ex6

ex6/code/task_a.py

@@ -0,0 +1,104 @@
+import numpy as np
+from adapt import *
+
+# PDE:
+#  -u''(x) = f(x)       x in (-1,1)
+#    u(-1) = -arctan(p)
+#    u'(1) = p / (p^2 + 1)
+# 
+# weak form
+# int u'v' dx = int f(x) * v(x) dx + p/(p^2+1) * v(1)
+# 
+
+# rhs
+def f(x):
+    return 2 * p**3 * x / (p**2 * x**2 + 1)**2
+
+# Stiffness and Load
+def K_loc(k, mesh):
+    K_loc = np.zeros((2,2))
+    K_loc[0,0] = integrate.quad(lambda x: phi_prime(k-1, mesh, x)**2, mesh[k-1], mesh[k])[0]
+    K_loc[1,0] = integrate.quad(lambda 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: 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: 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(-1) = -arctan(p)
+    K[0,:] = 0
+    K[0,0] = 1
+    F[0]   = -np.arctan(p)
+    # Neumann: u'(1) = p / (p^2 + 1)
+    F[-1] += p/(p**2+1) * phi(n, mesh, 1)
+    return K,F
+
+def plotting(mesh, u, comment):
+    exact_x = np.linspace(-1,1,1000)
+    exact = np.arctan(p*exact_x)
+    plt.plot(exact_x, exact, "--", linewidth=1, color="red", label="exact")
+    plt.title(f"p = {p} | 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_a.png", dpi=300)
+    plt.show()
+    return 0
+
+##############################################################################
+
+# parameters
+p_list = [5,10,20,100]
+for p in p_list:
+
+    # h-adaptivity
+    mesh = np.array([-1.0, -0.2, 0, 0.7, 1.0])            # with 0 as node
+    # mesh = np.array([-1.0,-0.131,0.372,1.0])              # without 0 as node
+
+    # r-adaptivity
+    # mesh = np.linspace(-1, 1, 11)                         # with 0 as node
+    # mesh = np.linspace(-1, 1, 10)                         # without 0 as node
+
+    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 = 6
+    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
+
+