The approach you have taken for the solution seems incorrect as the plot for the errors for half points is a straight line parallel to x-axis while the error for cell averages is downward sloping. This implies that while there is no change in error of half points, the error rate of averages is constantly decreasing with the increase in grid points which seems counter-intuitive.
The reason for this error seems to be due to the way half point arrays are defined using vectorization which does not handle the periodic boundary point indices. To resolve this issue, handle the periodic boundary points in the code by defining them using a loop and a modulo operation on indices of the half point arrays as that would wrap the indices over the periodic boundaries.
% Outer loop for grid refinements
for l = 2:j
% Operations
% Scheme (1): Point values
% Looping to fill the half points array for every point
phi_half_points = zeros(1, N-1); % N is the grid value at current index l
for i = 1:N-1
phi_half_points(i) = -1/8 * phi_values(mod(i-2, N) + 1) + ...
3/4 * phi_values(i) + ...
3/8 * phi_values(i+1);
end
% Calculate error for point values
errors_points(l) = max(abs(phi_half_points - phi_exact_half));
% Scheme (2): Cell averages
% Looping to fill half cells array for every point
phi_cell_averages = (phi_values(1:end-1) + phi_values(2:end)) / 2;
phi_half_cells = zeros(1, N-2);
for i = 1:N-2
phi_half_cells(i) = -1/6 * phi_cell_averages(mod(i-2, N-1) + 1) + ...
5/6 * phi_cell_averages(i) + ...
1/3 * phi_cell_averages(i+1);
end
% Calculate error for cell averages
errors_cells(l) = max(abs(phi_half_cells - phi_exact_half(1:end-1)));
end
