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3.4 Nonlinear minimization problems¶

We consider minimization problems of the form

find u \in V s.t. E (u) \leq E (v) \forall v \in V .

[1]:

# imports
from ngsolve import *
from ngsolve.webgui import Draw

Similar to the previous unit we want solve the (minimization) problem using Newton’s method. However this time we don’t start with an equation but a minimization problem. We will let NGSolve derive the corresponding expressions for minimization conditions.

To solve the problem we use the Variation integrator of NGSolve and formulate the problem through the symbolic description of an energy functional.

Let $E (u)$ be the energy that is to be minimized for the unknown state $u$ .

Then a necessary optimality condition is that the derivative at the minimizer $u$ in all directions $v$ vanishes, i.e.

δ E (u) (v) = 0 \forall v \in V

At this point we are back in business of the previous unit as we now have an equation that we need to solve.

Let’s continue with a concrete example.

Scalar minimization problems¶

As a first example we take $V = H_{0}^{1}$ and

E (u) = \int_{Ω} \frac{1}{2} | \nabla u |^{2} + \frac{1}{12} u^{4} - f u d x .

The minimization is equivalent (due to convexity) to solving the nonlinear PDE (cf. unit 3.3 with $f = 10$ )

δ E (u) (v) = 0 \forall v \in V \Leftrightarrow - Δ u + \frac{1}{3} u^{3} = f in V^{'}

with $Ω = (0, 1)^{2}$ .

[2]:

# define geometry and generate mesh
from ngsolve import *
from ngsolve.webgui import *
from netgen.occ import *
shape = Rectangle(1,1).Face()
shape.edges.Min(X).name="left"
shape.edges.Max(X).name="right"
shape.edges.Min(Y).name="bottom"
shape.edges.Max(Y).name="top"
geom = OCCGeometry(shape, dim=2)
mesh = Mesh(geom.GenerateMesh(maxh=0.3))

We solve the PDE with a Newton iteration.

[3]:

V = H1(mesh, order=4, dirichlet=".*")
u = V.TrialFunction()

We define the semi-linear form expression through the energy functional using the Variation-keyword:

[4]:

a = BilinearForm (V, symmetric=True)
a += Variation ( (0.5*grad(u)*grad(u) + 1/12*u**4-10*u) * dx)

Now NGSolve applies the derivative on the functional so that the previous statement corresponds to:

a += (grad(u) * grad(v) + 1/3*u**3*v - 10 * v)*dx

(which has the same form as the problems in the nonlinear example)

We recall the Newton iteration (cf. unit-3.3 ) and now apply the same loop:

Given an initial guess $u^{0}$
loop over $i = 0, . .$ until convergence:
- Compute linearization: $A (u^i) + \delta `A(u^i) :nbsphinx-math:Delta `u^{i} = 0 $:
  - $f^{i} = A (u^{i})$
  - $B^{i} = δ A (u^{i})$
  - Solve $B^{i} Δ u^{i} = - f^{i}$
- Update $u^{i + 1} = u^{i} + Δ u^{i}$
- Evaluate stopping criteria
Evaluate $E (u^{i + 1})$

As a stopping criteria we take $⟨ A u^{i}, Δ u^{i} ⟩ = ⟨ A u^{i}, A u^{i} ⟩_{(B^{i})^{- 1}} < ε$ .

Note that $A (u^{i})$ (a.Apply(...)) and $δ A (u^{i})$ (a.AssembleLinearization(...)) are now derived from $A$ which is defined as $A = δ E$ through the energy functional $E (\cdot)$ .

We obtain a similar Newton solver with the two additional advantages:

We don’t have to form $δ E$ manually, but let NGSolve do the job and
we can use the energy functional to interprete the success of iteration steps

[5]:

def SolveNonlinearMinProblem(a,gfu,tol=1e-13,maxits=10, callback=lambda gfu: None):
    res = gfu.vec.CreateVector()
    du  = gfu.vec.CreateVector()
    callback(gfu)
    for it in range(maxits):
        print ("Newton iteration {:3}".format(it),
               ", energy = {:16}".format(a.Energy(gfu.vec)),end="")

        #solve linearized problem:
        a.Apply (gfu.vec, res)
        a.AssembleLinearization (gfu.vec)
        du.data = a.mat.Inverse(V.FreeDofs()) * res

        #update iteration
        gfu.vec.data -= du
        callback(gfu)

        #stopping criteria
        stopcritval = sqrt(abs(InnerProduct(du,res)))
        print ("<A u",it,", A u",it,">_{-1}^0.5 = ", stopcritval)
        if stopcritval < tol:
            break
        Redraw(blocking=True)

So, let’s try it out:

[6]:

gfu = GridFunction(V)
gfu.Set((x*(1-x))**4*(y*(1-y))**4) # initial guess
gfu_it = GridFunction(gfu.space,multidim=0)
cb = lambda gfu : gfu_it.AddMultiDimComponent(gfu.vec) # store current state
SolveNonlinearMinProblem(a,gfu, callback = cb)
print ("energy = ", a.Energy(gfu.vec))

Newton iteration   0 , energy = -2.519306187470002e-05<A u 0 , A u 0 >_{-1}^0.5 =  1.8746349780341207
Newton iteration   1 , energy = -1.7525754148962955<A u 1 , A u 1 >_{-1}^0.5 =  0.010379563430868316
Newton iteration   2 , energy = -1.7526292864385133<A u 2 , A u 2 >_{-1}^0.5 =  1.1229094979196609e-06
Newton iteration   3 , energy = -1.7526292864391435<A u 3 , A u 3 >_{-1}^0.5 =  1.3185782591558818e-14
energy =  -1.752629286439144

[7]:

Draw(gfu,mesh,"u", deformation = True)
#Draw(gfu_it,mesh,"u", deformation = True)

[7]:

BaseWebGuiScene

Again, a Newton for minimization is also shipped with NGSolve (actually it is the same with the additional knowledge about the Energy and the possibility to do a line search for a given search direction):

[8]:

from ngsolve.solvers import *
gfu.Set((x*(1-x))**4*(y*(1-y))**4) # initial guess
NewtonMinimization(a,gfu)
#Draw(gfu,mesh,"u", deformation = True)

Newton iteration  0
Energy:  -2.519306187470002e-05
err =  1.8746349780341207
Newton iteration  1
Energy:  -1.7525754148962955
err =  0.010379563430868316
Newton iteration  2
Energy:  -1.7526292864385133
err =  1.1229094979196609e-06
Newton iteration  3
Energy:  -1.7526292864391435
err =  1.3185782591558818e-14

[8]:

(0, 4)

Nonlinear elasticity¶

We consider a beam which is fixed on one side and is subject to gravity only. We assume a Neo-Hookean hyperelastic material. The model is a nonlinear minimization problem with

E (v) := \int_{Ω} \frac{μ}{2} (tr (F^{T} F - I) + \frac{2 μ}{λ} \det (F^{T} F)^{\frac{λ}{2 μ}} - 1) - γ (f, v) d x

where $μ$ and $λ$ are the Lamé parameters and $F = I + D v$ where $v : Ω \to R^{2}$ is the sought for displacement.

We fix the domain to $Ω = (0, 1) \times (0, 0.1)$

[9]:

# define geometry and generate mesh
shape = Rectangle(1,0.1).Face()
shape.edges.Min(X).name="left"
shape.edges.Min(X).maxh=0.01
shape.edges.Max(X).name="right"
shape.edges.Min(Y).name="bot"
shape.edges.Max(Y).name="top"
geom = OCCGeometry(shape, dim=2)
mesh = Mesh(geom.GenerateMesh(maxh=0.05))

[10]:

# E module and poisson number:
E, nu = 210, 0.2
# Lamé constants:
mu  = E / 2 / (1+nu)
lam = E * nu / ((1+nu)*(1-2*nu))

V = VectorH1(mesh, order=2, dirichlet="left")
u  = V.TrialFunction()

#gravity:
force = CoefficientFunction( (0,-1) )

Now, we recall the energy

E (v) := \int_{Ω} \frac{μ}{2} (tr (F^{T} F - I) + \frac{2 μ}{λ} \det (F^{T} F)^{\frac{λ}{2 μ}} - 1) - γ (f, v) d x

[11]:

def Pow(a, b):
    return exp (log(a)*b)

def NeoHook (C):
    return 0.5 * mu * (Trace(C-I) + 2*mu/lam * Pow(Det(C), -lam/2/mu) - 1)

I = Id(mesh.dim)
F = I + Grad(u)
C = F.trans * F

factor = Parameter(1.0)

a = BilinearForm(V, symmetric=True)
a += Variation(  NeoHook (C).Compile() * dx
                -factor * (InnerProduct(force,u) ).Compile() * dx)

We want to solve the minimization problem for $γ = 5$ . Due to the high nonlinearity in the problem, the Newton iteration will not convergence for any initial guess. We approach the case $γ = 5$ by solving problems with $γ = i / 10$ for $i = 1, . ., 50$ and taking the solution of the previous problem as an initial guess.

