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2.1.7 Multigrid for hybrid methods

Mixed methods for second order problems can often be reduced to the mesh facet, so called hybrid mixed methods. Simiar, hybrid DG methods introduce new variables on the facets, such that the bulk of element variables can be condensed out.

We show how to setup a multigrid preconditioner for hybrid methods. Interesting applications are nearly incompressible materials, or Stokes, discretized by \(H(\operatorname{div})\)-conforming HDG or hybrid mixed methods.

from ngsolve import *
from ngsolve.webgui import Draw
from ngsolve.la import EigenValues_Preconditioner

The hybrid DG method:

On \(V_h = V_T \times V_F = P^k({\mathcal T}) \times P^k ({\mathcal F})\) we define the bilinear form

\[\DeclareMathOperator{\Div}{div} \sum_T \int_T \nabla u \nabla v - \sum_T \int_{\partial T} \tfrac{\partial u}{\partial n} (v-\widehat v) - \sum_T \int_{\partial T} \tfrac{\partial v}{\partial n} (u-\widehat u) + \frac{\alpha p^2}{h} \sum_F \int_F (u-\widehat u)(v-\widehat v)\]

Element variables can be condensed out, which leads to a system reduced to the Skeleton.

When splitting a large triangle \(T_H\) into small trianles, the functions on \(\partial T_H\) have a canonical representation on the facets of the fine triangles. However, facet variables on internal edges of \(T_H\) are not defined by embedding. The HarmonicProlongation provides the energy optimal extension to the internal edges. To define energy optimal we need the energy defined by a bilinear form.

ngmesh = unit_square.GenerateMesh(maxh=2)
mesh = Mesh(ngmesh)

order = 3
fes = L2(mesh, order=order) * FacetFESpace(mesh, order=order, hoprolongation=True, dirichlet=".*")
(u,uhat), (v,vhat) = fes.TnT()

n = specialcf.normal(2)
h = specialcf.mesh_size
dS = dx(element_vb=BND)

HDGform = u*v*dx+ grad(u)*grad(v)*dx - n*grad(u)*(v-vhat)*dS - n*grad(v)*(u-uhat)*dS + 5*(order+1)**2/h*(u-uhat)*(v-vhat)*dS
bfa = BilinearForm(HDGform, condense=True).Assemble()
fes.SetHarmonicProlongation(bfa, inverse="sparsecholesky")
pre = preconditioners.MultiGrid(bfa, blocktype=["vertexpatch"], cycle=1)

with TaskManager():
    for l in range(7):
        mesh.Refine()
        bfa.Assemble()
        # pre.Update()
        lam = EigenValues_Preconditioner(bfa.mat, pre)
        print ("l =", l, "ndof =", fes.ndof, "lam_min/lam_max = ", lam[0], lam[-1])

l = 0 ndof = 144 lam_min/lam_max =  0.9999999999999993 0.9999999999999993
l = 1 ndof = 544 lam_min/lam_max =  0.9573848616699543 0.9999918601826863
l = 2 ndof = 2112 lam_min/lam_max =  0.9261189386281288 0.9999692236038735
l = 3 ndof = 8320 lam_min/lam_max =  0.9325349652513784 0.9998939800629592
l = 4 ndof = 33024 lam_min/lam_max =  0.92945510895588 0.9997230751035944
l = 5 ndof = 131584 lam_min/lam_max =  0.9282844459834527 0.9996349409051806
l = 6 ndof = 525312 lam_min/lam_max =  0.9280114490358107 0.9995334765161776

f = LinearForm (x*v*dx).Assemble()
gfu = GridFunction(fes)
gfu.vec[:]=0
with TaskManager():
    Solve (bfa*gfu==f, pre, lin_solver=solvers.CGSolver, printrates=True)

CG iteration 1, residual = 0.09459461645928992
CG iteration 2, residual = 0.0005944516866628238
CG iteration 3, residual = 7.695100604718358e-06
CG iteration 4, residual = 2.0699972197723346e-07
CG iteration 5, residual = 3.8945463326402356e-09
CG iteration 6, residual = 4.9504671345225695e-11
CG iteration 7, residual = 1.0763303055548899e-12
CG iteration 8, residual = 2.096450968451275e-14

Draw (gfu.components[0]);

Hybrid-mixed methods:

Find \(\sigma, u, \widehat u \in \Sigma_h \times V_h \times F_h\):

\[\begin{split}\DeclareMathOperator{\Div}{div} \begin{array}{ccccccll} \int a \sigma \tau & + & \sum_T \int_T \Div \tau \, u & + & \sum_F \int_F [\tau_n] \widehat u & = & 0 & \forall \, \tau \in \Sigma \\ \int \Div \sigma \, v &&&&& = & \int f v & \forall \, v \in V_h \\ \int [ \sigma_n ] \, \widehat v &&&&& = & \int_{\Gamma_n} g \widehat v & \forall \, \widehat v \in F_h \end{array}\end{split}\]

where \(\Sigma_h\) is an discontinuous \(H(div)\) finite element space, \(V_h\) a sub-space of \(L_2\), and \(F_h\) consists of polynomials on every facet.

ngmesh = unit_square.GenerateMesh(maxh=0.2)
mesh = Mesh(ngmesh)

order = 2

fesSigma = PrivateSpace(Discontinuous(HDiv(mesh, order=order, RT=True)))
fesL2 = L2(mesh, order=order)
fesFacet = FacetFESpace(mesh, order=order, hoprolongation=True, dirichlet=".*")
fes = fesSigma*fesL2*fesFacet

(sigma, u,uhat), (tau, v,vhat) = fes.TnT()
n = specialcf.normal(2)
dS = dx(element_vb=BND)
mixedform = -sigma*tau*dx
mixedform += div(sigma)*v*dx - sigma*n*vhat*dS
mixedform += div(tau)*u*dx - tau*n*uhat*dS

bfa = BilinearForm(mixedform, condense=True).Assemble()
fes.SetHarmonicProlongation(bfa, inverse="sparsecholesky")
pre = preconditioners.MultiGrid(bfa, blocktype=["vertexpatch"], cycle=1)

for l in range(5):
    mesh.Refine()
    bfa.Assemble()
    lam = EigenValues_Preconditioner(bfa.mat, pre)
    print ("l =", l, "ndof =", fes.ndof, "lam_min/lam_max = ", lam[0], lam[-1])

l = 0 ndof = 2328 lam_min/lam_max =  0.8830690135570511 1.0000826609163065
l = 1 ndof = 9192 lam_min/lam_max =  0.8695401255366377 0.9995622254215335
l = 2 ndof = 36528 lam_min/lam_max =  0.848063940231164 0.9985740487773187
l = 3 ndof = 145632 lam_min/lam_max =  0.830706421034594 0.9979298221317767
l = 4 ndof = 581568 lam_min/lam_max =  0.8389819990458989 0.9976337192727811

f = LinearForm(x*v*dx).Assemble()
gfu = GridFunction(fes)
Solve(bfa*gfu==f, pre, solvers.CGSolver, printrates=True)
Draw (gfu.components[1]);

CG iteration 1, residual = 0.09771383816444197
CG iteration 2, residual = 0.001196480641099551
CG iteration 3, residual = 3.137555063707439e-05
CG iteration 4, residual = 9.521672795304109e-07
CG iteration 5, residual = 3.248449810410441e-08
CG iteration 6, residual = 1.6499700712880606e-09
CG iteration 7, residual = 8.248735706513305e-11
CG iteration 8, residual = 3.983130055423988e-12
CG iteration 9, residual = 2.07728681577481e-13
CG iteration 10, residual = 1.2019646044004085e-14

Nearly incompressible materials, H(div)-conforming HDG

[Lehrenfeld+Schöberl, 2016]

ngmesh = unit_square.GenerateMesh(maxh=0.3)
mesh = Mesh(ngmesh)

order = 3

fesT = HDiv(mesh, order=order, hoprolongation=True, dirichlet=".*")
fesF = TangentialFacetFESpace(mesh, order=order, hoprolongation=True, highest_order_dc=True, dirichlet=".*")
fes = fesT*fesF

(u,uhat), (v,vhat) = fes.TnT()
n = specialcf.normal(2)
def tang(v): return v-(v*n)*n
h = specialcf.mesh_size
dS = dx(element_vb=BND)

HDGform = InnerProduct(Grad(u),Grad(v))*dx - (Grad(u)*n)*tang(v-vhat)*dS - (Grad(v)*n)*tang(u-uhat)*dS \
    + 1*(order+1)**2/h*tang(u-uhat)*tang(v-vhat)*dS

bfa = BilinearForm(HDGform + 1e3*div(u)*div(v)*dx, condense=True).Assemble()
fes.SetHarmonicProlongation(bfa)
pre = preconditioners.MultiGrid(bfa, smoother="block", smoothingsteps=1, blocktype=["vertexpatch"], cycle=1)

with TaskManager():
    for l in range(4):
        mesh.Refine()
        bfa.Assemble()
        lam = EigenValues_Preconditioner(bfa.mat, pre)
        print ("l =", l, "ndof =", fes.ndof, lam[0], lam[-1])

l = 0 ndof = 2190 0.8031555871568414 1.0389867675724214
l = 1 ndof = 8622 0.6881351203081374 1.0480292176047754
l = 2 ndof = 34158 0.6819544900463874 1.0702830065295148
l = 3 ndof = 135918 0.6926495255892826 1.0752680621327033

with TaskManager():
    f = LinearForm ((0.5-y)*v[0]*dx).Assemble()
gfu = GridFunction(fes)
gfu.vec[:]=0
with TaskManager(pajetrace=10**8):
    solvers.BVP(bfa, f, gfu, pre, print=True)

