On numerical regularity of the face-to-face longest-edge bisection algorithm for tetrahedral partitions
The finite element method usually requires regular or strongly regular families of partitions in order to get guaranteed a priori or a posteriori error estimates. In this paper we examine the recently invented longest-edge bisection algorithm that always produces only face-to-face simplicial partiti...
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| Published in: | Science of computer programming Vol. 90; pp. 34 - 41 |
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| Language: | English |
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15.09.2014
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| ISSN: | 0167-6423, 1872-7964 |
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| Abstract | The finite element method usually requires regular or strongly regular families of partitions in order to get guaranteed a priori or a posteriori error estimates. In this paper we examine the recently invented longest-edge bisection algorithm that always produces only face-to-face simplicial partitions. First, we prove that the regularity of the family of partitions generated by this algorithm is equivalent to its strong regularity in any dimension. Second, we present a number of 3d numerical tests, which demonstrate that the technique seems to produce regular (and therefore strongly regular) families of tetrahedral partitions. However, a mathematical proof of this statement is still an open problem.
•We examine the longest-edge bisection algorithm that refines simplicial partitions.•The resulting families of partitions are regular iff they are strongly regular.•The longest-edge bisection algorithm is very easy to implement in any dimension.•Numerical tests seem to produce regular families of tetrahedral partitions. |
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| AbstractList | The finite element method usually requires regular or strongly regular families of partitions in order to get guaranteed a priori or a posteriori error estimates. In this paper we examine the recently invented longest-edge bisection algorithm that always produces only face-to-face simplicial partitions. First, we prove that the regularity of the family of partitions generated by this algorithm is equivalent to its strong regularity in any dimension. Second, we present a number of 3d numerical tests, which demonstrate that the technique seems to produce regular (and therefore strongly regular) families of tetrahedral partitions. However, a mathematical proof of this statement is still an open problem.
•We examine the longest-edge bisection algorithm that refines simplicial partitions.•The resulting families of partitions are regular iff they are strongly regular.•The longest-edge bisection algorithm is very easy to implement in any dimension.•Numerical tests seem to produce regular families of tetrahedral partitions. |
| Author | Hannukainen, Antti Křížek, Michal Korotov, Sergey |
| Author_xml | – sequence: 1 givenname: Antti surname: Hannukainen fullname: Hannukainen, Antti email: antti.hannukainen@aalto.fi organization: Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland – sequence: 2 givenname: Sergey surname: Korotov fullname: Korotov, Sergey email: korotov@bcamath.org organization: BCAM – Basque Center for Applied Mathematics, Mazarredo, 14, E-48009 Bilbao, Basque Country, Spain – sequence: 3 givenname: Michal surname: Křížek fullname: Křížek, Michal email: krizek@math.cas.cz organization: Institute of Mathematics, Academy of Sciences, Žitná 25, CZ-115 67 Prague 1, Czech Republic |
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| Keywords | Bisection algorithm Simplicial elements Conforming finite element method Nested tetrahedral partitions Regular family of partitions |
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| SubjectTerms | Bisection algorithm Conforming finite element method Nested tetrahedral partitions Regular family of partitions Simplicial elements |
| Title | On numerical regularity of the face-to-face longest-edge bisection algorithm for tetrahedral partitions |
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