1 Introduction

Consider a family \(\mathcal {F} = \{\mathcal {T}_h\}_{h \to 0}\) of face-to-face triangulations \(\mathcal {T}_h\) of a bounded polygonal domain. In 1957, J. Synge proved that linear triangular finite elements yield the optimal interpolation order in the C-norm provided the maximum angle condition is satisfied, i.e., there exists a constant γ 0 < π such that for any triangulation \(\mathcal {T}_h \in \mathcal {F}\) and any triangle \(T \in \mathcal {T}_h\) one has (see [27])

$$\displaystyle \begin{aligned} \gamma_T \le \gamma_0, {} \end{aligned} $$
(1)

where γ T is the maximum angle of T. In 1975/1976, Babuška and Aziz [3], Barnhill and Gregory [4], and Jamet [14] independently derived the optimal interpolation order in the energy norm of finite element approximations under the condition (1). Later the maximum angle condition was investigated in various norms in [1, 2, 15,16,17,18, 22, 24, 25].

In 1992, the condition (1) was generalized by Křížek [19] to tetrahedral elements as follows:

There exists a constant γ 0 < π such that for any face-to-face tetrahedralization \(\mathcal {T}_h \in \mathcal {F}\) and any tetrahedron \(T \in \mathcal {T}_h\) one has

$$\displaystyle \begin{aligned} \gamma_{\mathrm{D}} \le \gamma_0 \quad \text{and} \quad \gamma_{\mathrm{F}} \le \gamma_0, {} \end{aligned} $$
(2)

where γ D is the maximum dihedral angles between faces of T and γ F is the maximum angle in all four triangular faces of T. According to [20], the associated finite element approximations preserve the optimal interpolation order in the H 1-norm under the condition (1).

Note that degenerated tetrahedral elements have a lot of real-life technical applications. For example, in calculation of physical fields in eletrical rotary machines, see [20]. Flat tetrahedral elements are also used to approximate thin slots, layers, or gaps. Moreover, they are suitable when the true solution of some problem changes more rapidly in one direction than in another direction (e.g. in anisotropic materials) [1].

From Fig. 1 we observe that the condition (2) is satisfied for a needle, splinter, and wedge tetrahedron. For the other degenerated tetrahedra from Fig. 1, the interpolation error may diverge in the H 1-norm. On the other hand, the finite element method may converge in the H 1-norm, see [12]. Hence, (2) represents only a sufficient (and not necessary) condition for the convergence of the finite element method, see [21,22,23, 25].

Fig. 1
figure 1

Classification of degenerated tetrahedra according to [7, 9]

Full size image

Let us point out that the two conditions in (2) are independent. For instance, for a sliver (and also cap) tetrahedron the condition γ D ≤ γ 0 does not hold while γ F ≤ γ 0 holds. On the other hand, for a spike tetrahedron γ D ≤ γ 0 holds and γ F ≤ γ 0 is violated.

Since there are six dihedral angles of each tetrahedron and twelve angles between its adjacent edges, a direct generalization of (2) into higher dimensions would be technically quite complicated. Therefore, in the next section we introduce another concept which is based on the d-dimensional sine for d > 1. We will survey the main results from our previous paper [13], see also [5, 6].

2 The Maximum Angle Condition in Higher Dimensions

Recall that a d-simplex S in R d, d ∈{1, 2, 3, … }, is the convex hull of d + 1 vertices A 0, A 1, …, A d that do not belong to the same (d − 1)-dimensional hyperplane, i.e.,

$$\displaystyle \begin{aligned}S = \mathrm{conv} \{A_0, A_1, \dots , A_{d}\}. \end{aligned}$$

Let

$$\displaystyle \begin{aligned}F_i = \mathrm{conv} \{A_0, \dots , A_{i-1}, A_{i+1}, \dots ,A_{d}\} \end{aligned}$$

be the facet of S opposite to the vertex A i for i ∈{0, …, d}.

In 1978, Eriksson has introduced a generalization of the sine function to an arbitrary d-dimensional spatial angle, see [10, p. 74].

