Estimation of the interpolation error for semiregular prismatic elements

Article history: Received 10 December 2019 Received in revised form 20 February 2020 Accepted 26 April 2020 Available online 4 May 2020


Introduction
The finite element method is one of the most flexible and powerful methods to solve numerically a wide variety of partial differential equations [3,13,11,14]. A fundamental problem is to estimate the error between the exact solution and its computable finite element approximation. This error can be bounded by the best approximation of the exact solution in the finite element space consisting of piecewise polynomial functions (see Céa's lemma [4]). Hence, it is important to estimate the interpolation errors.
In the process of estimation of the interpolation error, some constant times a power of the discretization parameter h appears. It is crucial that this constant does not blow up when h tends to zero. For linear elliptic boundary value problems in 2-dimensional space, Zlámal [15] introduced the minimum angle condition that guarantees a bound on the constant in the final error which comes from the estimation error of the defined interpolation operator. See also Synge [12]. Babuška and Aziz [2] proposed that the minimum angle condition can be relaxed to the maximum angle condition. In 3-dimensional space, the natural extension of the maximum angle condition for tetrahedral elements was proposed by Křížek [9]. Recently, the generalization of the maximum angle condition in d-dimensional spaces (d ≥ 2), by means of sin d [5], for d-simplices is introduced and extended in [7,8] and also the equivalence of the maximum angle condition and its generalized version is proved.
The maximum angle condition enables us to keep an optimal error whereas we are allowed to consider degenerating families of elements in order to cover the narrow or flat parts of a given bounded domain. For instance, in geophysical simulations [10], where the domain consists of horizontal triangles as a base and regular vertical layer, all finite prismatic elements are produced by the Cartesian product of triangles and the closed intervals called triangular prisms. For such simulation, high aspect-ratio for the elements must be allowed. Therefore, we [6] analyzed the behavior of the interpolation error under the maximum angle condition on the above prisms.
The aim of this paper is to estimate the interpolation error for a more general class of prismatic elements than previously considered in [6]. This class of elements naturally appear e.g. in some standard geometric models. In Fig. 1 an example of a frustum is given. We interpolate a given function by quadratic polynomials which are bilinear on the reference prism. To introduce general prismatic element, similar to [6] we consider the maximum angle condition for all dihedral angles. In addition, we assume that the ratios between the three edges that connect the triangular faces is bounded from below by some positive constant. Note that these ratios for triangular prisms are one. We relax the conditions from [6] to allow e.g. slanted or skew elements. In particular small deformations of the geometry from [6] are covered. We refer to [6] for further motivation and context.
We use the technique of reference element in several parts of the main proof in order to demonstrate that the interpolation error is of order O (h) in the H 1 -norm for sufficiently smooth functions and sufficiently small h. In that proof we use a positive lower bound for the Jacobian determinant. In our case, this determinant is a quadratic polynomial in three variables whose coefficients are expressed in terms of volumes of tetrahedra formed by the vertices of the prism. The outline of the paper is as follows. First, in Section 2, we introduce notations and give some definitions. In addition, we propose an extension of the semiregularity property that allows us to consider some degenerate families of prismatic elements. In Section 3, we obtain a positive lower bound for the Jacobian determinant in Theorem 6, since we use the technique of the reference element to prove the main result. In Section 4 we prove Theorem 7 which states that the interpolation error is of the order O (h) under the extended semiregularity condition, followed by some conclusions in Section 5.

Main definitions and geometric preliminaries
We will consider meshes whose elements are defined in this definition: Definition 1. A straight-side, triangular based prism is a convex polyhedron with six vertices, two triangular faces and three quadrilateral (convex planar) faces. Furthermore, each quadrilateral is incident to the other four faces. The two triangles are not incident. See Fig. 2 (right). In this paper, we will refer to this as a general prism.
We define general prismatic meshes as follows: Definition 2. A general prismatic mesh P h of a bounded polyhedral domain is a face-to-face partition whose elements are general prisms, where h is the maximum diameter of all elements in the mesh.
The following lemma helps us to order the vertices of the prism P. For more details, see also Remark 1.

Lemma 1.
The three edges which connect the two triangular faces of P are either parallel or if we extend these edges in one direction then they meet each other at some point.
Proof. The planes containing the three quadrilateral faces intersect in one point or this intersection is empty.
The reference prismP (left) and an arbitrary prismatic element P (right). Further, the mapping F is given by formula (9).