CG iteration 1, residual = 0.019746505238806333
CG iteration 2, residual = 0.00020770356871207895
CG iteration 3, residual = 1.354162972015256e-05
CG iteration 4, residual = 1.20951248170296e-06
CG iteration 5, residual = 1.1795588285486284e-07
CG iteration 6, residual = 1.2588391515974417e-08
CG iteration 7, residual = 1.4899585820389396e-09
CG iteration 8, residual = 1.670719075847171e-10

Draw (gfu.components[0])

BaseWebGuiScene

Nearly incompressible materials / Stokes in 3D

ngmesh = unit_cube.GenerateMesh(maxh=2)
mesh = Mesh(ngmesh)

order = 2

fesT = HDiv(mesh, order=order, hoprolongation=True, dirichlet=".*")
fesF = TangentialFacetFESpace(mesh, order=order, hoprolongation=True, highest_order_dc=True, dirichlet=".*")
fes = fesT*fesF

(u,uhat), (v,vhat) = fes.TnT()
n = specialcf.normal(3)
def tang(v): return v-(v*n)*n
h = specialcf.mesh_size
dS = dx(element_vb=BND)

HDGform = 0.001*u*v*dx+InnerProduct(Grad(u),Grad(v))*dx - (Grad(u)*n)*tang(v-vhat)*dS - (Grad(v)*n)*tang(u-uhat)*dS \
    + 5*(order+1)**2/h*tang(u-uhat)*tang(v-vhat)*dS

bfa = BilinearForm(HDGform + 1e3*div(u)*div(v)*dx, condense=True).Assemble()
fes.SetHarmonicProlongation(bfa)
pre = preconditioners.MultiGrid(bfa, smoother="block", smoothingsteps=3, blocktype=["edgepatch"], cycle=1)

with TaskManager():
    for l in range(3):
        mesh.Refine()
        bfa.Assemble()
        # pre.Update()
        lam = EigenValues_Preconditioner(bfa.mat, pre)
        print ("l =", l, "ndof =", fes.ndof, "lam min/max = ", lam[0], lam[-1])

l = 0 ndof = 5574 lam min/max =  0.5060508739232159 1.211263015326649
l = 1 ndof = 43470 lam min/max =  0.25806967546034504 1.2905858121466443
l = 2 ndof = 342894 lam min/max =  0.23767448244474096 1.4487474122388475

with TaskManager():
    f = LinearForm ((0.5-y)*v[0]*dx).Assemble()
gfu = GridFunction(fes)
gfu.vec[:]=0

with TaskManager():
    Solve(bfa*gfu==f, pre, solvers.CGSolver, printrates=True)

CG iteration 1, residual = 0.016109906113163756
CG iteration 2, residual = 0.003663119730314514
CG iteration 3, residual = 0.0013763841180129733
CG iteration 4, residual = 0.0005505700530463958
CG iteration 5, residual = 0.00023667946218135173
CG iteration 6, residual = 0.00010014395420475485
CG iteration 7, residual = 4.3494048121134424e-05
CG iteration 8, residual = 1.659018620943908e-05
CG iteration 9, residual = 6.187485725235501e-06
CG iteration 10, residual = 2.3330442207179555e-06
CG iteration 11, residual = 9.509298878745049e-07
CG iteration 12, residual = 3.7770775958341955e-07
CG iteration 13, residual = 1.6068815303153608e-07
CG iteration 14, residual = 6.952194561498379e-08
CG iteration 15, residual = 2.782312459469958e-08
CG iteration 16, residual = 1.0924337753523798e-08
CG iteration 17, residual = 4.48787107621357e-09
CG iteration 18, residual = 1.86272302537129e-09
CG iteration 19, residual = 7.402872804340236e-10
CG iteration 20, residual = 2.9942322000877634e-10
CG iteration 21, residual = 1.2726933117906297e-10
CG iteration 22, residual = 5.4302541953658056e-11
CG iteration 23, residual = 2.2027345474593457e-11
CG iteration 24, residual = 8.9340862967483e-12
CG iteration 25, residual = 3.678442279217592e-12
CG iteration 26, residual = 1.4579105634728734e-12
CG iteration 27, residual = 5.886329949371207e-13
CG iteration 28, residual = 2.568065072296372e-13
CG iteration 29, residual = 1.0908752441054404e-13
CG iteration 30, residual = 4.422174497477047e-14
CG iteration 31, residual = 1.8045747213227776e-14
CG iteration 32, residual = 7.616932077830456e-15

clipping = { "function" : True,  "pnt" : (0.5,0.5,0.51), "vec" : (0,0,-1) }
Draw (gfu.components[0], order=2, clipping=clipping);