Definition 1

Let \(\hat A_i\) be the angle at the vertex A i of the simplex S. Then d-sine of the angle \(\hat A_i\) for d > 1 is given by

$$\displaystyle \begin{aligned} \sin_d (\hat A_{i} | A_0 A_1 \dots A_{d}) = \frac{ d^{d-1} \, (\mathrm{meas}\, _d S)^{d-1} }{(d-1) ! \, \boldsymbol{\Pi}_{j = 0, j \neq i}^{d} \mathrm{meas}\, _{d-1} F_j }. {} \end{aligned} $$
(3)

Remark 2

Let us show that d-sine is really a generalization of the classical sine function. Set d = 2 and consider an arbitrary triangle A 0 A 1 A 2. Denote by \(\hat A_{0}\) its angle at the vertex A 0. Then, obviously,

$$\displaystyle \begin{aligned} \mathrm{meas}\, _2 (A_0 A_1 A_2) = {1\over 2} |A_0 A_1| |A_0 A_2| \sin \hat A_0. {} \end{aligned} $$
(4)

Comparing this relation with (3), we find that

$$\displaystyle \begin{aligned} \sin \hat A_0 = \sin_2 (\hat A_0, A_0 A_1 A_2).\vspace{-3pt} {} \end{aligned} $$
(5)

Definition 3

A family \(\mathcal {F} = \{\mathcal {T}_h \}_{h \to 0}\) of partitions of a polytope into d-simplices is said to satisfy the generalized minimum angle condition if there exists C > 0 such that for any \(\mathcal {T}_h \in \mathcal {F}\) and any \(S = \mathrm {conv} \{ A_0, \dots , A_{d} \} \in \mathcal {T}_h\) one has

$$\displaystyle \begin{aligned} \forall \ i \in \{0, 1, \dots , d \} \qquad \sin_d ( \hat A_i | A_{0} A_{1} \dots A_{d}) \ge C > 0. {}\end{aligned} $$
(6)

This condition is investigated in the paper [6]. It generalizes the well-known Zlámal minimum angle condition for triangles (see [8, 28, 29]), which is stronger than (1).

Definition 4

A family \(\mathcal {F} = \{\mathcal {T}_h \}_{h \to 0}\) of partitions of a polytope into d-simplices is said to satisfy the generalized maximum angle condition if there exists C > 0 such that for any \(\mathcal {T}_h \in \mathcal {F}\) and any \(S = \mathrm {conv} \{ A_0, \dots , A_{d} \} \in \mathcal {T}_h\) one can always choose d edges of S, which, when considered as vectors, constitute a (higher-dimensional) angle whose d-sine is bounded from below by the constant C.

Remark 5

From (4) and (5) we observe that the condition stated in Definition 4 for d = 2 is equivalent to the maximum angle condition (1).

Remark 6

Let us show that in case of tetrahedra the validity of the maximum angle condition (2) implies the desired property in Definition 4, i.e. it is really a generalization. So, let (2) be valid for a given tetrahedron T. Then one can always find, see the proof of Theorem 7 in [19, pp. 517–518], three unit vectors t 1, t 2, and t 3 parallel to three edges of T, so that the volume of the parallelepiped \(\mathcal {P}(t_1, t_2, t_3)\) generated by t 1, t 2, t 3 is bounded from below by some constant c > 0. Now we use formula (3) to estimate the 3-dimensional sine of the angle formed by the vectors t 1, t 2, t 3 as follows

(7)

where S(t 1, t 2, t 3) is the tetrahedron made by t 1, t 2, t 3 originating at 0, and the area of each of the three faces F j involved is bounded from above by 1∕2 due to the fact that t 1, t 2, t 3 are unit vectors. The constant c can be, in fact, estimated from below by

$$\displaystyle \begin{aligned}m:= \min \Bigl( \sin \frac{\pi - \gamma_0}{2}, \sin \gamma_0 \Bigr),\end{aligned} $$

see [19, p. 518].

Now we present the main interpolation theorem of this paper using the standard Sobolev space notation.

Theorem 7

Let \(\mathcal {F}\) be a family of partitions of a polytope into d-simplices satisfying the generalized maximum angle condition. Then there exists a constant C > 0 such that for any \(\mathcal {T}_h \in \mathcal {F}\) and any \(S \in \mathcal {T}_h\) we have

$$\displaystyle \begin{aligned} \| v - \pi_S v\|{}_{1, \infty} \le C h_S |v|{}_{2, \infty} \qquad \forall v \in \mathcal{C}^2(S), {} \end{aligned} $$
(8)

where π S is the standard Lagrange linear interpolant and h S = diam S.

