def test4():
eps = 1e-12
func = lambda x: x**2
orig_mesh = Mesh1D((-5, -4, 3, 10), (1, 5, 1))
mesh1 = Mesh1D((-5, -4, 3, 10), (1, 1, 1))
f = Function(func, orig_mesh)
g = f.project_onto(mesh1)
h = Function(func, mesh1)
assert g == Function(func, mesh1)
assert h == h.project_onto(orig_mesh)
def test4():
eps = 1e-12
func = lambda x: x**2
orig_mesh = Mesh1D((-5, -4, 3, 10), (1, 5, 1))
mesh1 = Mesh1D((-5, -4, 3, 10), (1, 1, 1))
f = Function(func, orig_mesh)
g = f.project_onto(mesh1)
h = Function(func, mesh1)
assert g == Function(func, mesh1)
assert h == h.project_onto(orig_mesh)
def test_l2_h1_proj5():
"""
Tests exact projections.
The exact results were generated using:
from sympy import sin, cos, integrate, var, pi, exp, log, E, S
var("x")
f = exp(x)
S(1)/2 * integrate(f, (x, -1, 1)) + S(3)/2*x*integrate(x*f, (x, -1, 1))
"""
f_exact = lambda x: exp(x)
f_exact_l2 = lambda x: e / 2 - exp(-1) / 2 + 3 * x * exp(-1)
# TODO: The constant term here has to be checked:
f_exact_h1 = lambda x: +3 * e / 8 + 3 * exp(-1) / 8 + 3 * x * (e + exp(-1)
) / 8
pts = [-1, -0.5, 0, 0.5, 1]
orders = [20] * (len(pts) - 1)
m = Mesh1D(pts, orders)
f = Function(f_exact, m)
pts = [-1, 1]
orders = [1] * (len(pts) - 1)
m = Mesh1D(pts, orders)
f_proj_l2_exact = Function(f_exact_l2, m)
f_proj_l2 = f.project_onto(m, proj_type="L2")
f_proj_h1_exact = Function(f_exact_h1, m)
f_proj_h1 = f.project_onto(m, proj_type="H1")
eps_l2 = 1e-3
eps_h1 = 0.03
assert (f_proj_l2 - f_proj_l2_exact).l2_norm() < eps_l2
assert (f_proj_h1 - f_proj_h1_exact).l2_norm() < eps_h1
# Make sure that if we exchange the L2 and H1 solutions, then the test
# fails:
assert (f_proj_l2 - f_proj_h1_exact).l2_norm() > max(eps_l2, eps_h1)
assert (f_proj_h1 - f_proj_l2_exact).l2_norm() > max(eps_l2, eps_h1)
def test_l2_h1_proj5():
"""
Tests exact projections.
The exact results were generated using:
from sympy import sin, cos, integrate, var, pi, exp, log, E, S
var("x")
f = exp(x)
S(1)/2 * integrate(f, (x, -1, 1)) + S(3)/2*x*integrate(x*f, (x, -1, 1))
"""
f_exact = lambda x: exp(x)
f_exact_l2 = lambda x: e/2 - exp(-1)/2 + 3*x*exp(-1)
# TODO: The constant term here has to be checked:
f_exact_h1 = lambda x: +3*e/8 + 3*exp(-1)/8 + 3*x*(e + exp(-1))/8
pts = [-1, -0.5, 0, 0.5, 1]
orders = [20]*(len(pts)-1)
m = Mesh1D(pts, orders)
f = Function(f_exact, m)
pts = [-1, 1]
orders = [1]*(len(pts)-1)
m = Mesh1D(pts, orders)
f_proj_l2_exact = Function(f_exact_l2, m)
f_proj_l2 = f.project_onto(m, proj_type="L2")
f_proj_h1_exact = Function(f_exact_h1, m)
f_proj_h1 = f.project_onto(m, proj_type="H1")
eps_l2 = 1e-3
eps_h1 = 0.03
assert (f_proj_l2 - f_proj_l2_exact).l2_norm() < eps_l2
assert (f_proj_h1 - f_proj_h1_exact).l2_norm() < eps_h1
# Make sure that if we exchange the L2 and H1 solutions, then the test
# fails:
assert (f_proj_l2 - f_proj_h1_exact).l2_norm() > max(eps_l2, eps_h1)
assert (f_proj_h1 - f_proj_l2_exact).l2_norm() > max(eps_l2, eps_h1)
def test_l2_h1_proj6():
"""
Tests exact projections.
Slightly more complicated example.
