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On Numerical Regularity of the Longest-Edge Bisection Algorithm

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On Numerical Regularity of the Longest-Edge Bisection Algorithm 1 On numerical regularity of the longest-edge bisection 2 algorithm 1 2, 3 4 3 Antti Hannukainen , Sergey Korotov , Michal Kˇr´ıˇzek 4 October 13, 2012 1 5 Department of Mathematics and Systems Analysis, Aalto University 6 P.O. Box 11100, FI–00076 Aalto, Finland 7 e-mail: antti.hannukainen@aalto.fi 2 8 BCAM – Basque Center for Applied Mathematics 9 Mazarredo, 14, E–48009 Bilbao, Basque Country–Spain 10 e-mail: [email protected] 3 11 IKERBASQUE, Basque Foundation for Science, E–48011, Bilbao, Spain 4 12 Institute of Mathematics, Academy of Sciences 13 Zitn´a25,ˇ CZ–115 67 Prague 1, Czech Republic 14 e-mail: [email protected] 15 Abstract: The finite element method usually requires regular or strongly regular families 16 of partitions in order to get guaranteed a priori or a posteriori error estimates. In this 17 paper we examine the recently invented longest-edge bisection algorithm that produces 18 face-to-face simplicial partitions of a given bounded polytopic domain in arbitrary space 19 dimension. First, we prove that the regularity of the family of partitions generated by 20 this algorithm is equivalent to its strong regularity. Second, we present 3d numerical 21 examples, which demonstrate that the technique seems to produce regular (and therefore 22 strongly regular) families of tetrahedral partitions. However, a mathematical proof of this 23 statement is still an open problem. 24 Keywords: bisection algorithm, conforming finite element method, regular family of 25 partitions, nested tetrahedral partitions, simplicial elements. 26 Mathematical Subject Classification: 65M50, 65N30, 65N50 27 1 Introduction 28 When solving problems of mathematical physics by the finite element method, we have to 29 decompose the investigated domain into small subdomains – called elements. To achieve a 30 high numerical accuracy, elements should be small. In addition, at those parts where the 31 exact solution of a particular problem possesses some singularities, we have to use smaller 32 elements than those at the parts, where the exact solution is smooth enough. In this 1 33 paper we shall deal only with face-to-face simplicial partitions, i.e., conforming partitions 34 into triangles, tetrahedra, 4-simplexes, etc. 35 The longest-edge bisection algorithm is one of the most popular refinement algorithm 36 in the finite element method, since it is very simple and can be easily applied also in 37 higher dimensions. Martin Stynes in his papers [26]–[29] (see also [1] and [24]) examined 38 the regularity of a family triangulations obtained by a recursive use of the longest-edge 39 bisection algorithm. This algorithm bisects simultaneously all triangles by medians to the 40 longest edge of each triangle in a given triangulation (see Figure 1). In this way, a family 41 of nested triangulations is obtained. Figure 1: The classical longest-edge bisection algorithm may produce triangulations that are not face-to-face. Broken lines stand for new bisections. 42 M. Stynes proved that during the infinite bisection process only a finite number of 43 similarity-distinct subtriangles are generated, which proves the regularity of the resulting 44 family of triangulations, since the Zl´amal’s minimum angle condition (see [30]) is satisfied, 45 i.e., all angles α of all triangles are not less than some fixed constant α0 > 0 when the 46 mesh size h tends to zero, α α0. (1) ≥ 47 As usual, the mesh size h stands for the length of the longest edge in a given triangulation. Figure 2: A modified longest-edge bisection algorithm that always produces face-to-face triangulations. 48 However, this may lead to the so-called hanging nodes and thus refined triangulations 49 may not all be face-to-face, in general (see Figure 1). Therefore, in [12] a modified version 50 of the bisection algorithm by medians was introduced, where only simplices sharing the 51 longest edge of the whole triangulation are bisected at each step (see Figure 2). In this 52 way, all produced triangulations are face-to-face and moreover, the corresponding family 53 of triangulations satisfies the minimum angle condition (1) with computable and known 54 α0, see [12]. In [8] and [9] we modified the above bisection algorithm so that it may 55 also generate locally refined simplicial face-to-face partitions. This goal was achieved by 56 introducing a Lipschitz continuous mesh density function which tell us, which simplices 57 have to be chosen for bisections. We proved that this generalized algorithm also produces 58 nested face-to-face partitions. 