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LetT be a tetrahedral mesh. We present a 3-D local renement algorithm forT which is mainly based on an 8-subtetrahedron subdivision pro- cedure, and discuss the quality of rened meshes generated by the algorithm. It is proved that any tetrahedron T2Tproduces a nite number of classes of similar tetrahedra, independent of the number of renement levels. Fur- thermore, (Tni ) c(T), where T2T , cis a positive constant independent of T and the number of renement levels, Tni is any rened tetrahedron of T ,a ndis a tetrahedron shape measure. It is also proved that local rene- ments on tetrahedra can be smoothly extended to their neighbors to maintain a conforming mesh. Experimental results show that the ratio of the number of tetrahedra actually rened to the number of tetrahedra chosen for renement is bounded above by a small constant. The solution of some types of partial dierential equations (PDEs) using the nite element method is an adaptive process, which in general consists of mesh generation, solution of linear equations, error estimation, and mesh renement. In the mesh renement step, varying element sizes are used to improve the solution of PDEs, and an ecient scheme, local renement, is adopted to locally rene some regions where approximation errors are unacceptable. Also, the adaptive mesh renement technique is needed in the multigrid method, in which the solution of a problem is obtained by alternately solving the problem on several levels of coarse to ne grids. It is desirable that the adaptive renement produces rened elements of good shape using a fast rening process. These are the goals of our 3-D local renement algorithm (LRA). Not many results have been published on the quality of meshes produced by 3-D local renement algorithms (LRAs), although some 2-D quality LRAs are available (17, 18, 19, 2, 1, 13) and have proved to be successful in adaptive nite element analysis (14). Some 3-D LRAs are used in practice without theoretical analysis on the quality of the rened meshes (16, 20). In the bisection method of (8), the longest edge is always chosen to be bisected and diameter bounds are given for simplices of any dimension. However, whether the shapes of subtetrahedra produced by repeated longest-edge bisection degrade arbitrarily is still an open problem. In (12), we presented an LRA for tetrahedral meshes based on a special bisection

Quality local refinement of tetrahedral meshes based on 8-subtetrahedron subdivision