For the proof see [13].

3 Examples

Example 1

Denoting by A 0, A 1, A 2, A 3 the vertices of spindle, spear, spike, spade, cap, or sliver tetrahedron from Fig. 1 in an arbitrary way, we find by (3) that

$$\displaystyle \begin{aligned}\sin_3 (\hat A_0, A_0 A_1 A_2 A_3) \to 0 \end{aligned}$$

as the discretization parameter tends to zero. The same is true for the splinter tetrahedron. However, by Definition 4 we may choose three edges which constitute a spatial angle whose 3-sine is bounded from below by a fixed constant C > 0. Choosing the two short edges of the splinter tetrahedron and one long edge, we find that the generalized maximum angle condition holds.

Example 2

We show that if sin3 of two trihedral angles (cf. [11]) are the same numbers, then the magnitude of these solid angles in steradians need not be the same. Let A 0 A 1 A 2 A 3 be the regular tetrahedron whose edges have length 1. Then the altitude of its faces is \(\sqrt {3}/2\) and the spatial altitude is \(\sqrt {6}/3\). Consequently, from (3) for the trihedral angle \(\hat A_0\) we get

Consider now the tetrahedron with vertices B 0 = (0, 0, 0), B 1 = (1, 0, 0), , and B 3 = (0, 0, 1). Then by (3) we also find that

Now by the Girard Theorem for the spherical excess we have (see [26, p. 83])

$$\displaystyle \begin{aligned}\hat B_0=\frac{\pi}{2}+\frac{\pi}{2}+ \mathrm{arcsin}\frac{4\sqrt{3}}{9}-\pi=0.8785\dots~\mathrm{steradians}, \end{aligned}$$

whereas

$$\displaystyle \begin{aligned}\hat A_0=3\,\mathrm{arccos}\frac{1}{3}-\pi=0.5512\dots~\mathrm{steradians,} \end{aligned}$$

where all dihedral angles α of A 0 A 1 A 2 A 3 are ≈ 70.52∘, since .

Example 3

The sliver element from Fig. 1 can be narrowed as follows. For h → 0 consider the tetrahedron with vertices

$$\displaystyle \begin{aligned}(0,0,0), \quad (h^2,0,0), \quad (0,h,0), \quad (h^2,h,h^3). \end{aligned}$$

Example 4

A higher-dimensional example can be constructed in the following way. Consider positive numbers r 1, r 2, …, r d and a simplex with vertices A 0, A 1, …, A d. We fix some number k so that 0 ≤ k ≤ d. The first k + 1 vertices of the simplex are defined as follows. Let A 0 = (0, 0, …, 0, …, 0). Further, let

$$\displaystyle \begin{aligned}A_1 = (r_1, 0, \dots, 0, \dots, 0), \ A_2 = (0, r_2, \dots, 0, \dots, 0), \ \dots, \ \end{aligned}$$
$$\displaystyle \begin{aligned}A_k = (0, \dots, 0, r_k, 0, \dots, 0).\end{aligned}$$

The remaining vertices are:

$$\displaystyle \begin{aligned}A_{k+1} = (0, \dots, 0, r_{k+1}, 0, 0, \dots , 0), \ A_{k+2} = (0, \dots, 0, r_{k+1}, r_{k+2}, 0, \dots , 0), \end{aligned}$$
$$\displaystyle \begin{aligned}A_{k+3} = (0, \dots, 0, r_{k+1}, r_{k+2}, r_{k+3}, 0, \dots , 0), \ \dots, \ \end{aligned}$$
$$\displaystyle \begin{aligned}A_d = (0, \dots, 0, r_{k+1}, r_{k+2}, r_{k+3}, \dots, r_{d}). \end{aligned}$$

Therefore, for k = 0, we get the path-simplex, and for k = d the hypercube-corner simplex. Allowing some of the r k’s to approach zero with different rates, in general, we get various degenerated simplices still satisfying the generalized maximum angle conditions.