How to subtract two column vectors properly?

I am trying to calculate the discretization error in my code which is simply the difference between my numerical solution and exact solution. This number should be real small (close to zero) which I expect here, but for some reason I am getting a large number.

Code:

N=16;
[D,x] = cheb(N); 
ylow = 0; %a
yupp = 16; %b
Ly = yupp-ylow;
eta_ygl  = 2/Ly;
x = (1/2)*((yupp-ylow)*x + (yupp+ylow));
D=D*eta_ygl; 
D2 = D^2;
D2 = D2(2:N,2:N);
u = (yupp*exp(4*ylow)-ylow*exp(4*yupp)+ylow*exp(4*x)-yupp*exp(4*x)-exp(4*ylow)*x+exp(4*yupp)*x)/(16*ylow-16*yupp);
n = x.^2+10;

dndx = D *n; 
dudx = D *u;
prod1 = dndx .* dudx;
du2dx = D * dudx;
prod2 = n .*du2dx;

invN = (1./n) ;

source = prod1(2:N) + prod2(2:N);
uold = ones(size(u(2:N)));
max_iter =500; 
err_max = 1e-8;

 for iterations = 1:max_iter
    phikmax_old = (max(abs(uold)));
    duoldx = D(2:N,2:N) *uold;
    dnudx = dndx(2:N) .* duoldx;
    ffh = source; 
    RHS = ffh - dnudx; 
    Stilde = invN(2:N) .* RHS;

    unew = D2\Stilde;
    phikmax = (max(abs(unew)));

    if phikmax < err_max 
        it_error = err_max /2;
    else
        it_error = abs( phikmax - phikmax_old) / phikmax;
    end
    if it_error < err_max
        break;
    end
    uold = unew;

 end
unew = [0;unew;0];
DEsol = (unew-u) %HERE

DEsol = 17×1

1.0e+19 * 0 0.0077 0.0340 0.0876 0.1811 0.3252 0.5203 0.7432 0.9367 1.0166

DEsol is large but should be small considering unew and u are the same. Am I subtracting the columns here incorrectly? Thanks

Answer 1

[Matt J]

The relative error is the only thing you can expect to be small, which it is:

DEsol = norm(unew-u)/norm(u)   %relative error

DEsol = 2.5021e-08

I don't know if 2.5e-8 is as small as you'd hoped, but it's hard to know how from looking at your code what the sources and magnitudes of the errors are.