"""
f_exact = lambda x: sin(x)*exp(x)
f_exact_l2 = lambda x: -e*cos(1)/4 + e*sin(1)/4 + exp(-1)*sin(1)/4 + 3*x*(e*sin(1)/2 - exp(-1)*sin(1)/2 - cos(1)*exp(-1))/2 + cos(1)*exp(-1)/4
pts = [-1, -0.5, 0, 0.5, 1]
orders = [20]*(len(pts)-1)
m = Mesh1D(pts, orders)
f = Function(f_exact, m)
pts = [-1, 1]
orders = [1]*(len(pts)-1)
m = Mesh1D(pts, orders)
f_proj_l2 = Function(f_exact_l2, m)
assert (f.project_onto(m, proj_type="L2") - f_proj_l2).l2_norm() < 0.03
def test_l2_h1_proj6():
"""
Tests exact projections.
Slightly more complicated example.
"""
f_exact = lambda x: sin(x)*exp(x)
f_exact_l2 = lambda x: -e*cos(1)/4 + e*sin(1)/4 + exp(-1)*sin(1)/4 + 3*x*(e*sin(1)/2 - exp(-1)*sin(1)/2 - cos(1)*exp(-1))/2 + cos(1)*exp(-1)/4
pts = [-1, -0.5, 0, 0.5, 1]
orders = [20]*(len(pts)-1)
m = Mesh1D(pts, orders)
f = Function(f_exact, m)
pts = [-1, 1]
orders = [1]*(len(pts)-1)
m = Mesh1D(pts, orders)
f_proj_l2 = Function(f_exact_l2, m)
assert (f.project_onto(m, proj_type="L2") - f_proj_l2).l2_norm() < 0.03
def test_l2_h1_proj4():
"""
Tests conversion to FE basis.
"""
pts = arange(0, 2 * pi, 0.4)
orders = [2] * (len(pts) - 1)
m = Mesh1D(pts, orders)
f = Function(lambda x: sin(x), m)
assert f.project_onto(m, proj_type="Fekete") == f
assert f.project_onto(m, proj_type="L2") == f
assert f.project_onto(m, proj_type="H1") == f
orders = [3] * (len(pts) - 1)
m = Mesh1D(pts, orders)
assert f.project_onto(m, proj_type="Fekete") == f
assert f.project_onto(m, proj_type="L2") == f
assert f.project_onto(m, proj_type="H1") == f
orders = [4] * (len(pts) - 1)
m = Mesh1D(pts, orders)
assert f.project_onto(m, proj_type="Fekete") == f
assert f.project_onto(m, proj_type="L2") == f
assert f.project_onto(m, proj_type="H1") == f
pts = arange(0, 2 * pi, 3)
orders = [2] * (len(pts) - 1)
m = Mesh1D(pts, orders)
pts = array(list(arange(0, pts[-1], 0.1)) + [pts[-1]])
orders = [6] * (len(pts) - 1)
f_exact = Function(lambda x: sin(x), Mesh1D(pts, orders))
sol_l2 = f_exact.project_onto(m, proj_type="L2")
sol_h1 = f_exact.project_onto(m, proj_type="H1")
assert (sol_l2 - f_exact).l2_norm() < 0.07
assert (sol_h1 - f_exact).l2_norm() < 0.07
def test_l2_h1_proj4():
"""
Tests conversion to FE basis.
"""
pts = arange(0, 2*pi, 0.4)
orders = [2]*(len(pts)-1)
m = Mesh1D(pts, orders)
f = Function(lambda x: sin(x), m)
assert f.project_onto(m, proj_type="Fekete") == f
assert f.project_onto(m, proj_type="L2") == f
assert f.project_onto(m, proj_type="H1") == f
orders = [3]*(len(pts)-1)
m = Mesh1D(pts, orders)
assert f.project_onto(m, proj_type="Fekete") == f
assert f.project_onto(m, proj_type="L2") == f
assert f.project_onto(m, proj_type="H1") == f
orders = [4]*(len(pts)-1)
m = Mesh1D(pts, orders)
assert f.project_onto(m, proj_type="Fekete") == f
assert f.project_onto(m, proj_type="L2") == f
assert f.project_onto(m, proj_type="H1") == f
pts = arange(0, 2*pi, 3)
orders = [2]*(len(pts)-1)
m = Mesh1D(pts, orders)
pts = array(list(arange(0, pts[-1], 0.1)) + [pts[-1]])
orders = [6]*(len(pts)-1)
f_exact = Function(lambda x: sin(x), Mesh1D(pts, orders))
sol_l2 = f_exact.project_onto(m, proj_type="L2")
sol_h1 = f_exact.project_onto(m, proj_type="H1")
assert (sol_l2 - f_exact).l2_norm() < 0.07
assert (sol_h1 - f_exact).l2_norm() < 0.07
