def test_l2_h1_2(): eps = 1e-9 func = lambda x: log(x) mesh1 = Mesh1D((1, 1.5, 2, 2.5, e), (20, 20, 20, 20)) f = Function(func, mesh1) l2_norm_exact = sqrt(e - 2) h1_norm_exact = sqrt(e - 1 - exp(-1)) assert abs(f.l2_norm(method="Fekete") - l2_norm_exact) < eps assert abs(f.l2_norm(method="FE") - l2_norm_exact) < eps assert abs(f.h1_norm() - h1_norm_exact) < eps
def test_l2_h1_2(): eps = 1e-9 func = lambda x: log(x) mesh1 = Mesh1D((1, 1.5, 2, 2.5, e), (20, 20, 20, 20)) f = Function(func, mesh1) l2_norm_exact = sqrt(e-2) h1_norm_exact = sqrt(e-1-exp(-1)) assert abs(f.l2_norm(method="Fekete")-l2_norm_exact) < eps assert abs(f.l2_norm(method="FE")-l2_norm_exact) < eps assert abs(f.h1_norm()-h1_norm_exact) < eps
def test_l2_h1_1(): eps = 1e-12 eps_low = 1e-10 func = lambda x: sin(x) mesh1 = Mesh1D((0, pi), (20,)) mesh2 = Mesh1D((0, pi/2, pi), (20, 20)) mesh3 = Mesh1D((0, pi/2), (20,)) f = Function(func, mesh1) g = Function(func, mesh2) h = Function(func, mesh3) assert abs(f.l2_norm(method="Fekete")-sqrt(pi/2)) < eps assert abs(g.l2_norm(method="Fekete")-sqrt(pi/2)) < eps assert abs(h.l2_norm(method="Fekete")-sqrt(pi/4)) < eps assert abs(f.l2_norm(method="FE")-sqrt(pi/2)) < eps assert abs(g.l2_norm(method="FE")-sqrt(pi/2)) < eps assert abs(h.l2_norm(method="FE")-sqrt(pi/4)) < eps assert abs(f.h1_norm()-sqrt(pi)) < eps_low assert abs(g.h1_norm()-sqrt(pi)) < eps_low assert abs(h.h1_norm()-sqrt(pi/2)) < eps_low func = lambda x: cos(x) mesh1 = Mesh1D((0, pi/4, pi/2, 3*pi/4, pi), (20, 20, 20, 20)) f = Function(func, mesh1) assert abs(f.l2_norm(method="Fekete")-sqrt(pi/2)) < eps assert abs(f.l2_norm(method="FE")-sqrt(pi/2)) < eps assert abs(f.h1_norm()-sqrt(pi)) < eps
def test_l2_h1_1(): eps = 1e-12 eps_low = 1e-10 func = lambda x: sin(x) mesh1 = Mesh1D((0, pi), (20, )) mesh2 = Mesh1D((0, pi / 2, pi), (20, 20)) mesh3 = Mesh1D((0, pi / 2), (20, )) f = Function(func, mesh1) g = Function(func, mesh2) h = Function(func, mesh3) assert abs(f.l2_norm(method="Fekete") - sqrt(pi / 2)) < eps assert abs(g.l2_norm(method="Fekete") - sqrt(pi / 2)) < eps assert abs(h.l2_norm(method="Fekete") - sqrt(pi / 4)) < eps assert abs(f.l2_norm(method="FE") - sqrt(pi / 2)) < eps assert abs(g.l2_norm(method="FE") - sqrt(pi / 2)) < eps assert abs(h.l2_norm(method="FE") - sqrt(pi / 4)) < eps assert abs(f.h1_norm() - sqrt(pi)) < eps_low assert abs(g.h1_norm() - sqrt(pi)) < eps_low assert abs(h.h1_norm() - sqrt(pi / 2)) < eps_low func = lambda x: cos(x) mesh1 = Mesh1D((0, pi / 4, pi / 2, 3 * pi / 4, pi), (20, 20, 20, 20)) f = Function(func, mesh1) assert abs(f.l2_norm(method="Fekete") - sqrt(pi / 2)) < eps assert abs(f.l2_norm(method="FE") - sqrt(pi / 2)) < eps assert abs(f.h1_norm() - sqrt(pi)) < eps
