ex4

ex4/code/task_a.py

@@ -0,0 +1,57 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+# PDE:
+#  -u''(x) + a*u(x) = f(x)         x in (0,1)
+#              u(0) = 0
+#             u'(1) = \alpha*(g_b - u(1))
+
+# parameters
+a = 1
+f = 3
+alpha = 1
+gb = 1
+n = 10       # elements
+
+# mesh
+m = n+1      # nodes
+h = 1.0/n
+x = np.linspace(0,1,m)
+
+# local stiffness matrix
+K_loc = np.zeros((2,2))
+A = (1.0/h) * np.array([[ 1,-1], [-1, 1]])
+B = (a*h/6) * np.array([[ 2, 1], [ 1, 2]])
+K_loc = A+B
+
+# Assembling
+K = np.zeros((m,m))
+F = np.zeros(m)
+for i in range(n):
+    K[i:i+2,i:i+2] += K_loc
+    F[i:i+2] += np.full(2, f*h/2)
+
+# Boundary conditions
+# Dirichlet: u(0) = 0
+K[0,:] = 0
+K[0,0] = 1
+F[0]   = 0   
+# Neumann: u'(1) = \alpha*(g_b - u(1))
+A[-1,-1] += alpha
+F[-1] += alpha*gb
+
+u = np.linalg.solve(K, F)
+plt.plot(x, u, "-o", label="u_h")
+plt.title(f"n = {n} | h = {h} | a = {a} | f(x) = {f} | alpha = {alpha} | gb = {gb}")
+plt.xlabel("x")
+plt.ylabel("u(x)")
+plt.legend()
+plt.grid(True)
+
+print("K = ", K)
+print("f = ", F)
+print("u = ", u)
+
+plt.tight_layout()
+plt.savefig("../task_a.png", dpi=300)
+plt.show()

ex4/code/task_b.py

@@ -0,0 +1,61 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+# PDE:
+#  -(lambda(x)u'(x))' = 0         x in (0,1)
+#                u(0) = 0
+#                u(1) = 1
+#           lambda(x) = |  1     x in (0,1/sqrt(2))
+#                       | 10     x in (1/sqrt(2),1)   
+
+# parameters
+n  = 2             # elements (must be even)
+
+# mesh
+m = n+1            # nodes
+nh = int(n/2)      # elements per subdomain   
+mh = nh+1          # nodes per subdomain
+jump = 1/np.sqrt(2)
+x1 = np.linspace(0,jump, mh)
+x2 = np.linspace(jump, 1, mh)
+x = np.concatenate((x1[:-1],x2))
+h1 = jump/nh
+h2 = (1-jump)/nh
+
+# local stiffness matrix
+K_loc1 = ( 1.0/h1) * np.array([[ 1,-1], [-1, 1]])
+K_loc2 = (10.0/h2) * np.array([[ 1,-1], [-1, 1]])
+
+# Assembling
+K = np.zeros((m,m))
+F = np.zeros(m)
+for i in range(nh):
+    K[i:i+2,i:i+2] += K_loc1
+for i in range(nh,n):
+    K[i:i+2,i:i+2] += K_loc2
+
+# Boundary conditions
+# Dirichlet: u(0) = 0
+K[0,:] = 0
+K[0,0] = 1
+F[0]   = 0   
+# Dirichlet: u(1) = 1
+K[-1,:] = 0
+K[-1,-1] = 1
+F[-1]   = 1
+
+u = np.linalg.solve(K, F)
+plt.plot(x, u, "-o", label="u_h")
+plt.title(f"n = {n} (already exact)")
+plt.xlabel("x")
+plt.ylabel("u(x)")
+plt.legend()
+plt.grid(True)
+
+print("K = ", K)
+print("f = ", F)
+print("u = ", u)
+
+plt.tight_layout()
+plt.savefig("../task_b.png", dpi=300)
+plt.show()

ex4/code/task_c.py

@@ -0,0 +1,61 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+# Peclet problem:
+#  -u''(x) + pu'(x) = 0    x in (0,1)
+#              u(0) = 0
+#              u(1) = 1
+
+# parameters
+p = 70
+n_values = [10,20,30,40,70]       # elements
+
+# exact solution
+x_exact = np.linspace(0,1,1000)
+u_exact = (np.exp(p*x_exact)-1)/(np.exp(p)-1)
+
+for n in n_values:
+    # mesh
+    m = n+1      # nodes
+    h = 1.0/n
+    x = np.linspace(0,1,m)
+
+    # local stiffness matrix
+    K_loc = np.zeros((2,2))
+    A = (1.0/h) * np.array([[ 1,-1], [-1, 1]])
+    B = (p/2) * np.array([[ -1, 1], [ -1, 1]])
+    K_loc = A+B
+
+    # Assembling
+    K = np.zeros((m,m))
+    F = np.zeros(m)
+    for i in range(n):
+        K[i:i+2,i:i+2] += K_loc
+
+    # Boundary conditions
+    # Dirichlet: u(0) = 0
+    K[0,:] = 0
+    K[0,0] = 1
+    F[0]   = 0   
+    # Dirichlet: u(1) = 1
+    K[-1,:] = 0
+    K[-1,-1] = 1
+    F[-1]   = 1
+
+    u = np.linalg.solve(K, F)
+    plt.plot(x, u, "-o", markersize=2, label=f"n = {n}")
+
+plt.plot(x_exact, u_exact, "black", label="exact")
+plt.title(f"p = {p}")
+plt.xlabel("x")
+plt.ylabel("u(x)")
+plt.legend()
+plt.grid(True)
+
+print("K = ", K)
+print("f = ", F)
+print("u = ", u)
+
+plt.tight_layout()
+plt.savefig("../task_c.png", dpi=300)
+plt.show()