This research deals with the construction, evaluation, andanalysis of algorithms to solve large linear and nonlinearsystems of equations arising from problems in mechanics onemerging parallel architectures. The research deals particularlywith the hypercube architecture and, to a lesser extent, with theCray Y.MP. The research focuses on the types of systems thatarise in linear elasticity, large displacement analysis of platesand shells, and nonlinear elasticity, as well as the types ofsystems arising in incompressible fluid flow.