[12]:

gfu = GridFunction(V)
gfu.vec[:] = 0
gfu_l = GridFunction(V,multidim=0)
gfu_l.AddMultiDimComponent(gfu.vec)
for loadstep in range(50):
    print ("loadstep", loadstep)
    factor.Set ((loadstep+1)/10)
    SolveNonlinearMinProblem(a,gfu)
    if (loadstep + 1) % 10 == 0:
        gfu_l.AddMultiDimComponent(gfu.vec)

loadstep 0
Newton iteration   0 , energy = 8.750000000000002<A u 0 , A u 0 >_{-1}^0.5 =  0.016677582605645856
Newton iteration   1 , energy = 8.750132555222002<A u 1 , A u 1 >_{-1}^0.5 =  0.023331359879402973
Newton iteration   2 , energy = 8.749861163856334<A u 2 , A u 2 >_{-1}^0.5 =  0.00010230629993558923
Newton iteration   3 , energy = 8.749861158623053<A u 3 , A u 3 >_{-1}^0.5 =  5.117101000324665e-08
Newton iteration   4 , energy = 8.749861158623048<A u 4 , A u 4 >_{-1}^0.5 =  2.700976392886091e-13
Newton iteration   5 , energy = 8.74986115862305<A u 5 , A u 5 >_{-1}^0.5 =  8.056635812758371e-16
loadstep 1
Newton iteration   0 , energy = 8.749583933129315<A u 0 , A u 0 >_{-1}^0.5 =  0.016595299053909628
Newton iteration   1 , energy = 8.749710107506655<A u 1 , A u 1 >_{-1}^0.5 =  0.02295643766896064
Newton iteration   2 , energy = 8.749447354442504<A u 2 , A u 2 >_{-1}^0.5 =  0.0001421582801509689
Newton iteration   3 , energy = 8.749447344443873<A u 3 , A u 3 >_{-1}^0.5 =  2.3255246954410186e-06
Newton iteration   4 , energy = 8.749447344441162<A u 4 , A u 4 >_{-1}^0.5 =  6.328258303136999e-10
Newton iteration   5 , energy = 8.749447344441165<A u 5 , A u 5 >_{-1}^0.5 =  1.0084181223730138e-15
loadstep 2
Newton iteration   0 , energy = 8.748898257854604<A u 0 , A u 0 >_{-1}^0.5 =  0.016356091853941414
Newton iteration   1 , energy = 8.749005344320112<A u 1 , A u 1 >_{-1}^0.5 =  0.021890729080673497
Newton iteration   2 , energy = 8.748766384857403<A u 2 , A u 2 >_{-1}^0.5 =  0.00020825243896718186
Newton iteration   3 , energy = 8.748766363495848<A u 3 , A u 3 >_{-1}^0.5 =  4.891625115122797e-06
Newton iteration   4 , energy = 8.748766363483883<A u 4 , A u 4 >_{-1}^0.5 =  2.6654449893034696e-09
Newton iteration   5 , energy = 8.748766363483883<A u 5 , A u 5 >_{-1}^0.5 =  1.8086213780001543e-15
loadstep 3
Newton iteration   0 , energy = 8.747955512559923<A u 0 , A u 0 >_{-1}^0.5 =  0.01598112098289731
Newton iteration   1 , energy = 8.748035610071788<A u 1 , A u 1 >_{-1}^0.5 =  0.02029003772240239
Newton iteration   2 , energy = 8.747830272447588<A u 2 , A u 2 >_{-1}^0.5 =  0.0002638495352051914
Newton iteration   3 , energy = 8.74783023813094<A u 3 , A u 3 >_{-1}^0.5 =  7.248678996775497e-06
Newton iteration   4 , energy = 8.747830238104656<A u 4 , A u 4 >_{-1}^0.5 =  4.7441717484143555e-09
Newton iteration   5 , energy = 8.747830238104669<A u 5 , A u 5 >_{-1}^0.5 =  2.695668760170589e-15
loadstep 4
Newton iteration   0 , energy = 8.74677136608456<A u 0 , A u 0 >_{-1}^0.5 =  0.015500028754946344
Newton iteration   1 , energy = 8.74682218106983<A u 1 , A u 1 >_{-1}^0.5 =  0.018357337758334515
Newton iteration   2 , energy = 8.746654051147162<A u 2 , A u 2 >_{-1}^0.5 =  0.0002991087145041139
Newton iteration   3 , energy = 8.746654006940766<A u 3 , A u 3 >_{-1}^0.5 =  8.721699482160068e-06
Newton iteration   4 , energy = 8.746654006902721<A u 4 , A u 4 >_{-1}^0.5 =  5.2421344675834136e-09
Newton iteration   5 , energy = 8.746654006902721<A u 5 , A u 5 >_{-1}^0.5 =  3.368446884720984e-15
loadstep 5
Newton iteration   0 , energy = 8.745363235443092<A u 0 , A u 0 >_{-1}^0.5 =  0.014945100392135344
