NUMERICAL SOLUTION OF LARGE NONLINEAR BOUNDARY VALUE PROBLEMS BY QUADRATIC MINIMIZATION TECHNIQUES

Abstract

This chapter reviews the numerical treatment of large highly nonlinear two- or three- dimensional boundary value problems by quadratic minimization techniques. In different situations where these techniques have been applied, the methodology remained the same. In those situations, the method followed were: (1) deriving a variational formulation of the original boundary value problem, which was then approximated by Galerkin methods; (2) transforming this variational formulation into a quadratic minimization problem or into a sequence of quadratic minimization problems; and (3) solving each quadratic minimization problem by a conjugate gradient method with preconditioning, the preconditioning matrix being sparse, positive definite, and fixed once for all in the iterative process. This chapter also discusses a methodology for the description of least squares solution methods and their application to the solution of the unsteady Navier-Stokes equations for incompressible viscous fluids; the description of augmented lagrangian decomposition techniques and their application to the solution of equilibrium problems in finite elasticity.