Abstract

The development of scalable linear solvers is essential to facilitating large scale computations of many engineering and scientific problems. Larger simulations can provide new scientific insights, and in the context of design can lead to improved product performance and system level design approaches. Advances in computer hardware cannot overcome the inherent algorithmic complexity of some common algorithms, such as LU factorisation (which is the workhorse of commercial finite element programs). The development of tools for scalable, large scale simulations for differential equations depends on mathematical and algorithmic developments. I will present, in tutorial style, some linear solvers for problems in solid mechanics and geophysics that are scalable and optimal. That is, the computational cost scales linearly with problem size and the methods are suitable for parallel computers. It is important to match the solution method to the equation(s) being solved; black box methods do not work. Specific examples will include gas turbine components and simplified models of subduction zones. The present examples have up to 12 billion degrees of freedom and have been solved using up to 25k processes.