Remark 1.
We order the vertices of P similar to [9, pp. 517-518], in such a way that the non-parallel edges A 3 A 0 , A 4 A 1 and A 5 A 2 satisfy Lemma 1 and the triangular face A 3 A 4 A 5 is closer to the intersection point than the triangle A 0 A 1 A 2 . See Fig. 2 (right). If the edges are parallel, then we do not need to order the vertices similar to the nonparallel case except that the vertices A 0 and A 3 are the end points of the same edge connecting the triangular faces. Furthermore, we assume that in any case the maximum angle for the triangular base is at vertex A 0 .
We now define the modification of the semiregularity property from [6] to our setting that will be used throughout the paper.

Definition 3.
A family of general prismatic meshes F = {P h } h→0 is semiregular if there exist constants γ < π , c 1 > 0, and c 2 > 0 such that the following conditions hold: a) Maximum angle condition : For any P ∈ P h and any P h ∈ F let γ P be the maximum angle of any triangular faces and dihedral angle between any two faces of P. Then γ P ≤ γ . (1) b) Edge ratio condition : For any P ∈ P h and any P h ∈ F let L min and L max be the minimum and maximum lengths of the three edges connecting the triangular faces. Then c) Tetrahedra ratio condition : For any P ∈ P h and any P h ∈ F let the vertices of P be ordered as in Remark 1. Then Consider first a case in which a) fails, but b) and c) hold, see Fig. 3.
, and A 5 = (h, 0, h) be the vertices of the prism. In this case, A 1 A 0 A 2 → π as h → 0, so condition a) fails. On the other hand, conditions b) and c) hold with c 1 = 1 and c 2 = 1. Consider next a case in which c) fails but a) and b) hold, see Fig. 4.
The triangles on the base and top of the prism are equilateral. Now, if h tends to zero, this family degenerates into a regular tetrahedron, so clearly condition a) holds. Further, condition b) with c 1 = 1 is fulfilled, meanwhile condition c) is violated, since the ratio of the volume of the

Fig. 5. Orthogonal projection of the prism onto ŷẑ-plane with vertices
Finally, we consider a case in which b) fails, but the two other conditions hold. 5. Now assume that h → 0. Then the family degenerates into a pyramid, so clearly condition a) holds. Moreover, the family satisfies condition c) with The condition c) in Definition 3 implies bounds on ratios of the volumes of other tetrahedra as well. We will see this in Lemma 5.
To prove Lemma 5, we need the following lemmas from [9]. Lemma 3. [9] Let ζ ≤ η ≤ τ be angles of an arbitrary face of an arbitrary tetrahedron. Assume furthermore that τ ≤ γ . Then τ ≥ π/3 and η, τ ∈ π − γ 2 , γ . Lemma 4. [9] Let A be an arbitrary vertex of an arbitrary tetrahedron T and let χ ≤ ψ ≤ ϕ be angles between faces passing through A. Assume furthermore that ϕ ≤ γ . Then ϕ > π/3 and 3, which depend only on c 1 and m such that iii) The ratio of the volumes of the tetrahedra T ( Note that for other possibilities we have similar results. Lemma 3 implies that sin(θ ) is bounded from below by the positive constant m as in [9]. Then and consequently Hence, and similarly We denote the angles between the edges A 3 A 0 and A 1 A 0 , and the edges A 3 A 1 and A 1 A 0 , by α and β, respectively, see Fig. 6. Now, according to [9, pp. 517-518], Lemmas 3-4, and condition b), if α is greater than or equal to β, we get Otherwise β is greater than α. In this case, either To obtain a lower bound for sin(α), it suffices to use the law of sines for the triangle A 0 A 1 A 3 (see Fig. 7), conditions a) and b), which implies and therefore Vol Note that for the above inequalities we used Lemma 3, since α < β + β 1 , and (2), respectively. Writing the law of sines for the triangle A 0 A 1 A 3 once again, leads to Substitute the right-hand side of (6) into (5), we have ii) To estimate a lower bound for the ratio of the volumes of the tetrahedra T ( Otherwise, exchanging the indices of the vertices A 0 , A 3 , A 4 , A 1 in Fig. 8 into 1, 4, 3, 0, respectively, and following the same proof as in part i), there exists a positive constant C * (c 1 , m) such that iii) The proof is same as in parts i) and ii). iv) From part i), (2) and (3), we have In what follows, we use the standard denotation W k p ( ), k = 0, 1, . . . , p ≥ 1, for Sobolev spaces with norms . k,p = . k,p, and seminorms |.| k.p = |.| k,p, . The symbol C ( ) stands for the space of continuous functions over .
To prove the main result of the paper we will employ the technique based on a transfer of the prism P ∈ P h onto the reference prism P =K ×Î, where K is the triangular base and Î is the altitude of P .
Equation (9) defines a mapping F :P → P, which is a bijection from the prism P onto the prism P. Hence we can define φ i on P such that With any prismatic mesh P h we associate the finite element space For similar cases, see [1], Section 5.3. Then the interpolation operator π h : C ( ) → V h is uniquely determined by the requirement Let Ĵ denote the Jacobian of the mapping F . Then In order to obtain the rate of convergence of the interpolation operator, we will estimate an upper bound for |det(Ĵ)| −1 , which plays the key role in the proof of Theorem 7. We will show that the lower bound of |det(Ĵ)| depends on the volumes of tetrahedra in the prism P.

