def bubblexnortho(n,h,kappa2,alpha,p,nel):
xwr = gauss(r_jacobi(n+2*alpha+2))
xwl = gauss(r_jacobi(n+2*alpha))
X = linspace(-mpf(0.5)*h,mpf(0.5)*h,nel+1)
Chat = [ppoly() for i in range(n)]
for k in range(n):
C = polyaff(conv(bw(alpha),mon(k)),-mpf(0.5)*h,mpf(0.5)*h,mpf(-1),mpf(1))
els,G,x,phi = fea1dh(X, lagrangecheby(p), kappa2, 0, [mpf(0),mpf(0)], rhsbubble(C, h, kappa2, xwl))
ppsol = ppolyfea1sol(els,G,x,phi)
Chat[k] = ppolyaxpy(mpf(-1),polytoppoly(C,ppsol.intv),ppsol)
h1ni = mpf(1)/sqrt(ppolyh1norm2(Chat[k],xwl))
Chat[k] = ppolyscale(Chat[k],h1ni)
# Execute Gram-Schmidt
D = []
for k in range(n):
Dk = Chat[k]
for l in range(k):
aCkDl = ppolyh1inner(Chat[k], D[l], xwl, kappa2)
aDlDl = ppolyh1inner(D[l], D[l], xwl, kappa2)
Dk = ppolyaxpy(-aCkDl/aDlDl, D[l], Dk)
D.append(Dk)
Dinner = zeros(1,n)
for k in range(n):
Dinner[k] = ppolyh1inner(D[k], D[k], xwl, kappa2)
return D,Dinner
def iapbuba(els,G,x,phi,kappa2,rhs,iapp):
nel = len(els)
uh = ppolyfea1sol(els,G,x,phi)
duh = ppolyder(uh)
res = ppolyaxpy(-kappa2,uh,ppolyder(duh)) # u_h''-kappa2 u_h
p = phi.cols
# Precompute bubble functions
mp.dps = 30
alpha = p/2+1
h = (els[0][1]-els[0][0]) # ASSUME UNIFORM ELEMENTS
nbub = iapp
xwl = gauss(r_jacobi(nbub+2*alpha))
P = prebub(nbub,h,kappa2,alpha,xwl)
bub = []
for i in range(P.rows):
bub.append(polytoppoly(P[i,:], [(-0.5*h, 0.5*h)]))
degbub = len(bub[0].poly[0])
# for the quadrature of the residual
xwl = gauss(r_jacobi(max(degbub,p)))
#nprint(loc)
mp.dps = 15
h1norm2 = []
eh = ppoly()
t1 = time.time()
k = 0
for el in els:
# Translate bubbles to the element
tbub = []
for b in bub:
bt = ppolytrans(b, 0.5*(el[0]+el[1]))
tbub.append(bt)
# Execute bubble kernel
erhs = lambda phii, el0, el1 : rhs(phii, el0, el1, el0, el1) +\
0.5*(el1-el0) * quadpqab(phii,res.poly[k],el0,el1, xwl)
G,x = iapbubaker(tbub, erhs)
eeh = ppolyiapbubsol(G,x,tbub)
h1norm2.append(ppolyh1norm2(eeh,xwl))
ppolyext(eh, eeh)
k += 1
return eh,h1norm2,time.time()-t1
def ppolyfea1sol(els,G,x,phi):
pp = ppoly(els)
e = 0
for el in els:
q = zeros(1)
for p in range(phi.rows):
i = G[e][p]
if i != -1:
q = polyaxpy(x[i],polyaff(phi[p,:],el[0],el[1],-1,1),q)
pp.poly[e] = q
e += 1
return pp
def bubblexn(n,h,kappa2,alpha,p,nel):
xwr = gauss(r_jacobi(n+2*alpha+2))
xwl = gauss(r_jacobi(n+2*alpha))
X = linspace(-mpf(0.5)*h,mpf(0.5)*h,nel+1)
Chat = [ppoly() for i in range(n)]
for k in range(n):
C = polyaff(conv(bw(alpha),mon(k)),-mpf(0.5)*h,mpf(0.5)*h,mpf(-1),mpf(1))
els,G,x,phi = fea1dh(X, lagrangecheby(p), kappa2, 0, [mpf(0),mpf(0)], rhsbubble(C, h, kappa2, xwl))
ppsol = ppolyfea1sol(els,G,x,phi)
Chat[k] = ppolyaxpy(mpf(-1),polytoppoly(C,ppsol.intv),ppsol)
h1ni = mpf(1)/sqrt(ppolyh1norm2(Chat[k],xwl))
Chat[k] = ppolyscale(Chat[k],h1ni)
return Chat
def ppolyfea1solpp(els,G,x,phi):
pp = ppoly()
r0 = phi[0].intv[0][0]
r1 = phi[0].intv[-1][1]
e = 0
for el in els:
qq = ppolyaff(ppolyzero(phi[0].intv), r0, r1, el[0], el[1])
for p in range(len(phi)):
if G[e][p] != -1:
qq = ppolyaxpy(x[G[e][p]], ppolyaff(phi[p],el[0],el[1],r0,r1), qq)
