Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes.

*(English)*Zbl 0855.65124Summary: Unlike the case of refining triangles in the multigrid method, even an equilateral tetrahedron cannot be subdivided into eight similar subtetrahedra. Special techniques are needed in successive refinement of tetrahedra in 3D multitgrid methods. In this paper, several methods are proposed and analyzed for refining tetrahedra. It is shown that by a special method the measure of quasiuniformity for the nested refined grids remains bounded where the bound is explicitly given. In fact, there are at most six different types of tetrahedra in the sequence of successively subdivided tetrahedra for any given tetrahedron by the method. Thorough study is given to the tetrahedra which can be refined into 8 congruent or equivalent subtetrahedra. Numerical implementation is described.

##### MSC:

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |