Ph.D. Thesis, November 1997. Specifically, in the first part of the dissertation, a method of optimal incremental function approximation is developed for the adaptive solution of differential equations. In this manner, the proposed method is adaptive in nature, although a grid is neither built nor adapted in the traditional sense using a-posteriori error estimates. Variational principles are utilized for the definition of the objective function to be extremized in the associated optimization problems, ensuring that the problem is well-posed. Complicated data structures and expensive remeshing algorithms and systems solvers are avoided.
Computational efficiency is increased by using low-order basis functions and concurrent computing.
Numerical results and convergence rates are reported for a range of steady-state problems, including linear and nonlinear differential equations associated with general boundary conditions, and illustrate the potential of the proposed method. Fluid dynamics applications are emphasized. Conclusions are drawn by discussing the method's limitations, advantages, and possible extensions. The VII mechanism is based on the concept of a transpiration velocity boundary condition, whose convergence to steady state is accelerated. The number of variables in the associated optimization problem is reduced by means of function approximation concepts to ensure high number of parallel processors to number of necessary function evaluations ratio.