From 0ca11d55a7105e0013f94958acefac91c8fee64d Mon Sep 17 00:00:00 2001 From: "jakob.schratter" Date: Wed, 26 Nov 2025 10:03:38 +0100 Subject: [PATCH] Upload files to "ex4" --- ex4/ex_4A.py | 76 +++++++++++++++++++++++++++++++++++++++ ex4/ex_4B.py | 83 +++++++++++++++++++++++++++++++++++++++++++ ex4/ex_4C.py | 76 +++++++++++++++++++++++++++++++++++++++ ex4/ex_4_results.pdf | Bin 0 -> 258144 bytes 4 files changed, 235 insertions(+) create mode 100644 ex4/ex_4A.py create mode 100644 ex4/ex_4B.py create mode 100644 ex4/ex_4C.py create mode 100644 ex4/ex_4_results.pdf diff --git a/ex4/ex_4A.py b/ex4/ex_4A.py new file mode 100644 index 0000000..e8dd84e --- /dev/null +++ b/ex4/ex_4A.py @@ -0,0 +1,76 @@ +import numpy as np +import matplotlib.pyplot as plt + +np.set_printoptions(precision=2) + + +N = 10 # number of elements +a = 1 +alpha = 1 +g_b = 1 +f_const = 1 + +x = np.linspace(0, 1, N + 1) + + +A = np.zeros((N + 1, N + 1)) +f_vec = np.zeros(N + 1) + +for i in range(1, N + 1): + h = x[i] - x[i - 1] + + + a_11 = 1./h + a*h/3. + a_12 = -1./h + a*h/6. + a_21 = -1./h + a*h/6. + a_22 = 1./h + a*h/3 + + A[i - 1, i - 1] += a_11 + A[i - 1, i] += a_12 + A[i, i - 1] += a_21 + A[i, i] += a_22 + + + f_vec[i] = f_const*h + +print("A =\n", A) + +# take Neumann data into account +A[N, N] += alpha +f_vec[N] += alpha*g_b + + +# take Dirichlet data into account +u_g = np.zeros(N + 1) +u_g[0] = 0 +print("u_g =\n", u_g) + +# remove first row of A +A_g = A[1:N+1, :] +#print("A_g =\n", A_g) + +# remove first row of f_vec +f_g = f_vec[1:N+1] + +# assemble RHS with dirichlet data +f_g -= A_g.dot(u_g) +#print(f_g) + +# matrix for the inner nodes (excluding nodes with dirichlet bcs) +A_0 = A[1:N+1, 1:N+1] +#print(A_0) + +# solve for u_0 (free dofs) +u_0 = np.linalg.solve(A_0, f_g) + +# assemble "u = u_0 + u_g" +u = np.concatenate([[0], u_0]) + + +print("u =\n", u) + +plt.plot(x, u, '-') +plt.xlabel('x') +plt.ylabel('u_h(x)') +plt.grid() +plt.show() \ No newline at end of file diff --git a/ex4/ex_4B.py b/ex4/ex_4B.py new file mode 100644 index 0000000..cd58310 --- /dev/null +++ b/ex4/ex_4B.py @@ -0,0 +1,83 @@ +import numpy as np +import matplotlib.pyplot as plt + +np.set_printoptions(precision=2) + + + +N = 2 # number of elements +split = np.sqrt(2)/2 +#split = 0.5 + + +N_lower = round(N*split) # number of elements in (0, split) +print(N_lower, "elements below sqrt(2)/2") +N_upper = N - N_lower # number of elements in (split, 1) +print(N_upper, "elements above sqrt(2)/2") + +assert(N == N_lower + N_upper) + + + +x_lower = np.linspace(0, split, N_lower + 1) +x_upper = np.linspace(split, 1, N_upper + 1) +x = np.concatenate([x_lower, x_upper[1:]]) +print(x) + + +A = np.zeros((N + 1, N + 1)) + +for i in range(1, N + 1): + h = x[i] - x[i - 1] + + lam = 1 + if(x[i] > split): + lam = 10 + + + a_11 = lam/h + a_12 = -lam/h + a_21 = -lam/h + a_22 = lam/h + + A[i - 1, i - 1] += a_11 + A[i - 1, i] += a_12 + A[i, i - 1] += a_21 + A[i, i] += a_22 + +print("A =\n", A) + + + +# take dirichlet data into account +u_g = np.zeros(N + 1) +u_g[0] = 0 +u_g[N] = 1 +print("u_g =\n", u_g) + +# remove first and last row of A +A_g = A[1:N, :] +#print("A_g =\n", A_g) + +# assemble RHS with dirichlet data +f = -A_g.dot(u_g) +#print(f) + +# matrix for the inner nodes (excluding nodes with dirichlet bcs) +A_0 = A[1:N, 1:N] +#print(A_0) + +# solve for u_0 (free dofs) +u_0 = np.linalg.solve(A_0, f) + +# assemble "u = u_0 + u_g" +u = np.concatenate([[0], u_0, [1]]) + + +print("u =\n", u) + +plt.plot(x, u, '-') +plt.xlabel('x') +plt.ylabel('u_h(x)') +plt.grid() +plt.show() \ No newline at end of file diff --git a/ex4/ex_4C.py b/ex4/ex_4C.py new file mode 100644 index 0000000..5c8184e --- /dev/null +++ b/ex4/ex_4C.py @@ -0,0 +1,76 @@ +import numpy as np +import matplotlib.pyplot as plt + +np.set_printoptions(precision=2) + + +p = 70 +N_vec = [10, 20, 30, 40, 70] +for N in N_vec: + + x = np.linspace(0, 1, N + 1) + + A = np.zeros((N + 1, N + 1)) + + for i in range(1, N + 1): + h = x[i] - x[i - 1] + + + a_11 = 1./h - p/2. + a_12 = -1./h + p/2. + a_21 = -1./h - p/2. + a_22 = 1./h + p/2. + + A[i - 1, i - 1] += a_11 + A[i - 1, i] += a_12 + A[i, i - 1] += a_21 + A[i, i] += a_22 + + print("A =\n", A) + + + + # take dirichlet data into account + u_g = np.zeros(N + 1) + u_g[0] = 0 + u_g[N] = 1 + print("u_g =\n", u_g) + + # remove first and last row of A + A_g = A[1:N, :] + #print("A_g =\n", A_g) + + # assemble RHS with dirichlet data + f = -A_g.dot(u_g) + #print(f) + + # matrix for the inner nodes (excluding nodes with dirichlet bcs) + A_0 = A[1:N, 1:N] + #print(A_0) + + # solve for u_0 (free dofs) + u_0 = np.linalg.solve(A_0, f) + + # assemble "u = u_0 + u_g" + u = np.concatenate([[0], u_0, [1]]) + + + print("u =\n", u) + + plt.plot(x, u, '-') + + +# plotting the exact solution +plt.plot(x, (np.exp(p*x) - 1.)/(np.exp(p) - 1.)) + +plt.xlabel('x') +plt.ylabel('u_h(x)') +plt.legend(N_vec + ['exact']) +plt.title("Comparing discrete solution for increasing number of elements") 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