Multigrid equation solvers for large-scale nonlinear finite element simulations

Abstract : The finite element method has grown, in the past 40 years, to be a popular method for the simulation of physical systems in science and engineering. The finite element method is used in a wide array of industries. In fact just about any enterprise that makes a physical product can, and probably does, use finite element technology. The success of the finite element method is due in large part to its ability to allow the use of accurate formulation of partial differential equations (PDEs), on arbitrarily general physical domains with complex boundary conditions. Additionally, the rapid growth in the computational power available in today's computers -- for an ever more affordable price -- has made finite element technology more accessible to a wider variety of industries and academic disciplines. As computational resources allow people to produce ever more accurate simulation of their systems -- allowing for the more efficient design and safety testing of everything from automobiles to nuclear weapons to artificial knee joints -- all aspects of the finite element simulation process are stressed. The largest bottleneck in the growth in the scale of finite element applications is the linear equation solver used in implicit time integration schemes. This is due to the fact that the direct solution methods -- popular in the finite element community as they are efficient, easy to use, and relatively unaffected by the underlying PDE and discretization -- do not scale well with increasing problem size. The scale of problems that are now becoming feasible demand that iterative methods be used. The performance issues of iterative methods is very different from those of direct methods, as their performance is highly sensitive to the underlying PDE and discretization; the construction of robust iterative methods for finite element matrices is a hard problem which is currently a very active area of research.