ex4

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()