76 lines
No EOL
1.2 KiB
Python
76 lines
No EOL
1.2 KiB
Python
import numpy as np
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import matplotlib.pyplot as plt
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np.set_printoptions(precision=2)
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N = 10 # number of elements
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a = 1
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alpha = 1
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g_b = 1
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f_const = 1
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x = np.linspace(0, 1, N + 1)
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A = np.zeros((N + 1, N + 1))
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f_vec = np.zeros(N + 1)
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for i in range(1, N + 1):
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h = x[i] - x[i - 1]
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a_11 = 1./h + a*h/3.
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a_12 = -1./h + a*h/6.
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a_21 = -1./h + a*h/6.
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a_22 = 1./h + a*h/3
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A[i - 1, i - 1] += a_11
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A[i - 1, i] += a_12
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A[i, i - 1] += a_21
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A[i, i] += a_22
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f_vec[i] = f_const*h
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print("A =\n", A)
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# take Neumann data into account
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A[N, N] += alpha
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f_vec[N] += alpha*g_b
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# take Dirichlet data into account
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u_g = np.zeros(N + 1)
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u_g[0] = 0
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print("u_g =\n", u_g)
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# remove first row of A
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A_g = A[1:N+1, :]
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#print("A_g =\n", A_g)
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# remove first row of f_vec
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f_g = f_vec[1:N+1]
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# assemble RHS with dirichlet data
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f_g -= A_g.dot(u_g)
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#print(f_g)
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# matrix for the inner nodes (excluding nodes with dirichlet bcs)
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A_0 = A[1:N+1, 1:N+1]
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#print(A_0)
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# solve for u_0 (free dofs)
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u_0 = np.linalg.solve(A_0, f_g)
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# assemble "u = u_0 + u_g"
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u = np.concatenate([[0], u_0])
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print("u =\n", u)
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plt.plot(x, u, '-')
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plt.xlabel('x')
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plt.ylabel('u_h(x)')
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plt.grid()
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plt.show() |