from sympy import Symbol, sin, S
from function import Expression, Function
from mesh import IntervalMesh
from cg_space import FunctionSpace
from lagrange_element import LagrangeElement
import numpy as np
x = Symbol("x")
f = Expression(sin(2 * x))
meshf = IntervalMesh(a=-1, b=1, n_cells=4)
# Just f
fig = plot(f, mesh=meshf, label="f")
# Two functions
mesh = IntervalMesh(a=-1, b=1, n_cells=100)
poly_set = [S(1), x, x ** 2, x ** 3]
dof_set = np.array([-1, -0.5, 0.5, 1])
element = LagrangeElement(poly_set, dof_set)
V = FunctionSpace(mesh, element)
# Here's one
g = V.interpolate(f)
# Fake the other
gg = Function(V)
np.copyto(gg.vector, g.vector)
gg.vector += 1
# Add
fig = plot(g, fig=fig, label="g", color="r")
fig = plot(gg, fig=fig, label="gg", color="g")
plt.legend(loc="best")
plt.show()
def _solve(mode, points, degree, n_cells, u, f):
'''
In mode == convergence:
Solve -u`` = f with dirichet bcs bdry of (-1, 1) given by exact solution.
The Vh space is CG_space of degree elements and n_cells. Return hmin, error
for convergence computation.
In mode == cond:
Just return h and the matrix A.
'''
# Element. The polynomial space is spanned by Legendre basis
poly_set = leg.basis_functions(degree)
dof_set = points(degree)
element = LagrangeElement(poly_set, dof_set)
# Mesh
mesh = IntervalMesh(a=-1, b=1, n_cells=n_cells)
# Space
V = FunctionSpace(mesh, element)
bc = DirichletBC(V, u)
# Need mass matrix to intefrate the rhs
Mpoly_matrix = leg.mass_matrix(degree)
Mget_geom_tensor = lambda cell: 1./cell.Jac
M = assemble_matrix(V, Mpoly_matrix, Mget_geom_tensor, timer=0)
# Stiffness matrix for Laplacian
Apoly_matrix = leg.stiffness_matrix(degree)
Aget_geom_tensor = lambda cell: cell.Jac
A = assemble_matrix(V, Apoly_matrix, Aget_geom_tensor, timer=0)
# Interpolant of source
fV = V.interpolate(f)
# Integrate in L2 to get the vector
b = M.dot(fV.vector)
# Apply boundary conditions
bc.apply(A, b, True)
x = spsolve(A, b)
if mode == 'condition':
return mesh.hmin(), A
# As function
uh = Function(V, x)
# Error norm
# Higher order DG element
fine_degree = degree + 3
poly_set = leg.basis_functions(fine_degree)
dof_set = chebyshev_points(fine_degree)
element = LagrangeElement(poly_set, dof_set)
# THe space
V_fine = FunctionSpace(mesh, element, 'L2')
# Interpolate exact solution to fine
u_fine = V_fine.interpolate(u)
# Interpolate approx solution fine
uh_fine = V_fine.interpolate(uh)
# Difference vector
e = u_fine.vector - uh_fine.vector
# Need matrix for integration of H10 norm
Apoly_matrix = leg.stiffness_matrix(fine_degree)
A_fine = assemble_matrix(V_fine, Apoly_matrix, Aget_geom_tensor, timer=1)
# Integrate the error
e = sqrt(np.sum(e*A_fine.dot(e)))
# Mesh size
hmin = mesh.hmin()
return hmin, e