def verify_matrix(self, u, p=2, mtype=0, plot=True):
from fealpy.mesh.simple_mesh_generator import triangle
if mtype == 0:
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)],
dtype=np.float)
cell = np.array([0, 1, 2, 3], dtype=np.int)
cellLocation = np.array([0, 4], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
elif mtype == 1:
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)],
dtype=np.float)
cell = np.array([0, 1, 2, 3, 0, 2], dtype=np.int)
cellLocation = np.array([0, 3, 6], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
elif mtype == 2:
h = 0.1
mesh = triangle([-1, 1, -1, 1], h, meshtype='polygon')
elif mtype == 3:
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)],
dtype=np.float)
cell = np.array([[0, 1, 2, 3]], dtype=np.int)
mesh = QuadrangleMesh(node, cell)
mesh.uniform_refine()
mesh = PolygonMesh.from_mesh(mesh)
uspace = ReducedDivFreeNonConformingVirtualElementSpace2d(mesh, p)
uspace.verify_matrix()
up = uspace.project(u)
up = uspace.project_to_smspace(up)
def project_test(self, u, p=2, mtype=0, plot=True):
from fealpy.mesh.simple_mesh_generator import triangle
if mtype == 0:
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)],
dtype=np.float)
cell = np.array([0, 1, 2, 3], dtype=np.int)
cellLocation = np.array([0, 4], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
elif mtype == 1:
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)],
dtype=np.float)
cell = np.array([0, 1, 2, 3, 0, 2], dtype=np.int)
cellLocation = np.array([0, 3, 6], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
elif mtype == 2:
h = 0.025
mesh = triangle([-1, 1, -1, 1], h, meshtype='polygon')
elif mtype == 3:
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)],
dtype=np.float)
cell = np.array([[0, 1, 2, 3]], dtype=np.int)
mesh = QuadrangleMesh(node, cell)
mesh.uniform_refine()
mesh = PolygonMesh.from_mesh(mesh)
elif mtype == 4:
node = np.array([(-1, -1), (1, -1), (1, 0), (1, 1), (-1, 1),
(-1, 0)],
dtype=np.float)
cell = np.array([0, 1, 2, 5, 2, 3, 4, 5], dtype=np.int)
cellLocation = np.array([0, 4, 8], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
if True:
cell, cellLocation = mesh.entity('cell')
edge = mesh.entity('edge')
cell2edge = mesh.ds.cell_to_edge()
bc = mesh.entity_barycenter('edge')
uspace = ReducedDivFreeNonConformingVirtualElementSpace2d(mesh, p)
up = uspace.project(u)
up = uspace.project_to_smspace(up)
print(up)
integralalg = uspace.integralalg
error = integralalg.L2_error(u, up)
print(error)
A = uspace.matrix_A()
P = uspace.matrix_P()
sio.savemat('A.mat', {"A": A.toarray(), "P": P.toarray()})
if plot:
mesh.print()
fig = plt.figure()
axes = fig.gca()
mesh.add_plot(axes)
mesh.find_node(axes, showindex=True)
mesh.find_edge(axes, showindex=True)
mesh.find_cell(axes, showindex=True)
plt.show()
]
errorMatrix = np.zeros((3, maxit), dtype=np.float)
dof = np.zeros(maxit, dtype=np.float)
for i in range(maxit):
#mesh = pde.init_mesh(n=i+2, meshtype='poly')
mesh = pde.init_mesh(n=i + 2, meshtype='quad')
mesh = PolygonMesh.from_mesh(mesh)
NE = mesh.number_of_edges()
NC = mesh.number_of_cells()
idof = (p - 2) * (p - 1) // 2
dof[i] = NC