Jacobian determinant
For prisms [6] the determinant of the Jacobian is a constant. We see that for the general prisms, according to (11), det(Ĵ) is a polynomial in terms of x, ŷ, and ẑ. To show that det(Ĵ) = 0, using the linearity property of the determinant, the Jacobian determinant has the explicit form Proof. For a fixed ẑ =ẑ 0 , det(Ĵ) is linear, and thus attains its maximum and minimum at vertices of the triangle 0 ≤x, ŷ ≤ 1, x +ŷ ≤ 1, ẑ =ẑ 0 . Therefore it is enough to consider the restriction of det(Ĵ) to the three vertical lines. Then the extremal values of det(Ĵ) can be found at one of these points: the six vertices of the prism Â i , i = 0, . . . , 5 On the right-hand side of (15), all terms are six times the volume of a tetrahedron. Indeed, and Now, Lemma 5 provides the lower bounds for A, B 1 , and C 1 . In addition, condition c) and part i) of Lemma 5 imply that Using the same proof as in Lemma 5 for (16) and (17) Due to the valid interval of ẑ, there are two possibilities for D and G, D > 0, G < 0 or D < 0, G > 0. When G < 0, we obtain det(Ĵ)(P (c) ) ≥ A. For D < 0, where When λ 1 tends to zero, consequently P (c) tends to (0, 0, 0), and due to Lemma 5, the family of functions det(Ĵ)(P (c) ) for all P ∈ P h ∈ F is equicontinuous and by (18) we obtain det(Ĵ)(P (c) ) → A.
Since in this case, condition 2G 1 − (D 1 + D 2 ) < 0 leads to P (c) be outside of the domain, then the valid condition is where When λ 2 tends to zero, G 1 and (D 1 + D 2 )/2 tend together, and from definition of G we have This means that P (c) tends to (0, 0, 1) and by (18) and condition c), the family of functions det(Ĵ)(P (c) ) for all P ∈ P h and P h ∈ F is equicontinuous, and det(Ĵ)(P (c) ) tends to For the case that D < E, Furthermore, and a same proof as in Lemma 5, parts i) and ii), to obtain a lower bound for Vol (3) (1, 0, 0), and according to Lemma 5, the family of Jacobian determinant at P (c) for all P ∈ P h ∈ F is equicontinuous. Therefore (24) yields The other case is D < E, G > 0 and G < B 1 . Since the third coordinate of P (c) must be in (0, 1), we have If λ 4 tends to zero, then G tends to zero. By (28), D → E (or conversely), and from (25) we conclude that det(Ĵ)(P (c) ) → B 1 . When λ 4 → 1, then G → B 1 and (24) tends to Note that for the last inequality we used (26). Now, the same argument for (27) implies (14).
Then we can obtain the lower bound for (31) as follows.
Extending the proof of Lemma 5, one comes to (14). Now, if λ 5 → 0, then F − D → 0, and as a result P (c) → (0, 1, 0). Similar to previous cases, due to Lemma 5, here the family of Jacobian determinant at P (c) for all prisms belonging to F is also equicontinuous and we get det(Ĵ)(P (c) ) → C 1 .
For the case that G < C 1 , If λ 6 approaches to zero, then G → 0, which implies (33).
To end the proof, let λ 6 tend to 1. Then G → C 1 , and consequently by (32), (30) tends to Proof. From the definition of semi-norm we have |u − π P u| 2 1,2,P = P ∂ ∂x To estimate (36), first we will estimate it on P . Then, from equation (40) in [6], we have For the last inequalities we used the so-called sum of squares inequality In the remaining computations, we will use C as an unspecified positive constant. It is not necessarily the same in two lines of a computation, for instance in equation (38). We get which imply and similarly Now the upper bounds of | J (i, j) |, i = 1, 2, 3, j = 1, 2 can be expressed in terms of a, b, and m as follows.
Similarly, we have From equation (45)