Convergence of Heat Equation?

%% Ut=Uxx,

%%solution: U(x,t)=sin(2*pi*x)*exp(-(2*pi).^2*t),

%%I.C : U(x,0)= sin(2*pi*x)

%%B.C: U(0,t)=U(1,t)=0,

L = 1 ; %Length of Domain

t_final = 0.1; % time limit

N = 32; % no of elements

dx = 1/N; % step size

dt = 0.001

% x = 0:dx:L; %spatial discretization

x = linspace(0,L,N+1);

t = 0:dt:t_final; %temporal discretization

a= length(x);

b = length(t);

U = zeros(1,b);

W = zeros(a,b);

X = sin(2*pi*x);

W(:,1) =X;

W(:, b)=0;

% t=0;

T = zeros(b,1);

T(1) = 0;

for i=1:length(t)

T= T+dt;

U = sin(2*pi*x)*(exp((-(2*pi).^2)*T(i,:)));

W(:,i) = U;

X = U;

U(1) =0;

U(end)=0;

T(i) = T(1);

figure(1)

plot(x,U)

hold on

end

%% Backward Euler for FEM solution

xx = linspace(0,L,N+1); %Fem spatial discretization

% xx = 0:dx:L;

nx = length(xx); %total nodes

M = spdiags([(dx/6)*ones(nx,1), (2*dx/3)*ones(nx,1), (dx/6)*ones(nx,1)],[-1, 0,1],nx,nx) ; %% Mass matrix

A = spdiags([(-1/dx)*ones(nx,1), (2/dx)*ones(nx,1), (-1/dx)*ones(nx,1)],[-1, 0, 1],nx,nx) ;%% Stiffness matrix

delta_t = 0.001;

m = t_final/delta_t;

delta_t = t_final/m;

tspan = 0:delta_t:t_final;

% store solution for each time in matrix u

u = zeros(nx,m+1);

tvec = zeros(1,m+1);

U_tk = sin(2*pi*xx);

u(:,1) = U_tk;

tt = 0 ;

tvec(1) = tt;

tt = 0;

b = zeros(1,nx); % Neumann boundary condition

b(nx) = 0;

for nt = 1:m

tt = tt+delta_t;

c = U_tk + b;

w = (M+delta_t*A)\M;

FEM = c*w;

FEM(1) = 0;

FEM(end)= 0;

u(:, nt) = FEM;

% for next time step:

U_tk = FEM;

tvec(nt) = tt;

figure(2)

plot(xx,FEM)

end

plot(x,U-FEM); )

ERROR = sqrt(sum( dx*(U-FEM).^2 ))