Galerkin method fix the linear two-point BVP

I am a newcomer to matlab,I want to use the Galerkin method with the hat function as the set of basis functions to calculate the solution to the linear two-point BVP :

The hat knots are evenly distributed with the interval h = 1/20 and 1/40.

Compare the results to those of the exact solution, , to evaluate the order of accuracy using

the absolute errors for the two knot intervals.

Hope someone can teach or guide me how to do it .

Answer 1

[Anurag Ojha]

Hey Cheng

Kindly find the code below to use the Galerkin method with hat functions to solve the given boundary value problem (BVP) in MATLAB.

Also have compared the results with the exact solution ( y = t^3 )

function galerkin_bvp()
    % Define the intervals
    intervals = [1/20, 1/40];

    % Exact solution
    exact_solution = @(t) t.^3;

    % Loop over each interval
    for h = intervals
        % Discretize the interval
        t = 0:h:1;
        n = length(t);
        
        % Initialize the matrix A and vector b
        A = zeros(n, n);
        b = zeros(n, 1);
        
        % Fill the matrix A and vector b using the Galerkin method
        for i = 2:n-1
            A(i, i-1) = 1/h^2;
            A(i, i) = -2/h^2;
            A(i, i+1) = 1/h^2;
            b(i) = 6 * t(i);
        end
        
        % Boundary conditions
        A(1, 1) = 1;
        b(1) = 0;
        A(n, n) = 1;
        b(n) = 1;
        
        % Solve the linear system
        y_approx = A \ b;
        
        % Compute the exact solution
        y_exact = exact_solution(t)';
        
        % Compute the absolute errors
        abs_errors = abs(y_exact - y_approx);
        
        % Display the results
        fprintf('Results for h = %f:\n', h);
        fprintf('Max Absolute Error: %e\n', max(abs_errors));
        fprintf('Mean Absolute Error: %e\n\n', mean(abs_errors));
        
        % Plot the results
        figure;
        plot(t, y_exact, 'b', 'LineWidth', 2); hold on;
        plot(t, y_approx, 'ro--');
        legend('Exact Solution', 'Galerkin Approximation');
        title(['Galerkin Method with h = ', num2str(h)]);
        xlabel('t');
        ylabel('y');
        grid on;
    end
end

Results for h = 0.050000:

Max Absolute Error: 2.722128e-15

Mean Absolute Error: 1.475812e-15

Results for h = 0.025000:

Max Absolute Error: 8.523030e-15

Mean Absolute Error: 4.261312e-15

[Torsten]

This is Finite Differencing, not Galerkin.

[John D'Errico]

@Anurag Ojha

Please don't do an obvious homework assignment for a student who has made no effort. This does not teach the student anything, except how to get someone else to do their assignments for them.

But even more silly, you did not even use the method the student needs.