def test_tri_mass_mat_trig():
"""Check the integral of some trig functions on a square using the mass matrix"""
from hedge.mesh.generator import make_square_mesh
from hedge.discretization.local import TriangleDiscretization
from math import pi, cos, sin
mesh = make_square_mesh(a=-pi, b=pi, max_area=(2 * pi / 10)**2 / 2)
discr = discr_class(mesh,
TriangleDiscretization(8),
debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(
lambda x, el: cos(x[0])**2 * sin(x[1])**2)
ones = discr.interpolate_volume_function(lambda x, el: 1)
from hedge.optemplate import MassOperator
mass_op = MassOperator()
num_integral_1 = numpy.dot(ones, mass_op.apply(discr, f))
num_integral_2 = numpy.dot(f, mass_op.apply(discr, ones))
true_integral = pi**2
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
#print err_1, err_2
assert err_1 < 1e-10
assert err_2 < 1e-10
def test_1d_mass_mat_trig():
"""Check the integral of some trig functions on an interval using the mass
matrix
"""
from hedge.mesh.generator import make_uniform_1d_mesh
from hedge.discretization.local import IntervalDiscretization
from math import pi, cos
from numpy import dot
mesh = make_uniform_1d_mesh(-4 * pi, 9 * pi, 17, periodic=True)
discr = discr_class(mesh,
IntervalDiscretization(8),
debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(lambda x, el: cos(x[0])**2)
ones = discr.interpolate_volume_function(lambda x, el: 1)
from hedge.optemplate import MassOperator
mass_op = MassOperator()
num_integral_1 = dot(ones, mass_op.apply(discr, f))
num_integral_2 = dot(f, mass_op.apply(discr, ones))
num_integral_3 = discr.integral(f)
true_integral = 13 * pi / 2
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
err_3 = abs(num_integral_3 - true_integral)
assert err_1 < 1e-10
assert err_2 < 1e-10
assert err_3 < 1e-10
def test_tri_mass_mat_trig():
"""Check the integral of some trig functions on a square using the mass matrix"""
from hedge.mesh.generator import make_square_mesh
from hedge.discretization.local import TriangleDiscretization
from math import sqrt, pi, cos, sin
mesh = make_square_mesh(a=-pi, b=pi, max_area=(2 * pi / 10) ** 2 / 2)
discr = discr_class(mesh, TriangleDiscretization(8), debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(lambda x, el: cos(x[0]) ** 2 * sin(x[1]) ** 2)
ones = discr.interpolate_volume_function(lambda x, el: 1)
from hedge.optemplate import MassOperator
mass_op = MassOperator()
num_integral_1 = numpy.dot(ones, mass_op.apply(discr, f))
num_integral_2 = numpy.dot(f, mass_op.apply(discr, ones))
true_integral = pi ** 2
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
# print err_1, err_2
assert err_1 < 1e-10
assert err_2 < 1e-10
def test_1d_mass_mat_trig():
"""Check the integral of some trig functions on an interval using the mass matrix"""
from hedge.mesh.generator import make_uniform_1d_mesh
from hedge.discretization.local import IntervalDiscretization
from math import sqrt, pi, cos, sin
from numpy import dot
mesh = make_uniform_1d_mesh(-4 * pi, 9 * pi, 17, periodic=True)
discr = discr_class(mesh, IntervalDiscretization(8), debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(lambda x, el: cos(x[0]) ** 2)
ones = discr.interpolate_volume_function(lambda x, el: 1)
from hedge.optemplate import MassOperator
mass_op = MassOperator()
num_integral_1 = dot(ones, mass_op.apply(discr, f))
num_integral_2 = dot(f, mass_op.apply(discr, ones))
num_integral_3 = discr.integral(f)
true_integral = 13 * pi / 2
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
err_3 = abs(num_integral_3 - true_integral)
assert err_1 < 1e-10
assert err_2 < 1e-10
assert err_3 < 1e-10
def test_quadrature_tri_mass_mat_monomial():
"""Check that quadrature integration on triangles is exact as designed."""
from hedge.mesh.generator import make_square_mesh
mesh = make_square_mesh(a=-1, b=1, max_area=4 * 1 / 8 + 0.001)
order = 4
discr = discr_class(mesh,
order=order,
debug=discr_class.noninteractive_debug_flags(),
quad_min_degrees={"quad": 3 * order})
m, n = 2, 1
f = Monomial((m, n))
f_vec = discr.interpolate_volume_function(lambda x, el: f(x))
from hedge.discretization import ones_on_volume
ones = ones_on_volume(discr)
if False:
from hedge.visualization import SiloVisualizer
vis = SiloVisualizer(discr)
visf = vis.make_file("test")
vis.add_data(visf, [("f", f_vec * f_vec)])
visf.close()
from hedge.optemplate import (MassOperator, Field, QuadratureGridUpsampler)
f_fld = Field("f")
mass_op = discr.compile(MassOperator()(f_fld * f_fld))
from hedge.optemplate.primitives import make_common_subexpression as cse
f_upsamp = cse(QuadratureGridUpsampler("quad")(f_fld))
quad_mass_op = discr.compile(MassOperator()(f_upsamp * f_upsamp))
num_integral_1 = numpy.dot(ones, mass_op(f=f_vec))
num_integral_2 = numpy.dot(ones, quad_mass_op(f=f_vec))
true_integral = 4 / ((2 * m + 1) * (2 * n + 1))
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
print num_integral_1, num_integral_2, true_integral
print err_1, err_2
assert err_1 > 1e-8
assert err_2 < 1e-14
def op_template(self, apply_minv, u=None, dir_bc=None, neu_bc=None):
from hedge.optemplate import Field
if u is None:
u = Field("u")
result = PoissonOperator.op_template(self, apply_minv, u, dir_bc,
neu_bc)
if apply_minv:
return result + self.k**2 * u
else:
from hedge.optemplate import MassOperator
return result + self.k**2 * MassOperator()(u)
def prepare_rhs(self, rhs):
"""Prepare the right-hand side for the linear system op(u)=rhs(f).
In matrix form, LDG looks like this:
.. math::
Mv = Cu + g
Mf = Av + Bu + h
where v is the auxiliary vector, u is the argument of the operator, f
is the result of the grad operator, g and h are inhom boundary data, and
A,B,C are some operator+lifting matrices.
.. math::
M f = A M^{-1}(Cu + g) + Bu + h
so the linear system looks like
.. math::
M f = A M^{-1} Cu + A M^{-1} g + Bu + h
M f - A M^{-1} g - h = (A M^{-1} C + B)u (*)
So the right hand side we're putting together here is really
.. math::
M f - A M^{-1} g - h
.. note::
Resist the temptation to left-multiply by the inverse
mass matrix, as this will result in a non-symmetric
matrix, which (e.g.) a conjugate gradient Krylov
solver will not take well.
"""
pop = self.poisson_op
from hedge.optemplate import MassOperator
return (
MassOperator().apply(self.discr, rhs) -
self.compiled_bc_op(u=self.discr.volume_zeros(),
dir_bc=pop.dirichlet_bc.boundary_interpolant(
self.discr, pop.dirichlet_tag),
neu_bc=pop.neumann_bc.boundary_interpolant(
self.discr, pop.neumann_tag)))