Aj. Meade et al., SEQUENTIAL FUNCTION APPROXIMATION FOR THE SOLUTION OF DIFFERENTIAL-EQUATIONS, Communications in numerical methods in engineering, 13(12), 1997, pp. 977-986
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.