This work is motivated by the success of the finite element method in simulating complex physical systems in science and engineering, coupled with the wide spread availability of ever more powerful computers, which has lead to the need for efficient equation solvers for implicit finite element applications. Finite element matrices are often poorly conditioned - this fact has made the use of direct solvers popular as they are relatively unaffected by the condition number of the matrix. As the scale of the problems increase, direct methods, however, possess sub-optimal time and space complexity when compared to iterative methods. Also, as larger and faster computers are becoming more widely available, to a larger number of research institutions and industries, the use of iterative methods will become increasingly more necessary. Thus, given the computational resources available today, and those that are continually being introduced, direct methods are inefficient in solving large problems - resulting in the need to resort to iterative methods.
Many iterative methods are notoriously unreliable on finite element problems of interest.
Multigrid is one of a family of highly optimal multilevel domain decomposition methods, and is known to be a highly effective method to solve finite element matrices. The application of multigrid techniques to unstructured meshes, that are the hallmark of the finite element method, has not been well developed, and is currently an active area of research. In particular, the development of practical multigrid methods for unstructured finite element problems, of arbitrary geometric complexity and size, is an open problem. This paper discusses methods for the effective application of classical multigrid methods in the solution of the large sparse system of equations that arise from unstructured finite element problems in 3D large deformation elasticity and elasticity. The purpose, of this algorithm, is to automatically construct a coarse grid, from a finer grid, for use in standard multigrid algorithms.