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MatLab file, Exercise 3

Sheet4/Ex3.m

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+clear; close all; clc;
+p = 70;
+Nvals = [10 20 30 40 70];
+
+figure; hold on; box on;
+
+for N = Nvals
+    h = 1/N;
+    x = linspace(0,1,N+1)'; 
+    A = zeros(N-1);
+    b = zeros(N-1,1);
+
+    for i = 1:N-1
+        A(i,i) = 2/h;
+
+        if i > 1
+            A(i,i-1) = -1/h - p/2;
+        end
+
+        if i < N-1
+            A(i,i+1) = -1/h + p/2;
+        end
+    end
+    b(end) = (1/h - p/2);
+    uh_inner = A\b;
+    uh = [0; uh_inner; 1];
+
+    plot(x, uh, 'o-', 'LineWidth', 2, ...
+         'DisplayName', ['FEM, N=', num2str(N)]);
+end
+
+x_exact = linspace(0,1,2000)';
+u_exact = (exp(p*x_exact) - 1)/(exp(p) - 1);
+plot(x_exact, u_exact, 'k', 'LineWidth', 1, 'DisplayName', 'Exact');
+
+title("FEM vs Analytical Solution for -u'' + p u' = 0,  p=70");
+xlabel('x'); ylabel('u(x)');
+legend('Location','best');