def test_power(): eps = 1e-12 func = lambda x: x mesh1 = Mesh1D((0, 1), (1,)) mesh2 = Mesh1D((0, 1), (2,)) mesh3 = Mesh1D((0, 1), (3,)) f = Function(func, mesh1) assert abs(f.l2_norm() - sqrt(1./3)) < eps assert f**2 != Function(lambda x: x**2, mesh1) assert f**2 == Function(lambda x: x**2, mesh2) assert f**2 == Function(lambda x: x**2, mesh3) func = lambda x: x mesh1 = Mesh1D((5, 6), (1,)) mesh2 = Mesh1D((5, 6), (2,)) mesh3 = Mesh1D((5, 6), (3,)) f = Function(func, mesh1) assert f**2 != Function(lambda x: x**2, mesh1) assert f**2 == Function(lambda x: x**2, mesh2) assert f**2 == Function(lambda x: x**2, mesh3) func = lambda x: x**3+x mesh1 = Mesh1D((5, 6), (3,)) mesh2 = Mesh1D((5, 6), (5,)) mesh3 = Mesh1D((5, 6), (6,)) mesh4 = Mesh1D((5, 6), (9,)) f = Function(func, mesh1) assert f**2 != Function(lambda x: x**6+2*x**4+x**2, mesh1) assert f**2 != Function(lambda x: x**6+2*x**4+x**2, mesh2) assert f**2 == Function(lambda x: x**6+2*x**4+x**2, mesh3) assert f**3 == Function(lambda x: x**3 + 3*x**5 + 3*x**7 + x**9, mesh4) func = lambda x: x**3+x mesh1 = Mesh1D((5, 5.1, 6), (3, 3)) mesh2 = Mesh1D((5, 5.1, 6), (5, 5)) mesh3 = Mesh1D((5, 5.1, 6), (6, 6)) mesh4 = Mesh1D((5, 5.1, 6), (9, 9)) mesh5 = Mesh1D((5, 5.1, 6), (9, 7)) mesh6 = Mesh1D((5, 5.1, 6), (6, 5)) mesh7 = Mesh1D((5, 6), (6,)) mesh8 = Mesh1D((5, 6), (9,)) f = Function(func, mesh1) assert f**2 != Function(lambda x: x**6+2*x**4+x**2, mesh1) assert f**2 != Function(lambda x: x**6+2*x**4+x**2, mesh2) assert f**2 != Function(lambda x: x**6+2*x**4+x**2, mesh6) assert f**2 == Function(lambda x: x**6+2*x**4+x**2, mesh3) assert f**3 == Function(lambda x: x**3 + 3*x**5 + 3*x**7 + x**9, mesh4) assert f**3 != Function(lambda x: x**3 + 3*x**5 + 3*x**7 + x**9, mesh5) assert f**2 == Function(lambda x: x**6+2*x**4+x**2, mesh7) assert f**3 == Function(lambda x: x**3 + 3*x**5 + 3*x**7 + x**9, mesh8)
def test_power(): eps = 1e-12 func = lambda x: x mesh1 = Mesh1D((0, 1), (1, )) mesh2 = Mesh1D((0, 1), (2, )) mesh3 = Mesh1D((0, 1), (3, )) f = Function(func, mesh1) assert abs(f.l2_norm() - sqrt(1. / 3)) < eps assert f**2 != Function(lambda x: x**2, mesh1) assert f**2 == Function(lambda x: x**2, mesh2) assert f**2 == Function(lambda x: x**2, mesh3) func = lambda x: x mesh1 = Mesh1D((5, 6), (1, )) mesh2 = Mesh1D((5, 6), (2, )) mesh3 = Mesh1D((5, 6), (3, )) f = Function(func, mesh1) assert f**2 != Function(lambda x: x**2, mesh1) assert f**2 == Function(lambda x: x**2, mesh2) assert f**2 == Function(lambda x: x**2, mesh3) func = lambda x: x**3 + x mesh1 = Mesh1D((5, 6), (3, )) mesh2 = Mesh1D((5, 6), (5, )) mesh3 = Mesh1D((5, 6), (6, )) mesh4 = Mesh1D((5, 6), (9, )) f = Function(func, mesh1) assert f**2 != Function(lambda x: x**6 + 2 * x**4 + x**2, mesh1) assert f**2 != Function(lambda x: x**6 + 2 * x**4 + x**2, mesh2) assert f**2 == Function(lambda x: x**6 + 2 * x**4 + x**2, mesh3) assert f**3 == Function(lambda x: x**3 + 3 * x**5 + 3 * x**7 + x**9, mesh4) func = lambda x: x**3 + x mesh1 = Mesh1D((5, 5.1, 6), (3, 3)) mesh2 = Mesh1D((5, 5.1, 6), (5, 5)) mesh3 = Mesh1D((5, 5.1, 6), (6, 6)) mesh4 = Mesh1D((5, 5.1, 6), (9, 9)) mesh5 = Mesh1D((5, 5.1, 6), (9, 7)) mesh6 = Mesh1D((5, 5.1, 6), (6, 5)) mesh7 = Mesh1D((5, 6), (6, )) mesh8 = Mesh1D((5, 6), (9, )) f = Function(func, mesh1) assert f**2 != Function(lambda x: x**6 + 2 * x**4 + x**2, mesh1) assert f**2 != Function(lambda x: x**6 + 2 * x**4 + x**2, mesh2) assert f**2 != Function(lambda x: x**6 + 2 * x**4 + x**2, mesh6) assert f**2 == Function(lambda x: x**6 + 2 * x**4 + x**2, mesh3) assert f**3 == Function(lambda x: x**3 + 3 * x**5 + 3 * x**7 + x**9, mesh4) assert f**3 != Function(lambda x: x**3 + 3 * x**5 + 3 * x**7 + x**9, mesh5) assert f**2 == Function(lambda x: x**6 + 2 * x**4 + x**2, mesh7) assert f**3 == Function(lambda x: x**3 + 3 * x**5 + 3 * x**7 + x**9, mesh8)
