def get_bubble(deg):
x = Symbol('x')
poly_set = leg.basis_functions(deg=deg)
pts = np.r_[-1, 1, chebyshev_points(deg)[1:-1]]
A = np.zeros((deg+1, deg+1))
for col, f in enumerate(poly_set):
A[:, col] = lambdify(x, poly_set[col], 'numpy')(pts)
b = np.zeros(deg+1)
nodal_pt_index = -1 if deg % 2 else -deg/2
b[nodal_pt_index] = 1
bubble = np.linalg.solve(A, b)
bubble_dof = pts[nodal_pt_index] # Point eval here
# Return coefs of bubble w.r.t legendre and the node
return bubble, bubble_dof
# 0 boundary values
x = Symbol('x')
poly_set = leg.basis_functions(deg=deg)
# f = a*l0 + b*l1 + c*l2
# Suppose we reure that the L-2 norm is 1
# [f(-1)=0] = [l0(-1), l1(-1), l2(-1) ] [a]
# [f(1)=0 ] = [l0(1), l1(1), l2(1) ] [b]
# [norm=1 ] = [(l0, l0), ... [c]
# suppose instead nodality at 0
# [f(0) = 1] = .....
A = np.zeros((deg+1, deg+1))
pts = np.r_[-1, 1, chebyshev_points(deg)[1:-1]]
for col, f in enumerate(poly_set):
A[:, col] = lambdify(x, poly_set[col], 'numpy')(pts)
# norm
# A[2] = [float(integrate(f*f, (x, -1, 1))) for f in poly_set]
b = np.zeros(deg+1)
b[-1 if deg % 2 else -deg/2] = 1
coefs = np.linalg.solve(A, b)
f = sum(c*f for c, f in zip(coefs, poly_set))
#plot(f, (x, -1, 1))
def solve(n_cells, degree=3, with_plot=False):
# Problem
w = 3 * np.pi
x = Symbol("x")
u = sin(w * x)
f = -u.diff(x, 2)
# As Expr
u = Expression(u)
f = Expression(f)
# Space
# element = HermiteElement(degree)
poly_set = leg.basis_functions(degree)
dof_set = chebyshev_points(degree)
element = LagrangeElement(poly_set, dof_set)
mesh = IntervalMesh(a=-1, b=1, n_cells=n_cells)
V = FunctionSpace(mesh, element)
bc = DirichletBC(V, u)
# Need mass matrix to intefrate the rhs
M = assemble_matrix(V, "mass", get_geom_tensor=None, timer=0)
# NOTE We cannot you apply the alpha transform idea because the functions
# are mapped with this selective weight on 2nd, 3rd functions. So some rows
# of alpha would have to be multiplied by weights which are cell specific.
# And then on top of this there would be a dx = J*dy term. Better just to
# use the qudrature representations
# Mpoly_matrix = leg.mass_matrix(degree)
# M_ = assemble_matrix(V, Mpoly_matrix, Mget_geom_tensor, timer=0)
# Stiffness matrix for Laplacian
A = assemble_matrix(V, "stiffness", get_geom_tensor=None, timer=0)
# NOTE the above
# Apoly_matrix = leg.stiffness_matrix(degree)
# 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)
# As function
uh = Function(V, x)
# This is a (slow) way of plotting the high order
if with_plot:
fig = plt.figure()
ax = fig.gca()
uV = V.interpolate(u)
for cell in Cells(mesh):
a, b = cell.vertices[0, 0], cell.vertices[1, 0]
x = np.linspace(a, b, 100)
y = uh.eval_cell(x, cell)
ax.plot(x, y, color=random.choice(["b", "g", "m", "c"]))
y = uV.eval_cell(x, cell)
ax.plot(x, y, color="r")
y = u.eval_cell(x, cell)
ax.plot(x, y, color="k")
plt.show()
# Error norm in CG high order
fine_degree = degree + 3
poly_set = leg.basis_functions(fine_degree)
dof_set = chebyshev_points(fine_degree)
element = LagrangeElement(poly_set, dof_set)
V_fine = FunctionSpace(mesh, element)
# 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
# L2
if False:
Apoly_matrix = leg.mass_matrix(fine_degree)
get_geom_tensor = lambda cell: 1.0 / cell.Jac
# Need matrix for integration of H10 norm
else:
Apoly_matrix = leg.stiffness_matrix(fine_degree)
get_geom_tensor = lambda cell: cell.Jac
A_fine = assemble_matrix(V_fine, Apoly_matrix, get_geom_tensor, timer=0)
# Integrate the error
e = sqrt(np.sum(e * A_fine.dot(e)))
# Mesh size
hmin = mesh.hmin()
# Add the cond number
kappa = np.linalg.cond(A.toarray())
return hmin, e, kappa, A.shape[0]
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
import sys
sys.path.append('../')
from mesh import IntervalMesh
from cg_space import FunctionSpace
from function import Constant, Expression
from lagrange_element import LagrangeElement
from polynomials import legendre_basis as leg
from points import chebyshev_points
from sympy import Symbol
from scipy.sparse import csr_matrix
import numpy as np
# Element
degree = 2
poly_set = leg.basis_functions(degree)
dof_set = chebyshev_points(degree)
element = LagrangeElement(poly_set, dof_set)
# Mesh
n_cells = 2
mesh = IntervalMesh(a=-1, b=4, n_cells=n_cells)
# Space
V = FunctionSpace(mesh, element)
x = Symbol('x')
bc = DirichletBC(V, Expression(x))
# Hope that the map is okay. Check if applied correctly
# No symmetry
A = csr_matrix(np.random.rand(V.dim, V.dim))