ppolyext(pp,qq)
e += 1
return pp
def iapneu(els,G,x,phi,kappa2,rhs,duX,iapnel,iapp):
nel = len(els)
uh = ppolyfea1sol(els,G,x,phi)
duh = ppolyder(uh)
res = ppolyaxpy(-kappa2,uh,ppolyder(duh)) # u_h''-kappa2 u_h
xwl = gauss(r_jacobi(phi.cols+iapp))
h1norm2 = []
eh = ppoly()
k = 0
for el in els:
psi = lagrangecheby(iapp)
# Approximate derivatives on the boundaries
if duX is None:
A = 0 if k == 0 else 0.5*(polyval(duh.poly[k],el[0])-polyval(duh.poly[k-1],el[0]))
B = 0 if k == nel-1 else 0.5*(polyval(duh.poly[k],el[1])-polyval(duh.poly[k+1],el[1]))
else:
A = duX[k]-polyval(duh.poly[k],el[0])
B = duX[k+1]-polyval(duh.poly[k],el[1])
#print "Ae={0:s}, Be={0:s}".format(nstr(Ae),nstr(Be))
erhs = lambda phii, r0, r1, el0, el1 : rhs(phii, r0, r1, el0, el1) +\
0.5*(el1-el0) * quadpq(phii,polyaff(res.poly[k],mpf(-1),mpf(1),el0,el1), xwl)
Xs = linspace(el[0],el[1],iapnel+1)
subels = zip(Xs[0:iapnel], Xs[1:iapnel+1])
eels,eG,ex,epsi = fea1dapel1(subels, psi, kappa2, 1, [A, B], erhs)
eeh = ppolyfea1sol(eels,eG,ex,epsi)
h1norm2.append(ppolyh1norm2(eeh,xwl))
ppolyext(eh, eeh)
k += 1
return eh, h1norm2
def bubble(n,h,kappa2,alpha,p,nel):
xwr = gauss(r_jacobi(n+2*alpha+2))
xwl = gauss(r_jacobi(n+2*alpha))
P = prebub(n,h,kappa2,alpha,xwl)
X = linspace(-mpf(0.5)*h,mpf(0.5)*h,nel+1)
Chat = [ppoly() for i in range(n)]
for k in range(n):
els,G,x,phi = fea1dh(X, lagrangecheby(p), kappa2, 0, [mpf(0),mpf(0)], rhsbubble(P[k,:], h, kappa2, xwl))
ppsol = ppolyfea1sol(els,G,x,phi)
Chat[k] = ppolyaxpy(mpf(-1),polytoppoly(P[k,:],ppsol.intv),ppsol)
h1ni = mpf(1)/sqrt(ppolyh1norm2(Chat[k],xwl))
Chat[k] = ppolyscale(Chat[k],h1ni)
#xx,yy = polyvalres(P[k,:],-mpf(0.5)*h,mpf(0.5)*h,100)
#pl.plot(xx,yy,label=r"$C_{0:d}$".format(k),linewidth=0.8)
#xx,yy = ppolyvalres(ppsol,100)
#pl.plot(xx,yy,label=r"$C^h_{0:d}$".format(k),linewidth=0.8)
return Chat
def iapbub(els,G,x,phi,kappa2,rhs,iapnel,iapp,sobolev=True):
nel = len(els)
uh = ppolyfea1sol(els,G,x,phi)
duh = ppolyder(uh)
res = ppolyaxpy(-kappa2,uh,ppolyder(duh)) # u_h''-kappa2 u_h
p = phi.cols
#print ">>>> iapbub on", els
# Precompute bubble functions
mp.dps = 20
alpha = p/2+1
h = (els[0][1]-els[0][0]) # ASSUME UNIFORM ELEMENTS
nbub = iapp
if sobolev == True:
bub = bubble(nbub,h,kappa2,alpha,p,iapnel)
else:
bub = bubblexn(nbub,h,kappa2,alpha,p,iapnel)
degbub = len(bub[0].poly[0])
# Determine local matrix
xw = gauss(r_jacobi(degbub))
dbub = [ppolyder(b) for b in bub]
loc = zeros(nbub)
for k,l in product(range(nbub),range(nbub)):
loc[k,l] = kappa2*quadppqq(bub[k], bub[l], xw) + quadppqq(dbub[k], dbub[l], xw)
# for the quadrature of the residual
xwl = gauss(r_jacobi(max(degbub,p)))
#nprint(loc)
mp.dps = 15
h1norm2 = []
eh = ppoly()
t1 = time.time()
k = 0
for el in els:
# Translate bubbles to the element
tbub = []
for b in bub:
bt = ppolytrans(b, 0.5*(el[0]+el[1]))
tbub.append(bt)
# Execute bubble kernel
erhs = lambda phii, el0, el1 : rhs(phii, el0, el1, el0, el1) +\
0.5*(el1-el0) * quadpqab(phii,res.poly[k],el0,el1, xwl)
