 
Summary: TETRAHEDRAL BISECTION AND ADAPTIVE FINITE
ELEMENTS
DOUGLAS N. ARNOLD AND ARUP MUKHERJEE
Abstract. An adaptive finite element algorithm for elliptic boundary value prob
lems in R3 is presented. The algorithm uses linear finite elements, aposteriori error
estimators, a mesh refinement scheme based on bisection of tetrahedra, and a multigrid
solver. We show that the repeated bisection of an arbitrary tetrahedron leads to only
a finite number of dissimilar tetrahedra, and that the recursive algorithm ensuring con
formity of the meshes produced terminates in a finite number of steps. A procedure for
assigning numbers to tetrahedra in a mesh based on aposteriori error estimates, indicat
ing the degree of refinement of the tetrahedron, is also presented. Numerical examples
illustrating the effectiveness of the algorithm are given.
Key words. finite elements, adaptive mesh refinement, error estimators, bisection
of tetrahedra
AMS(MOS) subject classifications. 65N50.
1. Introduction. In this article, we describe an algorithm for the
accurate and efficient computation of solutions to elliptic boundary value
problems in R3
. The algorithm creates a sequence of adapted tetrahedral
meshes starting form an initial coarse tetrahedral mesh, and the solution is