def test8(): eps = 1e-12 func = lambda x: x**2 mesh1 = Mesh1D((-5, -4, 3, 10), (2, 5, 2)) mesh2 = Mesh1D((-5, -4, 3, 10), (2, 2, 2)) mesh3 = Mesh1D((-5, -4, 3, 10), (2, 2, 1)) mesh4 = Mesh1D((-5, 10), (2,)) mesh5 = Mesh1D((-5, 10), (3,)) mesh6 = Mesh1D((-5, 10), (1,)) f = Function(func, mesh1) g = Function(func, mesh2) h = Function(func, mesh3) l = Function(func, mesh4) zero = Function(lambda x: 0., Mesh1D((-5, 10), (1,))) assert zero.l2_norm() < eps assert Function(lambda x: 0., mesh1) == zero assert Function(lambda x: 0., mesh2) == zero assert Function(lambda x: 0., mesh3) == zero assert Function(lambda x: 0., mesh4) == zero assert Function(lambda x: 0., mesh5) == zero assert Function(lambda x: 0., mesh6) == zero assert f - g == zero assert (f-g).l2_norm() < eps assert g - f == zero assert (g - f).l2_norm() < eps assert f - l == zero assert (f - l).l2_norm() < eps assert g - l == zero assert (g - l).l2_norm() < eps assert f - h != zero assert (f - h).l2_norm() > eps assert h - f != zero assert (h - f).l2_norm() > eps assert g - h != zero assert (g - h).l2_norm() > eps assert h - g != zero assert (h - g).l2_norm() > eps assert f - Function(lambda x: x**2, mesh1) == zero assert f - Function(lambda x: x**3, mesh1) != zero assert f - Function(lambda x: x**2, mesh2) == zero assert f - Function(lambda x: x**2, mesh4) == zero assert f - Function(lambda x: x**2, mesh5) == zero assert f - Function(lambda x: x**2, mesh6) != zero
def test8(): eps = 1e-12 func = lambda x: x**2 mesh1 = Mesh1D((-5, -4, 3, 10), (2, 5, 2)) mesh2 = Mesh1D((-5, -4, 3, 10), (2, 2, 2)) mesh3 = Mesh1D((-5, -4, 3, 10), (2, 2, 1)) mesh4 = Mesh1D((-5, 10), (2, )) mesh5 = Mesh1D((-5, 10), (3, )) mesh6 = Mesh1D((-5, 10), (1, )) f = Function(func, mesh1) g = Function(func, mesh2) h = Function(func, mesh3) l = Function(func, mesh4) zero = Function(lambda x: 0., Mesh1D((-5, 10), (1, ))) assert zero.l2_norm() < eps assert Function(lambda x: 0., mesh1) == zero assert Function(lambda x: 0., mesh2) == zero assert Function(lambda x: 0., mesh3) == zero assert Function(lambda x: 0., mesh4) == zero assert Function(lambda x: 0., mesh5) == zero assert Function(lambda x: 0., mesh6) == zero assert f - g == zero assert (f - g).l2_norm() < eps assert g - f == zero assert (g - f).l2_norm() < eps assert f - l == zero assert (f - l).l2_norm() < eps assert g - l == zero assert (g - l).l2_norm() < eps assert f - h != zero assert (f - h).l2_norm() > eps assert h - f != zero assert (h - f).l2_norm() > eps assert g - h != zero assert (g - h).l2_norm() > eps assert h - g != zero assert (h - g).l2_norm() > eps assert f - Function(lambda x: x**2, mesh1) == zero assert f - Function(lambda x: x**3, mesh1) != zero assert f - Function(lambda x: x**2, mesh2) == zero assert f - Function(lambda x: x**2, mesh4) == zero assert f - Function(lambda x: x**2, mesh5) == zero assert f - Function(lambda x: x**2, mesh6) != zero