def mixedpoissondual(k,N, g, f, points):
tag = "B1"
hdiveltsA = HdivElements(k)
hdiveltsB = HdivElements(k)
hdiveltsBt = HdivElements(k)
l2eltsB = L2Elements(k)
l2eltsBt = L2Elements(k)
quadrule = pyramidquadrature(k+1)
meshevents = lambda m: buildcubemesh(N,m,tag)
Asystem = SymmetricSystem(hdiveltsA, quadrule, meshevents, [])
Bsystem = AsymmetricSystem(l2eltsB, hdiveltsB, quadrule, meshevents, [],[])
Btsystem = AsymmetricSystem(hdiveltsBt, l2eltsBt, quadrule, meshevents, [],[])
A = Asystem.systemMatrix(False)
# Bt = Btsystem.systemMatrix(True, False)
B = Bsystem.systemMatrix(False, True)
print A.shape, B.shape
F = Bsystem.loadVector(f, False)
gn = lambda x,n: g(x).reshape(-1,1,1) * n.reshape(-1,1,3)
G = Btsystem.boundaryLoad({tag:gn}, squarequadrature(k+1), trianglequadrature(k+1), False)
u,p = ps.directsolve(A, B, G[tag], F)
um,pm = ps.mixedcg(A, B, G[tag], F)
# print numpy.hstack((UU[:len(u)], u))
print math.sqrt(numpy.sum((u - um)**2)/len(u))
print math.sqrt(numpy.sum((p - pm)**2)/len(p))
return Btsystem.evaluate(points, p, {}, False)
def laplaceeigs(k,N,n):
import scipy.linalg as sl
tag = "B1"
elements = H1Elements(k)
quadrule = pyramidquadrature(k+1)
system = SymmetricSystem(elements, quadrule, lambda m: buildcubemesh(N, m, tag), [tag])
SM = system.systemMatrix(True)
S, SIBs, Gs = system.processBoundary(SM, {tag:lambda p: numpy.zeros((len(p),1))})
print S.shape
MM = system.systemMatrix(False)
M, _, _ = system.processBoundary(MM, {tag:lambda p: numpy.zeros((len(p),1))})
MLU = ssl.splu(M)
L = A = ssl.LinearOperator( M.shape, matvec=lambda x: MLU.solve(S* x), dtype=float)
return ssl.eigen(L, k=n, which='SM', return_eigenvectors=False)
def poissonneumann(k,N,g,f, points):
tag = "B1"
elements = H1Elements(k)
quadrule = pyramidquadrature(k+1)
system = SymmetricSystem(elements, quadrule, lambda m: buildcubemesh(N,m,tag), [])
SM = system.systemMatrix(True)
S = SM[1:,:][:,1:] # the first basis fn is guaranteed to be associated with an external degree, so is a linear comb of all the others
F = 0 if f is None else system.loadVector(f)
G = 0 if g is None else system.boundaryLoad({tag:g}, squarequadrature(k+1), trianglequadrature(k+1), False)
U = numpy.concatenate((numpy.array([0]), spsolve(S, G[tag][1:]-F[1:])))[:,numpy.newaxis]
return system.evaluate(points, U, {}, False)
def testSymmetry(self):
tag = "B1"
for k in range(1,3):
quadrule = pyramidquadrature(k+1)
for N in range(1,3):
for elements in [H1Elements(k), HcurlElements(k), HdivElements(k)]:
system = SymmetricSystem(elements, quadrule, lambda m: buildcubemesh(N, m, tag), [tag])
for deriv in [False, True]:
SM = system.systemMatrix(deriv)
g = lambda x: np.zeros((len(x),1))
S, SIBs, Gs = system.processBoundary(SM, {tag:g})
np.testing.assert_array_almost_equal(SM.todense(), SM.transpose().todense())
np.testing.assert_array_almost_equal(S.todense(), S.transpose().todense())
def laplacedirichlet(k, N, g, points):
tag = "B1"
elements = H1Elements(k)
quadrule = pyramidquadrature(k+1)
system = SymmetricSystem(elements, quadrule, lambda m: buildcubemesh(N, m, tag), [tag])
SM = system.systemMatrix(True)
S, SIBs, Gs = system.processBoundary(SM, {tag:g})
SG = SIBs[tag] * Gs[tag]
print S.shape
U = spsolve(S, -SG)[:,numpy.newaxis]
return system.evaluate(points, U, Gs, False)
def poissondirichlet(k,N,g,f, points):
tag = "B1"
elements = H1Elements(k)
quadrule = pyramidquadrature(k+1)
system = SymmetricSystem(elements, quadrule, lambda m: buildcubemesh(N, m, tag), [tag])
SM = system.systemMatrix(True)
S, SIBs, Gs = system.processBoundary(SM, {tag:g})
SG = SIBs[tag] * Gs[tag]
if f:
F = system.loadVector(f)
else: F = 0
print SM.shape, S.shape, SIBs[tag].shape, Gs[tag].shape, F.shape, SG.shape
t = Timer().start()
U = spsolve(S, -F-SG)[:,numpy.newaxis]
t.split("spsolve").show()
return system.evaluate(points, U, Gs, False)