From d3cadd62028b5f863d83b7db79e234f34cf2c0d6 Mon Sep 17 00:00:00 2001
From: "g.mandl" <g.mandl@noreply.imsc.uni-graz.at>
Date: Wed, 14 Jan 2026 18:42:43 +0100
Subject: [PATCH] =?UTF-8?q?Dateien=20nach=20=E2=80=9ESheet=5F6=E2=80=9C=20?=
 =?UTF-8?q?hochladen?=
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---
 Sheet_6/ex_6_a.m | 80 ++++++++++++++++++++++++++++++++++++++++++++
 Sheet_6/ex_6_b.m | 71 +++++++++++++++++++++++++++++++++++++++
 Sheet_6/ex_6_c.m | 86 ++++++++++++++++++++++++++++++++++++++++++++++++
 3 files changed, 237 insertions(+)
 create mode 100644 Sheet_6/ex_6_a.m
 create mode 100644 Sheet_6/ex_6_b.m
 create mode 100644 Sheet_6/ex_6_c.m

diff --git a/Sheet_6/ex_6_a.m b/Sheet_6/ex_6_a.m
new file mode 100644
index 0000000..a5ebd44
--- /dev/null
+++ b/Sheet_6/ex_6_a.m
@@ -0,0 +1,80 @@
+% Sheet 6 / Ex A
+clf, clear, close all
+
+% you need to play around with the parameters (enough starting nodes,
+% refinements etc)
+
+%p_vec = [5,10,20,100];
+p = 100; % choose your p
+n_nodes = 10;    % number of nodes on the coarse mesh (even - x=0 not included, odd - x=0 included)
+nodes = linspace(-1,1,n_nodes);
+f = @(x) 2*p^3.*x./(p^2.*x.^2+1).^2;
+u_exact = @(x) atan(p*x);
+lambda = @(x) 1+0*x;
+
+% h-adaptivity
+it_h = 4;   % number of refinements with h-adaptivity
+for it = 1:it_h
+    [K,f_vec] = assembling(nodes,lambda,f);
+    % adaption for boundary (dirichlet left)
+    f_vec(1) = f_vec(1) + K(1,1)*1e6*(-atan(p));
+    K(1,1) = K(1,1)*(1+1e6);
+    % adaption for boundary (neumann right)
+    f_vec(end) = f_vec(end) + p/(p^2+1);
+    
+    % solving the system
+    u = K\f_vec;
+    
+    % adapting the mesh
+    if it < it_h
+        nodes = adapt_h(nodes,u,lambda);
+    end
+end
+
+figure(1)
+hold on
+plot(nodes,u,'-o')
+fplot(u_exact,[-1,1])
+title(['h-adapt: u_h, u for p=' num2str(p) ' after ' num2str(it_h) ' refinements, nstart=' num2str(n_nodes) ', nend=' num2str(length(nodes))]);
+xlabel('x values');
+ylabel('u');
+legend('u_h', 'u');
+grid on;
+hold off
+file_name_1 = ['ex_6_A_h-adap_p' num2str(p) '_ref' num2str(it_h) '_n_nodes' num2str(n_nodes) '.jpg'];
+saveas(figure(1), file_name_1);
+
+
+% r-adaptivity
+n_nodes_r = 20;    % number of nodes on the coarse mesh (even - x=0 not included, odd - x=0 included)
+nodes_r = linspace(-1,1,n_nodes_r);
+it_r = 3;   % number of refinements with r-adaptivity
+for it = 1:it_r
+    [K,f_vec] = assembling(nodes_r,lambda,f);
+    % adaption for boundary (dirichlet left)
+    f_vec(1) = f_vec(1) + K(1,1)*1e6*(-atan(p));
+    K(1,1) = K(1,1)*(1+1e6);
+    % adaption for boundary (neumann right)
+    f_vec(end) = f_vec(end) + p/(p^2+1);
+    
+    % solving the system
+    u_r = K\f_vec;
+    
+    % adapting the mesh
+    if it < it_r
+        nodes_r = adapt_r(nodes_r,u_r,lambda);
+    end
+end
+
+figure(2)
+hold on
+plot(nodes_r,u_r,'-o')
+fplot(u_exact,[-1,1])
+title(['r-adapt: u_h, u for p=' num2str(p) ' after ' num2str(it_r) ' refinements and n=' num2str(n_nodes_r)]);
+xlabel('x values');
+ylabel('u');
+legend('u_h', 'u');
+grid on;
+hold off
+file_name_2 = ['ex_6_A_r-adap_p' num2str(p) '_ref' num2str(it_r) '_n_nodes' num2str(n_nodes_r) '.jpg'];
+saveas(figure(2), file_name_2);
\ No newline at end of file
diff --git a/Sheet_6/ex_6_b.m b/Sheet_6/ex_6_b.m
new file mode 100644
index 0000000..6949978
--- /dev/null
+++ b/Sheet_6/ex_6_b.m
@@ -0,0 +1,71 @@
+% Sheet 6 / Ex B
+clf, clear, close all
+
+% you need to play around with the parameters (enough starting nodes,
+% refinements etc)
+
+n_nodes = 10;    % number of nodes on the coarse mesh
+nodes = linspace(0,1,n_nodes);
+f = @(x) 0;
+lambda = @(x) (x<1/sqrt(2))*1 + (x>=1/sqrt(2))*10;
+
+% h-adaptivity
+it_h = 10;   % number of refinements with h-adaptivity
+for it = 1:it_h
+    [K,f_vec] = assembling(nodes,lambda,f);
+    % adaption for dirichlet boundary
+    K(1,1) = K(1,1)*(1 + 1e6);
+    f_vec(end) = K(end,end)*1e6;
