Problem with double-mesh and error estimation

Question

I am writing a Matlab code for a first-order nonlinear singularly perturbed parameterized problem with integral boundary condition. I have successfully achieved the desired order of convergence for the solution u(x) but same is not happening with the parameter \lambda.

At first, I calculated the solution $\{u(x),\lambda\}$ at N=32. Since the exact solution to the problem is not available, therefore I employed the double-mesh principle. For this, I created a double mesh, say xx and obtained the solution $\{U,\Lambda\}$. Then I stored their corresponding errors. In this way, one loop is completed. Then I fixed $N=64$, and repeated the same steps. I observed the convergence rate of $u$ is quadratic but not for $\lambda$. What am i doing wrong? Below is the code.

for r=1:1
    epsln=2^(-2*r);
    for n=1:1
        N=32*2^(n);
        %% Shishkin Mesh
        alph=2;%min. value of g_u
        sigm=min(0.5,2*(epsln/alph)*log(N));%Transition Point
        t=unique([linspace(0,sigm,N/2+1),linspace(sigm,1,N/2+1)]);
        h=diff(t);%step-size
        % Initial Solution
        u=1-t.^2;
        l=-0.4;
        %% Iterated Solution
        errU=1;
        errL=1;
        iter=1;
        while (errU >Tol || errL>Tol)
            [U,l_new]=Solution_Fun_ex1_Kudu(N,t,u,h,l,epsln);
            errU=max(abs(u-U))
            errL=abs(l-l_new)
            u=U;
            l=l_new
            iter=iter+1;
        end
        iter
        %Final Solution is stored in [u,l]
        %%Doublemesh Function
        z=Doublemesh_Fun_kudu(t,N);
        hh=diff(z);
        %Initial Solution
        y=1-z.^2;
        ll=-0.4;
        %%Iterated Solution
        errY=1;
        errLd=1;
        iterr=1;
        while (errY>Tol || errLd>Tol)
            [Y,ll_new]=Solution_Fun_ex1_Kudu(2*N,z,y,hh,ll,epsln);
            errY=max(abs(y-Y));
            errLd=abs(ll-ll_new);
            y=Y;
            ll=ll_new;
            iterr=iterr+1;
        end
        %Final Double mesh Solution is stored in [y,ll]
        
        %%Calculating Errors
        Err=zeros(1,N+1);
        for i=1:N+1
            Err(i)=abs(u(i)-y(2*i-1));
        end
        E(r,n)=max(Err)
        E_l(r,n)=abs(l-ll);
        %%Calculating Rate of Convergence
        if n>1
            R(r,n-1)=log(E(r,n-1)/E(r,n))/log(2);
            R_l(r,n-1)=log(E_l(r,n-1)/E_l(r,n))/log(2);
        end
        Meshpoints(n)=N;
    end
    Val_epsln(r)=epsln;
end