ex6

ex6/code/task_b.py

@@ -0,0 +1,100 @@
+import numpy as np
+from adapt import *
+
+# 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)   
+
+# rhs
+def f(x):
+    return 0
+
+def lam(x):
+    if x >= 0 and x <= 1/np.sqrt(2):
+        return 1
+    elif x <= 1 and x > 1/np.sqrt(2):
+        return 10
+    else:
+        return print("lambda undefined")
+
+# Stiffness and Load
+def K_loc(k, mesh):
+    K_loc = np.zeros((2,2))
+    K_loc[0,0] = integrate.quad(lambda x: lam(x)*phi_prime(k-1, mesh, x)**2, mesh[k-1], mesh[k])[0]
+    K_loc[1,0] = integrate.quad(lambda x: lam(x)*phi_prime(k-1, mesh, x)*phi_prime(k, mesh, x), mesh[k-1], mesh[k])[0]
+    K_loc[0,1] = integrate.quad(lambda x: lam(x)*phi_prime(k, mesh, x)*phi_prime(k-1, mesh, x), mesh[k-1], mesh[k])[0]
+    K_loc[1,1] = integrate.quad(lambda x: lam(x)*phi_prime(k, mesh, x)**2, mesh[k-1], mesh[k])[0]
+    return K_loc
+def F_loc(k, mesh):
+    F_loc = np.zeros(2)
+    F_loc[0] = integrate.quad(lambda x: f(x) * phi(k-1, mesh, x), mesh[k-1], mesh[k])[0]
+    F_loc[1] = integrate.quad(lambda x: f(x) * phi(k, mesh, x), mesh[k-1], mesh[k])[0]
+    return F_loc
+
+# Assembling
+def Assemble(mesh):
+    m = len(mesh)
+    n = m-1
+
+    K = np.zeros((m,m))
+    F = np.zeros(m)
+    for k in range(1,m):
+        K[k-1:k+1,k-1:k+1] += K_loc(k, mesh)
+        F[k-1:k+1] += F_loc(k, mesh)
+
+    # 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
+    return K,F
+
+def plotting(mesh, u, comment):
+    exact_x = [0, 1/np.sqrt(2), 1]
+    exact = [0, 10/(np.sqrt(2)+9), 1]
+    plt.plot(exact_x, exact, "--", linewidth=1, color="red", label="exact")
+    plt.title(f"n = {n} | {comment}")
+    plt.xlabel("x")
+    plt.ylabel("u(x)")
+    plt.plot(mesh, u, "-o", label="u_h")
+    plt.xticks(mesh, labels=[])
+    plt.legend()
+    plt.grid(True)
+    plt.tight_layout()
+    plt.savefig("task_b.png", dpi=300)
+    plt.show()
+    return 0
+
+##############################################################################
+
+mesh = np.linspace(0, 1, 10)
+m = len(mesh)
+n = m-1
+
+K, F = Assemble(mesh)                                 # assemble
+u = np.linalg.solve(K, F)                             # solve
+jumps = flux_jumps(mesh, u)                           # flux jumps
+plotting(mesh, u, "before adapting")                  # plotting
+
+iterations = 3
+for it in range(iterations):
+    # mesh = adapt_h(mesh, jumps)                       # h-adaptivity
+    mesh = adapt_r(mesh, np.abs(jumps))               # r-adaptivity (positive density (jumps)!)
+    m = len(mesh)
+    n = m-1
+
+    K, F = Assemble(mesh)                             # assemble
+    u = np.linalg.solve(K, F)                         # solve
+    jumps = flux_jumps(mesh, u)                       # flux jumps
+    # plotting(mesh, u, f"iteration {it+1}")            # plotting each iteration
+
+# print(jumps)
+plotting(mesh, u, f"after {iterations} iterations")   # plotting
+