 
Summary: LOCALLY ADAPTED TETRAHEDRAL MESHES USING BISECTION
DOUGLAS N. ARNOLD, ARUP MUKHERJEE, AND LUC POULY§
SIAM J. SCI. COMPUT. c 2000 Society for Industrial and Applied Mathematics
Vol. 22, No. 2, pp. 431448
Abstract. We present an algorithm for the construction of locally adapted conformal tetrahedral
meshes. The algorithm is based on bisection of tetrahedra. A new data structure is introduced, which
simplifies both the selection of the refinement edge of a tetrahedron and the recursive refinement to
conformity of a mesh once some tetrahedra have been bisected. We prove that repeated application
of the algorithm leads to only finitely many tetrahedral shapes up to similarity, and we bound the
amount of additional refinement that is needed to achieve conformity. Numerical examples of the
effectiveness of the algorithm are presented.
Key words. bisection, tetrahedral meshes, adaptive refinement, similarity classes, finite ele
ments
AMS subject classification. 65N50
PII. S1064827597323373
1. Introduction. The generation of locally adapted conforming tetrahedral
meshes is an important component of many modern algorithms, for example, in the
finite element solution of partial differential equations. Typically, such meshes are
produced by starting with a coarse tetrahedral mesh, selecting certain elements for
refinement, somehow refining those elements and others as necessary to maintain
