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,
$$begin{aligned} bigl(u – Pi ^{{F M}}_{h} ubigr) (p_{i}) = 0,qquad int _{e_{i}} frac{partial}{partial n}bigl(u – Pi ^{{F M}}_{h} ubigr) ,mathrm{d}s = 0quad (i = 1,2,3). end{aligned}$$
The Fujino–Morley interpolation has the property that (see, e.g., [4, 5])
$$begin{aligned} bigllangle u – Pi ^{{F M}}_{h} u, v_{h} bigrrangle _{h} = 0, quadforall v_{h} in V^{{F M}}_{h}(K). end{aligned}$$
(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.
$$begin{aligned} bigllVert u – Pi ^{{scriptsize F M}}_{h} u bigrrVert _{ infty,K} le C^{{F M}}_{h} bigllvert u – Pi ^{{scriptsize F M}}_{h} u bigrrvert _{2,K},quad forall u in V^{L}(K). end{aligned}$$
(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
$$begin{aligned} lVert u rVert _{infty,K}& = bigllVert Pi _{h}^{{ scriptsize F M}}u + uPi _{h}^{{scriptsize F M}}u bigrrVert _{ infty,K} \ & le bigllVert Pi _{h}^{{scriptsize F M}}u bigrrVert _{ infty,K} + bigllVert uPi _{h}^{{scriptsize F M}}u bigrrVert _{infty,K} \ & le frac{ lvert Pi _{h}^{{scriptsize F M}}u rvert _{2,K}}{sqrt{lambda _{h}}} + C_{h}^{{F M}} bigllvert uPi _{h}^{{scriptsize F M}}u bigrrvert _{2,K} \ & le sqrt{frac{1}{lambda _{h}} + bigl(C_{h}^{{F M}} bigr)^{2}} sqrt{ bigllvert Pi _{h}^{{scriptsize F M}}u bigrrvert ^{2}_{2,K} + bigllvert uPi _{h}^{{scriptsize F M}}u bigrrvert ^{2}_{2,K}} . end{aligned}$$
From the orthogonality in (9), we have
$$begin{aligned} bigllvert Pi _{h}^{{scriptsize F M}}u bigrrvert ^{2}_{2,K} + bigllvert uPi _{h}^{{scriptsize F M}}u bigrrvert ^{2}_{2,K} = lvert u rvert _{2,K}^{2}. end{aligned}$$
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 )).
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}):
$$begin{aligned} C^{{F M}}_{mathrm{res}}({K_{h}}):= sup _{u in H^{2}( {K_{h}})} frac{ lVert uPi _{h}^{{scriptsize F M}}u rVert _{infty, {K_{h}}}}{ lvert uPi _{h}^{{scriptsize F M}}u rvert _{2,{K_{h}}}} = sup_{w in W_{1}} frac{ lVert w rVert _{infty, {K_{h}}}}{ lvert w rvert _{2,{K_{h}}}}. end{aligned}$$
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.
Estimation of (lambda _{h}) by solving the finitedimensional 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
$$begin{aligned} lambda _{h} =operatorname{min} mathbf{x}^{T} mathbf{A} mathbf{x}, quadmbox{subject to}quad BiggllVert sum _{i=1}^{M} mathbf{x}_{i} phi _{i} BiggrrVert _{infty,K} ge 1 , end{aligned}$$
(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
\mathbf{x}\in {}^{}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 convexhull 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 convexhull property; refer to, e.g., [18, 19] for detailed discussion.
Convexhull 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
$$begin{aligned} J^{(n)}_{i,j,k}ge 0, qquadsum_{i+j+k = n} J^{(n)}_{i,j,k}=1, end{aligned}$$
we can easily obtain the following convexhull 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=6times # {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 convexhull property of the Bernstein polynomials, the following inequality holds:
$$begin{aligned} 1 le BiggllVert sum_{i=1}^{M} mathbf{x}_{i} phi _{i} BiggrrVert _{infty,K} le lVert mathbf{Bx} rVert _{ infty}. end{aligned}$$
Based on this inequality, we propose a new optimization by relaxing the constraint condition of (13):
$$begin{aligned} lambda _{h,B} = operatorname{min} mathbf{x}^{T} mathbf{Ax}, quadmbox{subject to}quad lVert mathbf{Bx} rVert _{infty} ge 1. end{aligned}$$
(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 positivedefinite, let us consider the Cholesky decomposition of A: (mathbf{A} = mathbf{R}^{T}mathbf{R}), where R is an (Mtimes 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 by^{Footnote 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
$$begin{aligned} lVert widehat{mathbf{B}}bar{mathbf{y}} rVert _{ infty }= max _{i=1,ldots,N} bigllvert {b^{T}_{i}} bar{ mathbf{y}} bigrrvert ge bigllvert {b^{T}_{mathrm{max}}} bar{ mathbf{y}} bigrrvert = 1. end{aligned}$$
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,
$$begin{aligned} 1 le max_{i=1,ldots,N} bigllvert {b_{i}}^{T} mathbf{y} bigrrvert le max_{i=1,ldots,N} lVert b_{i} rVert _{2} lVert mathbf{y} rVert _{2} = lVert b_{mathrm{max}} rVert _{2} lVert mathbf{y} rVert _{2}. end{aligned}$$
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:
$$begin{aligned} frac{lambda _{h}}{1+ (C_{h}^{{F M}})^{2} lambda _{h} } ge lambda _{h} bigl(1(1/N)^{2}bigr) quadiffquad 1 ge bigl(N^{2}1bigr) bigl(C_{h}^{{F M}}bigr)^{2} lambda _{h}. end{aligned}$$
(19)
For a small value of (h=1/N), we have
$$begin{aligned} bigl(N^{2}1bigr) bigl(C_{h}^{{F M}} bigr)^{2} approx bigl(N C_{h}^{{F M}} bigr)^{2} = bigl(C_{mathrm{res}}^{ {F M}}( {K_{h}})bigr)^{2}, lambda _{h}approx lambda =bigl(C^{L}( {K_{h}})bigr)^{2}. end{aligned}$$
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:
$$begin{aligned} bigllVert uPi ^{L} u bigrrVert _{infty,K_{1,pi /2,h}} le 0.41596 h lvert u rvert _{2,K_{1,pi /2,h}}. end{aligned}$$
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.
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:
$$begin{aligned} min_{f in P_{d}(K)} sum_{i=1}^{n} bigllvert f(p_{i}) – u_{h}(p_{i}) bigrrvert ^{2} quadbigl(n: # {mbox{nodes of triangulation}} bigr), end{aligned}$$
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:
$$begin{aligned} 0.40432 le C^{L} biggl(1,frac{pi}{2} biggr) le 0.41596. end{aligned}$$
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