Newton iteration   1 , energy = 8.745386960171833<A u 1 , A u 1 >_{-1}^0.5 =  0.016290257519009144
Newton iteration   2 , energy = 8.745254520464078<A u 2 , A u 2 >_{-1}^0.5 =  0.0003132391521607688
Newton iteration   3 , energy =  8.7452544718526<A u 3 , A u 3 >_{-1}^0.5 =  9.043578969804919e-06
Newton iteration   4 , energy = 8.745254471811707<A u 4 , A u 4 >_{-1}^0.5 =  4.215318820646109e-09
Newton iteration   5 , energy = 8.745254471811702<A u 5 , A u 5 >_{-1}^0.5 =  2.8997312377221248e-15
loadstep 6
Newton iteration   0 , energy = 8.743749097806283<A u 0 , A u 0 >_{-1}^0.5 =  0.014346493694390759
Newton iteration   1 , energy = 8.743750467097106<A u 1 , A u 1 >_{-1}^0.5 =  0.014246300602241949
Newton iteration   2 , energy = 8.743649143781806<A u 2 , A u 2 >_{-1}^0.5 =  0.00030936494189159165
Newton iteration   3 , energy = 8.743649096250385<A u 3 , A u 3 >_{-1}^0.5 =  8.38460692811568e-06
Newton iteration   4 , energy = 8.743649096215233<A u 4 , A u 4 >_{-1}^0.5 =  2.692147723215961e-09
Newton iteration   5 , energy = 8.743649096215234<A u 5 , A u 5 >_{-1}^0.5 =  2.706088905270728e-15
loadstep 7
Newton iteration   0 , energy = 8.741946608055962<A u 0 , A u 0 >_{-1}^0.5 =  0.013729428634298936
Newton iteration   1 , energy = 8.74193113281337<A u 1 , A u 1 >_{-1}^0.5 =  0.012330807534039926
Newton iteration   2 , energy = 8.74185519957042<A u 2 , A u 2 >_{-1}^0.5 =  0.00029213298828293104
Newton iteration   3 , energy = 8.741855157102606<A u 3 , A u 3 >_{-1}^0.5 =  7.128970076876137e-06
Newton iteration   4 , energy = 8.741855157077195<A u 4 , A u 4 >_{-1}^0.5 =  1.4420557564279676e-09
Newton iteration   5 , energy = 8.741855157077197<A u 5 , A u 5 >_{-1}^0.5 =  3.2635314084219904e-15
loadstep 8
Newton iteration   0 , energy = 8.739972524644806<A u 0 , A u 0 >_{-1}^0.5 =  0.013113224428771376
Newton iteration   1 , energy = 8.739945347610659<A u 1 , A u 1 >_{-1}^0.5 =  0.01060136635533351
Newton iteration   2 , energy = 8.739889202270737<A u 2 , A u 2 >_{-1}^0.5 =  0.0002663059128308429
Newton iteration   3 , energy = 8.739889166926455<A u 3 , A u 3 >_{-1}^0.5 =  5.661221886341418e-06
Newton iteration   4 , energy = 8.739889166910432<A u 4 , A u 4 >_{-1}^0.5 =  6.711437099260478e-10
Newton iteration   5 , energy = 8.739889166910432<A u 5 , A u 5 >_{-1}^0.5 =  2.8224446340844284e-15
loadstep 9
Newton iteration   0 , energy = 8.737842388930513<A u 0 , A u 0 >_{-1}^0.5 =  0.012511583560594706
Newton iteration   1 , energy = 8.737807753740729<A u 1 , A u 1 >_{-1}^0.5 =  0.009079612968653486
Newton iteration   2 , energy = 8.737766557197903<A u 2 , A u 2 >_{-1}^0.5 =  0.00023601511099922696
Newton iteration   3 , energy = 8.737766529405949<A u 3 , A u 3 >_{-1}^0.5 =  4.256830257610049e-06
Newton iteration   4 , energy = 8.737766529396884<A u 4 , A u 4 >_{-1}^0.5 =  2.7743063557914185e-10
Newton iteration   5 , energy = 8.737766529396882<A u 5 , A u 5 >_{-1}^0.5 =  3.1054566479418343e-15
loadstep 10
Newton iteration   0 , energy = 8.73557038769295<A u 0 , A u 0 >_{-1}^0.5 =  0.011933492662442846
Newton iteration   1 , energy = 8.73553152420747<A u 1 , A u 1 >_{-1}^0.5 =  0.007763816791529964
Newton iteration   2 , energy = 8.73550139375871<A u 2 , A u 2 >_{-1}^0.5 =  0.00020446417278053672
Newton iteration   3 , energy = 8.735501372884304<A u 3 , A u 3 >_{-1}^0.5 =  3.0633187978332092e-06
Newton iteration   4 , energy = 8.735501372879611<A u 4 , A u 4 >_{-1}^0.5 =  1.0296498508873415e-10
