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% Make the mesh:
domain = [2,3]; nElement = 100;
Mesh = mesh_generator_1d(domain, nElement);

% Form a data structure to access the finite element basis:
iDegree = 3; Fem = femherm1d(Mesh, iDegree); hDegree = 1;

% Create an array with the first 100 entries corresponding to the values
% of sin(2*pi*x) evaluated at the Mesh nodes:
uGlobal = sin(2*pi*Fem.point(1:nElement+1)); iDerivative = 0;

% Extend the array so that the last 100 entries correspond to the values of
% 2*pi*cos(2*pi*x) evaluated at the Mesh nodes:
uGlobal(nElement+2:2*(nElement+1)) = 2*pi*cos(2*pi*Fem.point(nElement+2:2*(nElement+1)));

% Set the resolution and prepare the figure:
reso = 20; figure(1); clf; hold on;
uInterpolate = zeros(1,(reso+1)*size(Mesh.element,2));

for k = 1:size(Mesh.element,2) % Loop over each element in the mesh
    element = Mesh.node(Mesh.element(:,k)); % Extract element endpoints
    uLocal = uGlobal(Fem.T(:,k)); % uLocal(1) = u1, uLocal(2) = u2, uLocal(3) = u1', uLocal(4) = u2'
    x = linspace(element(1), element(2), reso+1); % x contains the points we are evaluating the shape functions at
    uInterpolate(((k-1)*(reso+1)+1):(k*(reso+1))) = evalfeherm1d(x, uLocal, ...
        element, hDegree, iDerivative); % Evaluate shape functions at x and scale by corresponding coefficients in uLocal 
    plot(x,uInterpolate(((k-1)*(reso+1)+1):(k*(reso+1)))); % Plot the result
end
hold off

% Make the mesh:
domain = [2,3]; nElement = 100;
Mesh = mesh_generator_1d(domain, nElement);

% Form a data structure to access the finite element basis:
iDegree = 3; Fem = femherm1d(Mesh, iDegree); hDegree = 1;

% Create an array with the first 100 entries corresponding to the values
% of sin(2*pi*x) evaluated at the Mesh nodes:
uGlobal = sin(2*pi*Fem.point(1:nElement+1)); iDerivative = 0;

% Extend the array so that the last 100 entries correspond to the values of
% 2*pi*cos(2*pi*x) evaluated at the Mesh nodes:
uGlobal(nElement+2:2*(nElement+1)) = 2*pi*cos(2*pi*Fem.point(nElement+2:2*(nElement+1)));

% Set the resolution and prepare the figure:
reso = 20; figure(1); clf; hold on;
uInterpolate = zeros(1,(reso+1)*size(Mesh.element,2));

for k = 1:size(Mesh.element,2) % Loop over each element in the mesh
    element = Mesh.node(Mesh.element(:,k)); % Extract element endpoints
    uLocal = uGlobal(Fem.T(:,k)); % uLocal(1) = u1, uLocal(2) = u2, uLocal(3) = u1', uLocal(4) = u2'
    x = linspace(element(1), element(2), reso+1); % x contains the points we are evaluating the shape functions at
    uInterpolate(((k-1)*(reso+1)+1):(k*(reso+1))) = evalfeherm1d(x, uLocal, ...
        element, hDegree, iDerivative); % uInterpolate = u1*N1 + u2*N2 + u1'*N3 + u2'*N4
    plot(x,uInterpolate(((k-1)*(reso+1)+1):(k*(reso+1)))); % Plot the result
end
hold off

% Make the mesh:
domain = [2,3]; nElement = 100;
Mesh = mesh_generator_1d(domain, nElement);

% Form a data structure to access the finite element basis:
iDegree = 3; Fem = femherm1d(Mesh, iDegree); hDegree = 1;

% Create an array with the first 100 entries corresponding to the values
% of sin(2*pi*x) evaluated at the Mesh nodes:
uGlobal = sin(2*pi*Fem.point(1:nElement+1)); iDerivative = 0;

% Extend the array so that the last 100 entries correspond to the values of
% 2*pi*cos(2*pi*x) evaluated at the Mesh nodes:
uGlobal(nElement+2:2*(nElement+1)) = 2*pi*cos(2*pi*Fem.point(nElement+2:2*(nElement+1)));