uspace = ReducedDivFreeNonConformingVirtualElementSpace2d(mesh, p, q=6)
pspace = ScaledMonomialSpace2d(mesh, 0)
isBdDof = uspace.boundary_dof()
udof = uspace.number_of_global_dofs()
pdof = pspace.number_of_global_dofs()
uh = uspace.function()
ph = pspace.function()
A = uspace.matrix_A()
P = uspace.matrix_P()
F = uspace.source_vector(pde.source)
AA = bmat([[A, P.T], [P, None]], format='csr')
from fealpy.functionspace import RaviartThomasFiniteElementSpace2d
from fealpy.functionspace import DivFreeNonConformingVirtualElementSpace2d
from fealpy.functionspace import ReducedDivFreeNonConformingVirtualElementSpace2d
from fealpy.functionspace import ScaledMonomialSpace2d
from fealpy.pde.stokes_model_2d import StokesModelData_0, StokesModelData_1
p = 2
n = 5
tmesh = MeshFactory.boxmesh2d([0, 1, 0, 1], nx=5, ny=5, meshtype='tri')
# RT 空间
space0 = RaviartThomasFiniteElementSpace2d(tmesh, p=p - 1)
# 三角形转化为多边形网格， 注意这里要求三角形的 0 号边转化后仍然是多边形的 0
# 号边
NC = tmesh.number_of_cells()
NV = 3
node = tmesh.entity('node')
cell = tmesh.entity('cell')[:, [1, 2, 0]]
cellLocation = np.arange(0, (NC + 1) * NV, NV)
pmesh = PolygonMesh(node, cell.reshape(-1), cellLocation)
# 缩减虚单元空间
space1 = ReducedDivFreeNonConformingVirtualElementSpace2d(pmesh, p=p)
b0 = space1.source_vector(pde.source)
b1 = space1.pressure_robust_source_vector(pde.source, space0)
plt.show()
def stokes_equation_test(self, p=2, maxit=4, mtype=1):
from scipy.sparse import bmat
from fealpy.pde.Stokes_Model_2d import CosSinData, PolyY2X2Data
from fealpy.mesh.simple_mesh_generator import triangle
h = 0.4
#pde = CosSinData()
pde = PolyY2X2Data()
error = np.zeros((maxit, ), dtype=np.float)
for i in range(maxit):
if mtype == 0:
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)],
dtype=np.float)
cell = np.array([0, 1, 2, 3], dtype=np.int)
cellLocation = np.array([0, 4], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
else:
mesh = pde.init_mesh(n=i + 2, meshtype='poly')
NE = mesh.number_of_edges()
NC = mesh.number_of_cells()
idof = (p - 2) * (p - 1) // 2
if True:
fig = plt.figure()
axes = fig.gca()
mesh.add_plot(axes)
uspace = ReducedDivFreeNonConformingVirtualElementSpace2d(mesh, p)
pspace = ScaledMonomialSpace2d(mesh, 0)
isBdDof = uspace.boundary_dof()
udof = uspace.number_of_global_dofs()
pdof = pspace.number_of_global_dofs()
uh = uspace.function()
ph = pspace.function()
uspace.set_dirichlet_bc(uh, pde.dirichlet)
A = uspace.matrix_A()
P = uspace.matrix_P()
F = uspace.source_vector(pde.source)
AA = bmat([[A, P.T], [P, None]], format='csr')
FF = np.block([F, np.zeros(pdof, dtype=uspace.ftype)])
x = np.block([uh, ph])
isBdDof = np.block(
[isBdDof, isBdDof,
np.zeros(NC * idof + pdof, dtype=np.bool)])
gdof = udof + pdof
FF -= AA @ x
bdIdx = np.zeros(gdof, dtype=np.int)
bdIdx[isBdDof] = 1
Tbd = spdiags(bdIdx, 0, gdof, gdof)
T = spdiags(1 - bdIdx, 0, gdof, gdof)
AA = T @ AA @ T + Tbd
FF[isBdDof] = x[isBdDof]
x[:] = spsolve(AA, FF)
uh[:] = x[:udof]
ph[:] = x[udof:]
up = uspace.project_to_smspace(uh)
integralalg = uspace.integralalg
error[i] = integralalg.L2_error(pde.velocity, up)
h /= 2
print(error)
print(error[0:-1] / error[1:])
plt.show()