Newton iteration   5 , energy = 8.735501372879613<A u 5 , A u 5 >_{-1}^0.5 =  3.855146896576513e-15
loadstep 11
Newton iteration   0 , energy = 8.733169335796159<A u 0 , A u 0 >_{-1}^0.5 =  0.011384291415803674
Newton iteration   1 , energy = 8.733128560905088<A u 1 , A u 1 >_{-1}^0.5 =  0.006639099721604443
Newton iteration   2 , energy = 8.733106521835378<A u 2 , A u 2 >_{-1}^0.5 =  0.0001739028994158834
Newton iteration   3 , energy = 8.733106506726692<A u 3 , A u 3 >_{-1}^0.5 =  2.128012816590016e-06
Newton iteration   4 , energy = 8.733106506724422<A u 4 , A u 4 >_{-1}^0.5 =  3.4241888147528354e-11
Newton iteration   5 , energy = 8.733106506724425<A u 5 , A u 5 >_{-1}^0.5 =  4.202632668754335e-15
loadstep 12
Newton iteration   0 , energy = 8.73065073088584<A u 0 , A u 0 >_{-1}^0.5 =  0.010866663649545746
Newton iteration   1 , energy = 8.730609627352983<A u 1 , A u 1 >_{-1}^0.5 =  0.005684550176890224
Newton iteration   2 , energy = 8.730593466066004<A u 2 , A u 2 >_{-1}^0.5 =  0.00014574158623094499
Newton iteration   3 , energy = 8.730593455450633<A u 3 , A u 3 >_{-1}^0.5 =  1.4373276293350514e-06
Newton iteration   4 , energy = 8.730593455449597<A u 4 , A u 4 >_{-1}^0.5 =  9.957065655159065e-12
Newton iteration   5 , energy = 8.730593455449597<A u 5 , A u 5 >_{-1}^0.5 =  4.553764333472074e-15
loadstep 13
Newton iteration   0 , energy = 8.728024846757634<A u 0 , A u 0 >_{-1}^0.5 =  0.010381451820192344
Newton iteration   1 , energy = 8.727984446186102<A u 1 , A u 1 >_{-1}^0.5 =  0.004877644309085625
Newton iteration   2 , energy = 8.727972544723219<A u 2 , A u 2 >_{-1}^0.5 =  0.00012072280174228327
Newton iteration   3 , energy = 8.72797253743792<A u 3 , A u 3 >_{-1}^0.5 =  9.497245790259865e-07
Newton iteration   4 , energy = 8.727972537437465<A u 4 , A u 4 >_{-1}^0.5 =  2.3171563308944815e-12
Newton iteration   5 , energy = 8.727972537437468<A u 5 , A u 5 >_{-1}^0.5 =  4.6085897268221025e-15
loadstep 14
Newton iteration   0 , energy = 8.725300843768164<A u 0 , A u 0 >_{-1}^0.5 =  0.009928279578345058
Newton iteration   1 , energy = 8.725261782651362<A u 1 , A u 1 >_{-1}^0.5 =  0.004196713092763844
Newton iteration   2 , energy = 8.725252970508553<A u 2 , A u 2 >_{-1}^0.5 =  9.910189073828725e-05
Newton iteration   3 , energy = 8.725252965598385<A u 3 , A u 3 >_{-1}^0.5 =  6.171499936508376e-07
Newton iteration   4 , energy = 8.725252965598198<A u 4 , A u 4 >_{-1}^0.5 =  2.6828932414872465e-13
Newton iteration   5 , energy = 8.725252965598195<A u 5 , A u 5 >_{-1}^0.5 =  4.372036196595596e-15
loadstep 15
Newton iteration   0 , energy = 8.722486883056932<A u 0 , A u 0 >_{-1}^0.5 =  0.009506005046403374
Newton iteration   1 , energy = 8.722449524170578<A u 1 , A u 1 >_{-1}^0.5 =  0.0036221383603452614
Newton iteration   2 , energy = 8.722442958756766<A u 2 , A u 2 >_{-1}^0.5 =  8.080932446903616e-05
Newton iteration   3 , energy = 8.722442955491687<A u 3 , A u 3 >_{-1}^0.5 =  3.9619412125728194e-07
Newton iteration   4 , energy = 8.722442955491614<A u 4 , A u 4 >_{-1}^0.5 =  1.533983247647122e-13
Newton iteration   5 , energy = 8.72244295549161<A u 5 , A u 5 >_{-1}^0.5 =  4.718261745185127e-15
loadstep 16
Newton iteration   0 , energy = 8.719590236980153<A u 0 , A u 0 >_{-1}^0.5 =  0.009113039187346133
Newton iteration   1 , energy = 8.719554758791887<A u 1 , A u 1 >_{-1}^0.5 =  0.0031367887187666406
Newton iteration   2 , energy = 8.719549834337263<A u 2 , A u 2 >_{-1}^0.5 =  6.558227828578711e-05
