A Discrete Least Squares Method - Semantic Scholar

MATHEMATICS OF COMPUTATION, VOLUME 31, NUMBER 137 JANUARY 1977, PAGES 60-65

A Discrete Least Squares Method By Peter H. Sammon* Abstract.

We consider a discrete least squares approximation

two-point

boundary

the approximation

value problem

to the solution

for a 2mth order elliptic operator.

space of piecewise

polynomials

of a

We describe

and devise a Gaussian quadrature

rule that is suitable for replacing the integrals in the usual least squares method. We then show that if the quadrature der of convergence

1. Introduction.

rule is of sufficient

accuracy,

the optimal

or-

is obtained.

Let a < b. We shall consider a scheme for finding an approxi-

mate solution to the following uniformly strongly elliptic boundary value problem: m

Lu(x)= (1-1)

L

(rlïDr(ars(x)Dsu(x))

= /(*)

on (a, b),

r,s=0

Lfuia) = Ifu{b~)= 0 for 0 2m, z > 2m - 1 and A = {x¡}f=0. (2) We have that

(2.2)

inf II? - xll2m < Ch"-2m\\g\\n for ûlgEV", where h = max{(x/+ j - x¡): 0 < i < TV}and C is independent of g and h.

We note that C is allowed to depend on an upper bound a for the mesh ratio, given by (/i/min{(xí+1 -x¡): 0 < / 0 and fix some interval (jc,., xi+1) in A. Let

{z«}jL, be the roots of the A'th Legendre polynomial on (xt, *I+1). Gaussian quadrature nodes in (x¡, *í+1).

These are the

It is well known that there exist (unique)

weights tv« > 0, 1 A.

Since b'(g - A'g, x) = 0, Schwarz's inequality (we recall that w.- > 0) and simple esti-

mates yield the following:

Cllx-A'g\\\m < b'(x - A'g, x-A'g) = b'(x - A'g, X'g) < [b'(x -A'g,x-

A'g)}ll2 [b'(x -g,X-g)V/2