 1 + (1/N)^{2} \lambda _{h}/\lambda (K)}. \end{aligned}$$
The conclusion is achieved by sorting the inequality. □
Remark 3.2
Theoretically, for a refined uniform triangulation, the lower bound (11) using \(C_{h}^{{F M}}\) is sharper (i.e., larger) than (18). This can be confirmed by utilizing the following relation:
$$\begin{aligned} \frac{\lambda _{h}}{1+ (C_{h}^{{F M}})^{2} \lambda _{h} } \ge \lambda _{h} \bigl(1-(1/N)^{2}\bigr) \quad\iff\quad 1 \ge \bigl(N^{2}-1\bigr) \bigl(C_{h}^{{F M}}\bigr)^{2} \lambda _{h}. \end{aligned}$$
(19)
For a small value of \(h=1/N\), we have
$$\begin{aligned} \bigl(N^{2}-1\bigr) \bigl(C_{h}^{{F M}} \bigr)^{2} \approx \bigl(N C_{h}^{{F M}} \bigr)^{2} = \bigl(C_{\mathrm{res}}^{ {F M}}( {K_{h}})\bigr)^{2}, \lambda _{h}\approx \lambda =\bigl(C^{L}( {K_{h}})\bigr)^{-2}. \end{aligned}$$
Thus, the second equality of (19) holds due to \(C_{\mathrm{res}}^{{F M}}({K_{h}}) < C^{L}({K_{h}})\). However, in practical computation, the raw estimate of \(C_{\mathrm{res}}^{{F M}}({K_{h}})\) will produce a worse bound of λ than (18).
Using Corollary 3.1, the following steps are modified from the algorithm to obtain a lower bound for λ, without using the quantity of \(C_{h}^{{F M}}\):
Revision of algorithm for calculating the lower bound of
\(\lambda (K)\)
-
c*.
Apply Lemma 3.1 or Lemma 3.2 to calculate \(\lambda _{h,B} (\le \lambda _{h})\).
-
d*.
Solve the lower bound for λ using Corollary 3.1 along with \(\lambda _{h,B}\).
Remark 3.3
To compare the efficiencies of the two formulas (11) and (18), we apply them to estimate λ for a unit right isosceles \(K_{1,\pi /2}\). By using uniform triangulation of size \(h=1/64\), the estimate (11) gives \(\lambda \ge 5.7659\) and (18) gives a sharper bound as \(\lambda \ge 5.7798\). Hence, a sharper upper bound is obtained using (18) and we have the following estimation:
$$\begin{aligned} \bigl\lVert u-\Pi ^{L} u \bigr\rVert _{\infty,K_{1,\pi /2,h}} \le 0.41596 h \lvert u \rvert _{2,K_{1,\pi /2,h}}. \end{aligned}$$
As a comparison, the result (5) will yield a raw bound as \(C^{L}(1,\pi /2,h) \le 1.3712h\).
For a triangle \(K_{\alpha,\theta}\) with two fixed vertices \(p_{1}(0,0), p_{2}(1,0)\), let us vary the vertex \(p_{3}(x, y)\) and calculate the approximate value of \(C^{L}(\alpha,\theta )\) for each position of \(p_{3}\). Note that \(C^{L}\) can be regarded as a function with respect to the coordinate \((x,y)\) of \(p_{3}\), which is denoted by \(C^{L}(x,y)\). In Fig. 9, we draw the contour lines of \(C^{L}(x,y)\), where the abscissa and the ordinate denote x- and y- coordinates of \(p_{3}\), respectively.
3.3 Lower bound of the constant
To confirm the precision of the obtained estimation for the Lagrange interpolation constant, the lower bounds of the constants are calculated. Let \(u_{h}\) be the function obtained by numerical computation solving the minimization problem. To obtain the lower bound, an appropriate polynomial f over K of higher degree d is selected by solving the minimization problem below:
$$\begin{aligned} \min_{f \in P_{d}(K)} \sum_{i=1}^{n} \bigl\lvert f(p_{i}) - u_{h}(p_{i}) \bigr\rvert ^{2} \quad\bigl(n: \# \{\mbox{nodes of triangulation}\} \bigr), \end{aligned}$$
where \(p_{i}\) denote the nodes of the triangulation of K. From the definition of \(\lambda (K)\) in (8) and the relation \(C^{L}(K)=1/\sqrt{\lambda (K)}\), we have a lower bound of \(C^{L}(K)\) as follows:
$$\begin{aligned} C^{L}(K) \ge \frac{ \lVert f \rVert _{\infty,K}}{ \lvert f \rvert _{2,K}}. \end{aligned}$$
Remark 3.4
For the unit right isosceles triangle \(K_{1, \pi /2}\), the upper bound for the constant is obtained by solving the optimization problem with mesh size \(1/64\). Meanwhile, the lower bound of the constant is obtained by using a polynomial of degree 9. The two side bounds read:
$$\begin{aligned} 0.40432 \le C^{L} \biggl(1,\frac{\pi}{2} \biggr) \le 0.41596. \end{aligned}$$