SEQUENTIAL FUNCTION APPROXIMATION FOR THE SOLUTION OF DIFFERENTIAL-EQUATIONS

A computational method for the solution of differential equations is p roposed. With this method an accurate approximation is built by increm ental additions of optimal local basis functions. The parallel direct search software package (PDS), that supports parallel objective functi on evaluations, is used to solve the associated optimization problem e fficiently. The advantage of the method is that, although it resembles adaptive methods in computational mechanics, an a priori grid is not necessary. Moreover, the traditional matrix construction and evaluatio ns are avoided. Computational cost is reduced while efficiency is enha nced by the low-dimensional parallel-executed optimization and paralle l function evaluations. In addition, the method should be applicable t o a broad class of interpolation functions. Results and global converg ence rates obtained for one-and two-dimensional boundary value problem s are satisfactorily compared to those obtained by the conventional Ga lerkin finite element method. (C) 1997 John Wiley & Sons, Ltd.