Delete Sheet4/Ex2.m

Sheet4/Ex2.m

@@ -1,79 +0,0 @@
-%% FEM vs analytical solution for -(lambda(x) u')' = 0 with Dirichlet BC
-clear; clc;
-
-%% Parameters
-n = 20;                 % number of elements (increase for nicer plot)
-h = 1/n;
-x = linspace(0,1,n+1)'; % node coordinates
-
-% Jump location
-x0 = 1/sqrt(2);
-
-% Piecewise lambda function
-lambda_fun = @(x) (x < x0).*1 + (x >= x0).*10;
-
-%% Assemble FEM matrix and RHS (lifting via u_D(x)=x)
-A = zeros(n+1,n+1);
-F = zeros(n+1,1);
-
-% Derivatives of linear shape functions are constant on each element
-phi_prime = [-1/h; 1/h];
-
-for e = 1:n
-    % element nodes
-    nodes = [e, e+1];
-    xL = x(nodes(1)); 
-    xR = x(nodes(2));
-
-    % CASE A: no jump inside the element
-    if (xR <= x0) || (xL >= x0)
-        lambda_e = lambda_fun((xL + xR)/2);  % constant lambda
-        A_e = lambda_e * (1/h) * [1 -1; -1 1];
-        % load due to u_D'(x)=1
-        F_e = [ lambda_e; -lambda_e ];
-
-    % CASE B: jump inside the element -> split integrals
-    else
-        h1 = x0 - xL;
-        h2 = xR - x0;
-
-        % Local stiffness contributions
-        A_e = [ (1*h1 + 10*h2)/h^2 , -(1*h1 + 10*h2)/h^2 ;
-               -(1*h1 + 10*h2)/h^2 ,  (1*h1 + 10*h2)/h^2 ];
-
-        % load vector components
-        F_e = [ (h1 + 10*h2)/h ;
-               -(h1 + 10*h2)/h ];
-    end
-
-    % Assemble
-    A(nodes,nodes) = A(nodes,nodes) + A_e;
-    F(nodes) = F(nodes) + F_e;
-end
-
-%% Apply Dirichlet BC: u(0)=0, u(1)=1
-% Dirichlet BC: w(0)=0, w(1)=0
-A(1,:) = 0; A(:,1) = 0; A(1,1) = 1;
-A(end,:) = 0; A(:,end) = 0; A(end,end) = 1;
-F(1) = 0;
-F(end) = 0;
-
-%% Solve FEM
-w = A \ F;
-u_fem = x + w;
-
-%% Analytical solution with piecewise λ(x)
-x0 = 1/sqrt(2);
-J = 1 / (x0 + (1-x0)/10);
-
-u_exact = @(x) (x <= x0) .* (J*x) + ...
-               (x >  x0) .* ( J*x0 + (J/10)*(x - x0) );
-u_an = u_exact(x);
-
-%% Plot
-plot(x,u_fem,'o-','LineWidth',2); hold on;
-plot(x,u_an,'r--','LineWidth',2);
-legend('FEM','Analytical','Location','Best');
-xlabel('x'); ylabel('u(x)');
-title('FEM vs Analytical Solution for (B)');
-grid on;