Newton iteration   3 , energy = 8.719549832186642<A u 3 , A u 3 >_{-1}^0.5 =  2.5225360548715454e-07
Newton iteration   4 , energy = 8.719549832186612<A u 4 , A u 4 >_{-1}^0.5 =  1.507813278565408e-13
Newton iteration   5 , energy = 8.719549832186614<A u 5 , A u 5 >_{-1}^0.5 =  5.745370916311974e-15
loadstep 17
Newton iteration   0 , energy = 8.716617391769223<A u 0 , A u 0 >_{-1}^0.5 =  0.008747562874567521
Newton iteration   1 , energy = 8.716583851994583<A u 1 , A u 1 >_{-1}^0.5 =  0.0027260365842272194
Newton iteration   2 , energy =  8.7165801324147<A u 2 , A u 2 >_{-1}^0.5 =  5.3062302055573826e-05
Newton iteration   3 , energy = 8.716580131006795<A u 3 , A u 3 >_{-1}^0.5 =  1.5981075076557974e-07
Newton iteration   4 , energy = 8.71658013100678<A u 4 , A u 4 >_{-1}^0.5 =  9.189275057309154e-14
loadstep 18
Newton iteration   0 , energy = 8.713574140655615<A u 0 , A u 0 >_{-1}^0.5 =  0.008407670778939549
Newton iteration   1 , energy = 8.713542520325504<A u 1 , A u 1 >_{-1}^0.5 =  0.0023775675324181665
Newton iteration   2 , energy = 8.713539690707933<A u 2 , A u 2 >_{-1}^0.5 =  4.28618645886888e-05
Newton iteration   3 , energy = 8.713539689789291<A u 3 , A u 3 >_{-1}^0.5 =  1.0101914350642276e-07
Newton iteration   4 , energy = 8.713539689789279<A u 4 , A u 4 >_{-1}^0.5 =  4.80099598315063e-14
loadstep 19
Newton iteration   0 , energy = 8.710465667024843<A u 0 , A u 0 >_{-1}^0.5 =  0.00809146379891827
Newton iteration   1 , energy = 8.710435900463288<A u 1 , A u 1 >_{-1}^0.5 =  0.0020811047922922286
Newton iteration   2 , energy = 8.710433732402823<A u 2 , A u 2 >_{-1}^0.5 =  3.460554942630233e-05
Newton iteration   3 , energy = 8.710433731804002<A u 3 , A u 3 >_{-1}^0.5 =  6.385680094378083e-08
Newton iteration   4 , energy =   8.710433731804<A u 4 , A u 4 >_{-1}^0.5 =  2.3754712346280936e-14
loadstep 20
Newton iteration   0 , energy = 8.70729661790546<A u 0 , A u 0 >_{-1}^0.5 =  0.007797106054476603
Newton iteration   1 , energy = 8.707268612789706<A u 1 , A u 1 >_{-1}^0.5 =  0.001828115920838295
Newton iteration   2 , energy = 8.707266939759124<A u 2 , A u 2 >_{-1}^0.5 =  2.7952479963162672e-05
Newton iteration   3 , energy = 8.707266939368424<A u 3 , A u 3 >_{-1}^0.5 =  4.043944269231185e-08
Newton iteration   4 , energy = 8.707266939368418<A u 4 , A u 4 >_{-1}^0.5 =  1.220615463902853e-14
loadstep 21
Newton iteration   0 , energy = 8.70407116848376<A u 0 , A u 0 >_{-1}^0.5 =  0.007522857909616671
Newton iteration   1 , energy = 8.704044819049004<A u 1 , A u 1 >_{-1}^0.5 =  0.001611535315643125
Newton iteration   2 , energy = 8.704043518933695<A u 2 , A u 2 >_{-1}^0.5 =  2.260598208530695e-05
Newton iteration   3 , energy = 8.704043518678155<A u 3 , A u 3 >_{-1}^0.5 =  2.5693535010444772e-08
Newton iteration   4 , energy = 8.70404351867816<A u 4 , A u 4 >_{-1}^0.5 =  7.391132015510724e-15
loadstep 22
Newton iteration   0 , energy = 8.700793078503203<A u 0 , A u 0 >_{-1}^0.5 =  0.007267093060730808
Newton iteration   1 , energy = 8.700768274070779<A u 1 , A u 1 >_{-1}^0.5 =  0.001425517167805448
Newton iteration   2 , energy = 8.70076725677529<A u 2 , A u 2 >_{-1}^0.5 =  1.831528979800765e-05
Newton iteration   3 , energy = 8.700767256607556<A u 3 , A u 3 >_{-1}^0.5 =  1.6396500158033528e-08
Newton iteration   4 , energy = 8.700767256607557<A u 4 , A u 4 >_{-1}^0.5 =  5.845030553756428e-15
loadstep 23
Newton iteration   0 , energy = 8.697465741449388<A u 0 , A u 0 >_{-1}^0.5 =  0.0070283052427454595
