.

Sheet6/Ex6A.m

@@ -0,0 +1,48 @@
+clear; clc; close all;
+pList = [5, 10, 20, 100];
+nIter = 9;
+
+figure;
+tiledlayout(2,2)
+for ip = 1:length(pList)
+    p = pList(ip);
+
+    f = @(x) 2*p^3*x./(p^2*x.^2+1).^2; %RHS
+    lambda = @(x) 1; %DIffusion coefficient 
+
+    u_exact = @(x) atan(p*x);
+
+    x = linspace(-1,1,10); %initial uniform mesh (includes x=0)
+
+    for it = 1:nIter %adaptive FEM loop
+        [K,F] = assemble_1D(x,lambda,f); %assemble global stiffness matrix K and load vector F
+        %Neumann boundary conditions at x=1
+        F(end) = F(end) + p/(p^2+1);
+        %Dirichlet boundary conditions at x=-1
+        K(1,:) = 0;
+        K(1,1) = 1;
+        F(1) = -atan(p);
+
+        u = K \ F;
+
+        if it < nIter % h-adaptivity loop
+            eta = flux_jump(x,u,lambda); 
+            x = h_adapt(x,eta,0.3);
+        end
+    end
+
+    u_ex = u_exact(x(:));        % force column vector
+    err_inf = max(abs(u - u_ex));
+    nexttile
+    plot(x,u,'-o','LineWidth',1.5); hold on
+    plot(x,u_ex,'--','LineWidth',1.5)
+    title(['p = ',num2str(p)])
+    xlabel('x'), ylabel('u')
+    legend('FEM','exact','Location','best')
+    grid on
+
+    fprintf('p = %d\n', p);
+    fprintf('error = %.3e\n\n', err_inf);
+end
+
+sgtitle('Exercise 6A: h-adaptivity and exact solution comparison')

Sheet6/Ex6B.m

@@ -0,0 +1,23 @@
+clear; clc;
+lambda = @(x) (x < 1/sqrt(2)) + 10*(x >= 1/sqrt(2)); %diffusion coefficient 
+f = @(x) 0; %homogeneous RHS
+x = linspace(0,1,6);   %coarse initial mesh, doesn't include x_m
+nIter = 6;
+
+for it = 1:nIter
+    [K,F] = assemble_1D(x,lambda,f); %assemble
+    K(1,:) = 0;  K(1,1) = 1;  F(1) = 0; % Dirichlet BC at x=0
+    K(end,:) = 0; K(end,end) = 1; F(end) = 1; % Dirichlet BC at x=1
+    u = K\F;
+
+    if it < nIter %h-adaptivity loop
+        eta = flux_jump(x, u, lambda);
+        x = h_adapt(x,eta,0.4);
+    end
+end
+
+figure
+plot(x,u,'-o','LineWidth',1.5)
+xlabel('x'), ylabel('u')
+title('Ex6B, h-adaptivity with coefficient jump')
+grid on

Sheet6/Ex6C.m

@@ -0,0 +1,43 @@
+clear; clc; close all;
+p = 70;
+%p=-70;
+lambda = @(x) 1; %constant diffusion
+f = @(x) 0; %homogeneous RHS
+
+N = 30;     % Initial mesh (uniform)
+x = linspace(0,1,N)';
+
+nIter = 8;
+
+for it = 1:nIter
+    [K,F] = assemble_1D(x,lambda,f); %assemble 
+    for i = 1:length(x)-1 %convection term added manually
+        h = x(i+1) - x(i);
+        Ke_conv = p/2 * [-1 1; -1 1]; 
+        K(i:i+1,i:i+1) = K(i:i+1,i:i+1) + Ke_conv;
+    end
+    K(1,:) = 0; K(1,1) = 1; F(1) = 0; % Dirichlet BC at x=0
+    K(end,:) = 0; K(end,end) = 1; F(end) = 1; % Dirichlet BC at x=1
+
+    u = K\F;
+
+    if it < nIter
+        Nn = length(x); %initialize indicator
+        eta = zeros(Nn,1);
+        for j = 2:Nn-1 %interior nodes only
+            ul = (u(j)-u(j-1))/(x(j)-x(j-1)); %left derivative
+            ur = (u(j+1)-u(j))/(x(j+1)-x(j)); %right derivative
+            %Jump in convection-diffusion flux
+            Jl = -ul + p*u(j);
+            Jr = -ur + p*u(j);
+            eta(j) = abs(Jr - Jl);
+        end
+        x = r_adapt(x, eta);
+    end
+end
+
+figure
+plot(x,u,'-o','LineWidth',1.5)
+xlabel('x'), ylabel('u')
+title(['Ex6C, r-adaptivity, Péclet p = ',num2str(p)])
+grid on

Sheet6/assemble_1D.m

@@ -0,0 +1,17 @@
+function [K,F] = assemble_1D(x,lambda,f)
+N = length(x);
+K = zeros(N);
+F = zeros(N,1);
+
+for i = 1:N-1
+    h = x(i+1)-x(i);
+    xm = (x(i)+x(i+1))/2; %mid point evaluation
+    lam = lambda(xm);
+    Ke = lam/h * [1 -1; -1 1]; %exact P1 stiffness matrix
+    xm = (x(i)+x(i+1))/2;
+    Fe = f(xm)*h/2 * [1;1]; %mid point quadrature for RHS
+    %assembly into global system
+    K(i:i+1,i:i+1) = K(i:i+1,i:i+1) + Ke; 
+    F(i:i+1) = F(i:i+1) + Fe;
+end
+end