% Set the resolution and prepare the figure:
reso = 20; figure(1); clf; hold on;
uInterpolate = zeros(1,(reso+1)*size(Mesh.element,2));

for k = 1:size(Mesh.element,2) % Loop over each element in the mesh
    element = Mesh.node(Mesh.element(:,k)); % Extract element endpoints
    uLocal = uGlobal(Fem.T(:,k)); % uLocal(1) = u1, uLocal(2) = u2, uLocal(3) = u1', uLocal(4) = u2'
    x = linspace(element(1), element(2), reso+1); % x contains the points we are evaluating the shape functions at
    uInterpolate(((k-1)*(reso+1)+1):(k*(reso+1))) = evalfeherm1d(x, uLocal, ...
        element, hDegree, 0); % Evaluate shape functions at x and scale by corresponding coefficients in uLocal 
    plot(x,uInterpolate(((k-1)*(reso+1)+1):(k*(reso+1)))); % Plot the result
end
hold off

% Make the mesh:
domain = [2,3]; nElement = 100;
Mesh = mesh_generator_1d(domain, nElement);

% Form a data structure to access the finite element basis:
iDegree = 3; Fem = femherm1d(Mesh, iDegree); hDegree = 1;

% Create an array with the first 100 entries corresponding to the values
% of sin(2*pi*x) evaluated at the Mesh nodes:
uGlobal = sin(2*pi*Fem.point(1:nElement+1)); iDerivative = 0;

% Extend the array so that the last 100 entries correspond to the values of
% 2*pi*cos(2*pi*x) evaluated at the Mesh nodes:
uGlobal(nElement+2:2*(nElement+1)) = 2*pi*cos(2*pi*Fem.point(nElement+2:2*(nElement+1)));

% Set the resolution and prepare the figure:
reso = 20; figure(1); clf; hold on;
uInterpolate = zeros(1,(reso+1)*size(Mesh.element,2));

for k = 1:size(Mesh.element,2) % Loop over each element in the mesh
    element = Mesh.node(Mesh.element(:,k)); % Extract element endpoints
    uLocal = uGlobal(Fem.T(:,k)); % uLocal(1) = u1, uLocal(2) = u2, uLocal(3) = u1', uLocal(4) = u2'
    x = linspace(element(1), element(2), reso+1); % x contains the points we are evaluating the shape functions at
    uInterpolate(((k-1)*(reso+1)+1):(k*(reso+1))) = evalfeherm1d(x, uLocal, ...
        element, hDegree, iDerivative); % Evaluate shape functions at x and scale by corresponding coefficients in uLocal 
    plot(x,uInterpolate(((k-1)*(reso+1)+1):(k*(reso+1)))); % Plot the result
end
hold off

One of the comments suggested to use the formula $u(x) \approx u_1 N_1(x) + u_2 N_2(x) + u_1' N_3(x) + u_2' N_4(x)$ instead of $u(x) \approx u_1 N_1(x) + u_2 N_2(x) + u_1 N_3(x) + u_2 N_4 (x)$. I do this by computing $u_1' = 2\pi \cos(2\pi a)$ and $u_2' = 2\pi \cos(2\pi b)$ for each element $K = [a,b]$. Then my Hermite approximation to $\sin(2 \pi x)$ looks like:

In case it helps, here is the MATLAB code for my problem.

One of the comments suggested to use the formula $u(x) \approx u_1 N_1(x) + u_2 N_2(x) + u_1' N_3(x) + u_2' N_4(x)$ instead of $u(x) \approx u_1 N_1(x) + u_2 N_2(x) + u_1 N_3(x) + u_2 N_4 (x)$. I agree that using this formula is what I should have done in this first place. I implement this by computing $u_1' = 2\pi \cos(2\pi a)$ and $u_2' = 2\pi \cos(2\pi b)$ for each element $K = [a,b]$. Then my Hermite approximation to $\sin(2 \pi x)$ looks like:

Which is clearly very wrong. In case it helps, here is the MATLAB code for my problem. I use a few of my own finite element computer programs here.