59 Bisection-type algorithms are very favourite in finite element community, since they 60 enable us to perform local mesh refinement quite easily. In [3, 13] local mesh refinements 61 are generated by means of the famous red refinement. A given tetrahedron is first sub- 62 divided into eight tetrahedra by mid-lines (see Figure 3). Then each of its neighbouring 2 63 tetrahedra has to be subdivided be means of the red-green refinement into four subte- 64 trahedra or by means of the green refinement into two subtetrahedra to get face-to-face 65 partitions. This process is quite complicated from a combinatorial and programming point 66 of view, although the corresponding family of partitions is regular (see [13] for the proof). 67 Thus, a practical realization of the longest-edge bisection algorithm is much simpler than 68 those of algorithms based on red, red-green, and green refinement. However, for the di- 69 mension d 3 it is not known whether the resulting family of partitions is regular (see ≥ 70 Definition 2.1 below). Therefore in this work we investigate the regularity properties of 71 the 3d meshes generated by the longest-edge bisection numerically. Figure 3: Red, red-green, and green refinements of several adjacent tetrahedra. 72 For arbitrary space dimension d, Kearfott [11] proved that the longest-edge algorithm 73 produces d-simplices whose largest diameter tends to zero. In this algorithm each d- 74 simplex is bisected by a (d 1)-dimensional hyperplane passing through the midpoint of − 75 its longest edge and through its vertices that do not belong to this edge. 76 In [2, 10, 14, 15, 16, 17, 18] several other bisection-like algorithms are investigated, 77 some of them do not necessarily halve the longest edge. The longest-edge algorithm was 78 originally designed for solving systems of nonlinear algebraic equations (see [25, 7]). Later, 79 mainly due to effort of Rivara [19, 20, 21, 22, 23], bisection methods started to become 80 popular also in practical use of the finite element method. 81 2 Theoretical background d 82 The convex hull of d + 1 points in R for d 1, 2, 3,... , which are not contained in a ∈{ } 83 hyperplane of dimension d 1, is called a d-simplex or just a simplex. The angles between − 84 its (d 1)-dimensional facets are called dihedral. − d 85 Next we define a simplicial partition h over a bounded polytope Ω R . We sub- T ⊂ 86 divide Ω into a finite number of simplices so that their union is Ω, any two simplices 87 have disjoint interiors and any facet of any simplex is a facet of another simplex from 88 the partition or belongs to the boundary ∂Ω of Ω. The discretization parameter h is the 89 length of the longest edge in a given simplicial partition. We see that simplicial partitions 90 are face-to-face. 91 The set = h h 0 is named a family of simplicial partitions if for any ε> 0 there F {T } → 92 exists h with h<ε. T ∈F 3 93 Definition 2.1 A family = h h 0 of simplicial partitions of a bounded polytope Ω F {T } → 94 is called regular if there exists a constant C > 0 such that for all partitions h and T ∈ F 95 for all simplices S h we have ∈ T d meas dS C(diam S) . (2) ≥ 96 97 It can be easily checked that the regularity for d = 2 is equivalent to condition (1). 98 Various equivalent definitions can be found in [4] and [5]. Lemma 2.1 Let = h h 0 be a regular family of simplicial partitions of a bounded F {T } → polytope Ω and let S be an arbitrary d-simplex from an arbitrary partition h . Denote T ∈F by a the length of the shortest edge of S and by e the length of its longest edge. Then a κe, ≥ 99 where κ> 0 is a fixed constant independent of h. Proof: Let A1, A2,...,Ad+1 be vertices of S, let Fi be the facet of S opposite to the vertex Ai, and let vi be the altitude from the vertex Ai to the facet Fi. Without loss of generality we may assume that the shortest edge is A1A2. Then clearly a v1 and from ≥ (2) we obtain d d 1 a a d 1 Ce = C(diam S) measdS = v1measd 1F1 measd 1F1 e − , ≤ d − ≤ d − ≤ d 100 where the last inequality is due to the fact that the (d 1)-simplex F1 is contained in − 101 a (d 1)-dimensional hypercube whose edge has length e. Dividing the left-hand and − d 1 102 right-hand sides of the above inequality by e − , the lemma follows. 103 Definition 2.2 A family = h h 0 of simplicial partitions of a bounded polytope Ω is F {T } → 104 called strongly regular if there exists a constant C > 0 such that for all partitions h T ∈F 105 and for all simplices S h we have ∈ T d meas dS Ch .
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