Flow channel in 3D

from ngsolve import *
from netgen.occ import *
from ngsolve.webgui import Draw
from ngsolve.krylovspace import CGSolver

box = Box((0,0,0), (2.5,0.41,0.41))
box.faces.name="wall"
box.faces.Min(X).name="inlet"
box.faces.Max(X).name="outlet"
cyl = Cylinder((0.5,0.2,0), Z, h=0.41,r=0.05)
cyl.faces.name="cyl"
shape = box-cyl

mesh = shape.GenerateMesh(maxh=0.2).Curve(3)
Draw (mesh);

order = 2

fesT = HDiv(mesh, order=order, hoprolongation=True, dirichlet="wall|inlet|cyl")
fesF = TangentialFacetFESpace(mesh, order=order, hoprolongation=True, highest_order_dc=True, dirichlet=".*")
fes = fesT*fesF

(u,uhat), (v,vhat) = fes.TnT()
n = specialcf.normal(3)
def tang(v): return v-(v*n)*n
h = specialcf.mesh_size
dS = dx(element_vb=BND)

HDGform = 0.001*u*v*dx+InnerProduct(Grad(u),Grad(v))*dx - (Grad(u)*n)*tang(v-vhat)*dS - (Grad(v)*n)*tang(u-uhat)*dS \
    + 5*(order+1)**2/h*tang(u-uhat)*tang(v-vhat)*dS

with TaskManager():
    bfa = BilinearForm(HDGform + 1e3*div(u)*div(v)*dx, condense=True).Assemble()
    fes.SetHarmonicProlongation(bfa)
    pre = preconditioners.MultiGrid(bfa, inverse="sparsecholesky", smoother="block", smoothingsteps=1, blocktype=["edgepatch"], cycle=1)

with TaskManager():
    for l in range(1):
        mesh.Refine()
        bfa.Assemble()
        lam = EigenValues_Preconditioner(bfa.mat, pre)
        print ("l =", l, "ndof =", fes.ndof, "lam min/max = ", lam[0], lam[-1])

l = 0 ndof = 1779310 lam min/max =  0.073952920181655 3.3368613888810534

gfu = GridFunction(fes)

uin = (1.5*4*y*(0.41-y)/(0.41*0.41)*z*(0.41-z)/0.41**2,0, 0)
gfu.components[0].Set(uin, definedon=mesh.Boundaries("inlet"))

inv = CGSolver(bfa.mat, pre.mat, printrates=True, tol=1e-5)

with TaskManager():
    gfu.vec.data -= inv@bfa.mat * gfu.vec
gfu.vec.data += bfa.harmonic_extension * gfu.vec

CG iteration 1, residual = 14.705439018174088
CG iteration 2, residual = 0.6535781335123036
CG iteration 3, residual = 0.29491228099867905
CG iteration 4, residual = 0.1501249968250978
CG iteration 5, residual = 0.11534735471253715
CG iteration 6, residual = 0.07205075926982736
CG iteration 7, residual = 0.05198208987552031
CG iteration 8, residual = 0.03870922114840071
CG iteration 9, residual = 0.022859269501352954
CG iteration 10, residual = 0.016697123231945464
CG iteration 11, residual = 0.010896954086043877
CG iteration 12, residual = 0.007552851135372963
CG iteration 13, residual = 0.005539446504942511
CG iteration 14, residual = 0.004139996160118522
CG iteration 15, residual = 0.003060493009303792
CG iteration 16, residual = 0.0021714417553234757
CG iteration 17, residual = 0.001557073093465616
CG iteration 18, residual = 0.001174192633463337
CG iteration 19, residual = 0.0008572928809041569
CG iteration 20, residual = 0.0006182722955587457
CG iteration 21, residual = 0.0005100286526487766
CG iteration 22, residual = 0.0003640054422596445
CG iteration 23, residual = 0.0002769501127044876
CG iteration 24, residual = 0.00020615710374347185
CG iteration 25, residual = 0.00014504450758834206

clipping = { "function" : True,  "pnt" : (1,0.2,0.2 ), "vec" : (0,0,-1.0) }
Draw (gfu.components[0], mesh, order=2, clipping=clipping);

Pressure:

Draw (div(gfu.components[0]), mesh, order=2, clipping=clipping, draw_surf=False);