+    K(end,end) = K(end,end)*(1 + 1e6);
+    
+    % solving the system
+    u = K\f_vec;
+    
+    % adapting the mesh
+    if it < it_h
+        nodes = adapt_h(nodes,u,lambda);
+    end
+end
+
+figure(1)
+hold on
+plot(nodes,u,'-o')
+title(['h-adapt: u_h after ' num2str(it_h) ' refinements, nstart=' num2str(n_nodes) ', nend=' num2str(length(nodes))]);
+xlabel('x values');
+ylabel('u');
+grid on;
+hold off
+file_name_1 = ['ex_6_B_h-adap_ref' num2str(it_h) '_n_nodes' num2str(n_nodes) '.jpg'];
+saveas(figure(1), file_name_1);
+
+
+% r-adaptivity
+n_nodes_r = 10;    % number of nodes on the coarse mesh
+nodes_r = linspace(0,1,n_nodes_r);
+it_r = 3;   % number of refinements with r-adaptivity
+for it = 1:it_r
+    [K,f_vec] = assembling(nodes_r,lambda,f);
+    % adaption for dirichlet boundary
+    K(1,1) = K(1,1)*(1 + 1e6);
+    f_vec(end) = K(end,end)*1e6;
+    K(end,end) = K(end,end)*(1 + 1e6);
+    
+    % solving the system
+    u_r = K\f_vec;
+    
+    % adapting the mesh
+    if it < it_r
+        nodes_r = adapt_r(nodes_r,u_r,lambda);
+    end
+end
+
+figure(2)
+hold on
+plot(nodes_r,u_r,'-o')
+title(['r-adapt: u_h after ' num2str(it_r) ' refinements and n=' num2str(n_nodes_r)]);
+xlabel('x values');
+ylabel('u');
+grid on;
+hold off
+file_name_2 = ['ex_6_B_r-adap_ref' num2str(it_r) '_n_nodes' num2str(n_nodes_r) '.jpg'];
+saveas(figure(2), file_name_2);
\ No newline at end of file
diff --git a/Sheet_6/ex_6_c.m b/Sheet_6/ex_6_c.m
new file mode 100644
index 0000000..bf07103
--- /dev/null
+++ b/Sheet_6/ex_6_c.m
@@ -0,0 +1,86 @@
+% Sheet 6 / Ex C
+clf, clear, close all
+
+% you need to play around with the parameters (enough starting nodes,
+% refinements etc)
+
+p = 70; % parameter
+%p = -70;
+
+n_nodes = 10;    % number of nodes on the coarse mesh
+nodes = linspace(0,1,n_nodes);
+f = @(x) 0;
+lambda = @(x) 1+0*x;
+
+% h-adaptivity
+it_h = 10;   % number of refinements with h-adaptivity
+for it = 1:it_h
+    [K,f_vec] = assembling(nodes,lambda,f);
+    % need to add the term for phi'*phi (convection term)
+    conv_loc = p/2*[-1,-1;1,1];
+    for k=1:length(nodes)-1
+        K(k:k+1,k:k+1) = K(k:k+1,k:k+1) + conv_loc;
+    end
+    
+    % adaption for dirichlet boundary
+    K(1,1) = K(1,1)*(1 + 1e6);
+    f_vec(end) = K(end,end)*1e6;
+    K(end,end) = K(end,end)*(1 + 1e6);
+    
+    % solving the system
+    u = K\f_vec;
+    
+    % adapting the mesh
+    if it < it_h
+        nodes = adapt_h(nodes,u,lambda);
+    end
+end
+
+figure(1)
+hold on
+plot(nodes,u,'-o')
+title(['h-adapt: u_h after ' num2str(it_h) ' refinements, nstart=' num2str(n_nodes) ', nend=' num2str(length(nodes))]);
+xlabel('x values');
+ylabel('u');
+grid on;
+hold off
+file_name_1 = ['ex_6_C_h-adap_ref' num2str(it_h) '_n_nodes' num2str(n_nodes) '.jpg'];
+saveas(figure(1), file_name_1);
+
+
+% r-adaptivity
+n_nodes_r = 20;    % number of nodes on the coarse mesh
+nodes_r = linspace(0,1,n_nodes_r);
+it_r = 3;   % number of refinements with r-adaptivity
+for it = 1:it_r
+    [K,f_vec] = assembling(nodes_r,lambda,f);
+    % need to add the term for phi'*phi (convection term)
+    conv_loc = p/2*[-1,-1;1,1];
+    for k=1:length(nodes_r)-1
+        K(k:k+1,k:k+1) = K(k:k+1,k:k+1) + conv_loc;
+    end
+    
+    % adaption for dirichlet boundary
+    K(1,1) = K(1,1)*(1 + 1e6);
+    f_vec(end) = K(end,end)*1e6;
+    K(end,end) = K(end,end)*(1 + 1e6);
+    
+    % solving the system
+    u_r = K\f_vec;
+    
+    % adapting the mesh
+    if it < it_r
+        nodes_r = adapt_r(nodes_r,u_r,lambda);
+    end
+end
+
+figure(2)
+hold on
+plot(nodes_r,u_r,'-o')
+title(['r-adapt: u_h after ' num2str(it_r) ' refinements and n=' num2str(n_nodes_r)]);
+xlabel('x values');
+ylabel('u');
+grid on;
+hold off
+file_name_2 = ['ex_6_C_r-adap_ref' num2str(it_r) '_n_nodes' num2str(n_nodes_r) '.jpg'];
+saveas(figure(2), file_name_2);
\ No newline at end of file