G,x = iapbubker(tbub, loc, erhs)
eeh = ppolyiapbubsol(G,x,tbub)
h1norm2.append(ppolyh1norm2(eeh,xwl))
ppolyext(eh, eeh)
k += 1
return eh,h1norm2,time.time()-t1
def ppolyiapbubsol(G,x,bub):
pp = ppoly(bub[0].intv)
for i in G[0]:
pp = ppolyaxpy(x[G[0][i]], bub[i], pp)
return pp
def ppolytrans(pp, a):
qq = ppoly()
for i in range(len(pp.intv)):
qq.intv.append((pp.intv[i][0]+a, pp.intv[i][1]+a))
qq.poly.append(polycomp(pp.poly[i], poly([1, -a])))
return qq
def iapbubspread(els,G,x,phi,kappa2,rhs,iapnel,iapp,sobolev=True):
nel = len(els)
uh = ppolyfea1sol(els,G,x,phi)
duh = ppolyder(uh)
res = ppolyaxpy(-kappa2,uh,ppolyder(duh)) # u_h''-kappa2 u_h
p = phi.cols
#print ">>>> iapbub on", els
# Precompute bubble functions
mp.dps = 20
alpha = p/2+1
# ASSUME UNIFORM ELEMENTS !!!
h1 = (els[0][1]-els[0][0]) # element size
h = iapnel*h1 # spread size
nbub = iapp
spread = (iapnel-1)/2
if sobolev == True:
bub1 = bubble(nbub,h1,kappa2,alpha,p,iapnel)
bub = bubble(nbub,h,kappa2,alpha,p,iapnel)
else:
bub1 = bubblexn(nbub,h1,kappa2,alpha,p,iapnel)
bub = bubblexn(nbub,h,kappa2,alpha,p,iapnel)
degbub = len(bub[0].poly[0])
# Determine local matrices
xw = gauss(r_jacobi(degbub))
dbub1 = [ppolyder(b) for b in bub1]
dbub = [ppolyder(b) for b in bub]
loc1 = zeros(nbub)
loc = zeros(nbub)
for k,l in product(range(nbub),range(nbub)):
loc1[k,l] = kappa2*quadppqq(bub1[k], bub1[l], xw) + quadppqq(dbub1[k], dbub1[l], xw)
loc[k,l] = kappa2*quadppqq(bub[k], bub[l], xw) + quadppqq(dbub[k], dbub[l], xw)
# for the quadrature of the residual
xwl = gauss(r_jacobi(max(degbub,p)))
#nprint(loc)
mp.dps = 15
h1norm2s = []
eh = ppoly()
t1 = time.time()
k = 0
for el in els:
spreadels = []
# Only use spreaded bubbles if the whole spread interval is in the domain
if k-spread < 0 or k+spread >= nel:
# Fall back to 0 spread case
# Translate bubbles to the element
tbub = []
for b in bub1:
bt = ppolytrans(b, 0.5*(el[0]+el[1]))
tbub.append(bt)
erhs = lambda phii, el0, el1 : rhs(phii, el0, el1, el0, el1) +\
0.5*(el1-el0) * quadpqab(phii,res.poly[k],el0,el1, xwl)
G,x = iapbubker(tbub, loc1, erhs)
eeh = ppolyiapbubsol(G,x,tbub)
h1norm2s.append(ppolyh1norm2(eeh,xwl))
ppolyext(eh, eeh)
else:
# Multiple element support
tbub = []
for b in bub:
bt = ppolytrans(b, 0.5*(el[0]+el[1]))
tbub.append(bt)
# xx,yy = ppolyvalres(bt, 100)
# pl.plot(xx,yy,label="spread",linewidth=1.5)
erhs = lambda phii, l, el0, el1 : rhs(phii, el0, el1, el0, el1) +\
0.5*(el1-el0) * quadpqab(phii,res.poly[k-spread+l],el0,el1, xwl)
G,x = iapbubspreadker(tbub, loc, erhs)
eeh = ppolyiapbubsol(G,x,tbub)
# We are only concerened with the restriction to the current element
h1norm2s.append(h1norm2ab(eeh.poly[spread],eeh.intv[spread][0],eeh.intv[spread][1],xwl))
eh.intv.append(eeh.intv[spread])
eh.poly.append(eeh.poly[spread])
#xx,yy = polyvalres(eeh.poly[spread], eeh.intv[spread][0], eeh.intv[spread][1], 1000)
#pl.plot(xx,yy,label="spread",linewidth=2)
#xx,yy = ppolyvalres(eeh, 1000)
#pl.plot(xx,yy,label="spread",linewidth=2)
k += 1
return eh,h1norm2s,time.time()-t1
