The solution of elliptic diffusion operators is the computational bottleneck in many simulations of engineering and scientific disciplines. We present a truly scalable - ultrascalable - linear solver for the diffusion operator in unstructured elasticity problems. Scalability is demonstrated with speedup studies of a non-linear analysis of a vertebral body with over a half a billion degrees of freedom on up to 4088 processors on the ASCI White machine. This work is significant becase in the domain of unstructured implicit finite element analysis in solid mechanics with complex geometry, this is the first demonstration of a highly parallel, and efficient, applicaiton of a mathematically optimal linear solution method on a common large scale computing platform - the IBM SP Power3.

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