Newton iteration   1 , energy = 8.697442371797747<A u 1 , A u 1 >_{-1}^0.5 =  0.0012652231229517356
Newton iteration   2 , energy = 8.697441570430097<A u 2 , A u 2 >_{-1}^0.5 =  1.4872780407049409e-05
Newton iteration   3 , energy = 8.697441570319485<A u 3 , A u 3 >_{-1}^0.5 =  1.0518632260622615e-08
Newton iteration   4 , energy = 8.697441570319487<A u 4 , A u 4 >_{-1}^0.5 =  4.405332255777376e-15
loadstep 24
Newton iteration   0 , energy = 8.694092227393039<A u 0 , A u 0 >_{-1}^0.5 =  0.006805108340818318
Newton iteration   1 , energy = 8.694070186017763<A u 1 , A u 1 >_{-1}^0.5 =  0.0011266436800926223
Newton iteration   2 , energy = 8.694069550595463<A u 2 , A u 2 >_{-1}^0.5 =  1.2109069700851704e-05
Newton iteration   3 , energy = 8.694069550522148<A u 3 , A u 3 >_{-1}^0.5 =  6.787740387661239e-09
Newton iteration   4 , energy = 8.694069550522144<A u 4 , A u 4 >_{-1}^0.5 =  6.254583601294689e-15
loadstep 25
Newton iteration   0 , energy = 8.690675320299126<A u 0 , A u 0 >_{-1}^0.5 =  0.0065962324653914065
Newton iteration   1 , energy = 8.690654506275454<A u 1 , A u 1 >_{-1}^0.5 =  0.001006449999127333
Newton iteration   2 , energy = 8.690653999211646<A u 2 , A u 2 >_{-1}^0.5 =  9.887408054086182e-06
Newton iteration   3 , energy = 8.690653999162762<A u 3 , A u 3 >_{-1}^0.5 =  4.4080650967048695e-09
Newton iteration   4 , energy = 8.690653999162764<A u 4 , A u 4 >_{-1}^0.5 =  5.758262191161789e-15
loadstep 26
Newton iteration   0 , energy = 8.68721755053038<A u 0 , A u 0 >_{-1}^0.5 =  0.006400517697529269
Newton iteration   1 , energy = 8.687197869461734<A u 1 , A u 1 >_{-1}^0.5 =  0.0009018719819724309
Newton iteration   2 , energy = 8.687197462311365<A u 2 , A u 2 >_{-1}^0.5 =  8.098192892916217e-06
Newton iteration   3 , energy = 8.687197462278565<A u 3 , A u 3 >_{-1}^0.5 =  2.8818016831866e-09
Newton iteration   4 , energy = 8.687197462278563<A u 4 , A u 4 >_{-1}^0.5 =  5.812600260020221e-15
loadstep 27
Newton iteration   0 , energy = 8.683721223189794<A u 0 , A u 0 >_{-1}^0.5 =  0.0062169066296662474
Newton iteration   1 , energy = 8.683702587568161<A u 1 , A u 1 >_{-1}^0.5 =  0.0008105984822188017
Newton iteration   2 , energy = 8.683702258669632<A u 2 , A u 2 >_{-1}^0.5 =  6.654005047155873e-06
Newton iteration   3 , energy = 8.683702258647491<A u 3 , A u 3 >_{-1}^0.5 =  1.8969669035519443e-09
Newton iteration   4 , energy = 8.683702258647495<A u 4 , A u 4 >_{-1}^0.5 =  6.063341880066008e-15
loadstep 28
Newton iteration   0 , energy = 8.680188442866307<A u 0 , A u 0 >_{-1}^0.5 =  0.006044436430119278
Newton iteration   1 , energy = 8.680170772062889<A u 1 , A u 1 >_{-1}^0.5 =  0.000730695854463049
Newton iteration   2 , energy = 8.680170504818914<A u 2 , A u 2 >_{-1}^0.5 =  5.485327885213798e-06
Newton iteration   3 , energy = 8.680170504803863<A u 3 , A u 3 >_{-1}^0.5 =  1.2574286430841388e-09
Newton iteration   4 , energy = 8.680170504803861<A u 4 , A u 4 >_{-1}^0.5 =  5.098360944895964e-15
loadstep 29
Newton iteration   0 , energy = 8.676621135273345<A u 0 , A u 0 >_{-1}^0.5 =  0.005882230892523314
Newton iteration   1 , energy = 8.676604355304637<A u 1 , A u 1 >_{-1}^0.5 =  0.0006605415495133434
Newton iteration   2 , energy = 8.676604136921796<A u 2 , A u 2 >_{-1}^0.5 =  4.536967917129385e-06
Newton iteration   3 , energy =  8.6766041369115<A u 3 , A u 3 >_{-1}^0.5 =  8.393603919837039e-10
Newton iteration   4 , energy = 8.676604136911497<A u 4 , A u 4 >_{-1}^0.5 =  5.809184305073015e-15
