Until recent times, a general lack of computational capacity has effectively limited
implicit finite element calculations in solid and structural mechanics to moderately sized
two-dimensional simulations. At the core of any implicit time-integration scheme,
whether for linear or nonlinear simulation, is the necessity of solving coupled
linear systems of equations. Direct methods of solving linear systems, i.e., algorithms
based upon Gaussian elimination, have proved quite adequate for this task in two-
dimensional applications, and their simplicity and robustness have led to near
universal adoption of these techniques in general-purpose finite element codes. However,
the increasing demands for three-dimensional simulation, as well as ever larger two-
dimensional applications, require us to confront the unfavorable rates of growth for
both storage and floating-point operations associated with traditional direct methods.
These difficulties have long inspired the study and development of iterative linear
equation solvers by numerical analysts. The computational mechanics community is
returning to this topic in search of cost-effective methods for application to large-scale
analysis. Furthermore, iterative methods are proving to be a natural avenue for the
parallel solution of systems of linear equations. The ongoing evolution of seemingly
all hardware towards parallel processing adds further impetus to this topic.
Element-by-element methods originally motivated by the global