loadstep 30
Newton iteration   0 , energy = 8.673021066202953<A u 0 , A u 0 >_{-1}^0.5 =  0.005729492751287068
Newton iteration   1 , energy = 8.673005109368702<A u 1 , A u 1 >_{-1}^0.5 =  0.0005987699748712484
Newton iteration   2 , energy = 8.673004929927785<A u 2 , A u 2 >_{-1}^0.5 =  3.765124006156311e-06
Newton iteration   3 , energy = 8.673004929920697<A u 3 , A u 3 >_{-1}^0.5 =  5.642222429306511e-10
Newton iteration   4 , energy = 8.673004929920696<A u 4 , A u 4 >_{-1}^0.5 =  5.9182491903632614e-15
loadstep 31
Newton iteration   0 , energy = 8.669389858159557<A u 0 , A u 0 >_{-1}^0.5 =  0.005585496424926984
Newton iteration   1 , energy = 8.66937466261523<A u 1 , A u 1 >_{-1}^0.5 =  0.0005442283166895947
Newton iteration   2 , energy = 8.669374514381587<A u 2 , A u 2 >_{-1}^0.5 =  3.1350221732245437e-06
Newton iteration   3 , energy = 8.66937451437667<A u 3 , A u 3 >_{-1}^0.5 =  3.8190916999844325e-10
Newton iteration   4 , energy = 8.669374514376667<A u 4 , A u 4 >_{-1}^0.5 =  6.584792678738834e-15
loadstep 32
Newton iteration   0 , energy = 8.665729004986108<A u 0 , A u 0 >_{-1}^0.5 =  0.005449581270422759
Newton iteration   1 , energy = 8.665714514290284<A u 1 , A u 1 >_{-1}^0.5 =  0.0004959404367353185
Newton iteration   2 , energy = 8.665714391199181<A u 2 , A u 2 >_{-1}^0.5 =  2.619025848326988e-06
Newton iteration   3 , energy = 8.66571439119575<A u 3 , A u 3 >_{-1}^0.5 =  2.6027987217186216e-10
Newton iteration   4 , energy = 8.66571439119575<A u 4 , A u 4 >_{-1}^0.5 =  6.310411503237572e-15
loadstep 33
Newton iteration   0 , energy = 8.662039884750648<A u 0 , A u 0 >_{-1}^0.5 =  0.005321145381082999
Newton iteration   1 , energy = 8.66202604741291<A u 1 , A u 1 >_{-1}^0.5 =  0.0004530773114986438
Newton iteration   2 , energy = 8.662025944683327<A u 2 , A u 2 >_{-1}^0.5 =  2.1951363000024995e-06
Newton iteration   3 , energy = 8.662025944680916<A u 3 , A u 3 >_{-1}^0.5 =  1.7858487098449372e-10
Newton iteration   4 , energy = 8.662025944680916<A u 4 , A u 4 >_{-1}^0.5 =  6.607679837108749e-15
loadstep 34
Newton iteration   0 , energy = 8.65832377112368<A u 0 , A u 0 >_{-1}^0.5 =  0.005199639928201556
Newton iteration   1 , energy = 8.658310540168342<A u 1 , A u 1 >_{-1}^0.5 =  0.0004149327733557936
Newton iteration   2 , energy = 8.658310454011463<A u 2 , A u 2 >_{-1}^0.5 =  1.8458080206835281e-06
Newton iteration   3 , energy = 8.65831045400976<A u 3 , A u 3 >_{-1}^0.5 =  1.2334028931169478e-10
Newton iteration   4 , energy = 8.65831045400976<A u 4 , A u 4 >_{-1}^0.5 =  6.680201512168534e-15
loadstep 35
Newton iteration   0 , energy = 8.654581843443761<A u 0 , A u 0 >_{-1}^0.5 =  0.005084564027367384
Newton iteration   1 , energy = 8.65456917599831<A u 1 , A u 1 >_{-1}^0.5 =  0.0003809035534945384
Newton iteration   2 , energy = 8.654569103396332<A u 2 , A u 2 >_{-1}^0.5 =  1.5570154927991207e-06
Newton iteration   3 , energy = 8.654569103395122<A u 3 , A u 3 >_{-1}^0.5 =  8.573891311263491e-11
Newton iteration   4 , energy = 8.654569103395124<A u 4 , A u 4 >_{-1}^0.5 =  7.059568199342222e-15
loadstep 36
Newton iteration   0 , energy = 8.650815195641044<A u 0 , A u 0 >_{-1}^0.5 =  0.00497546009900596
Newton iteration   1 , energy = 8.650803052553723<A u 1 , A u 1 >_{-1}^0.5 =  0.0003504728204257288
Newton iteration   2 , energy = 8.650802991091053<A u 2 , A u 2 >_{-1}^0.5 =  1.3175190450571487e-06
Newton iteration   3 , energy = 8.650802991090185<A u 3 , A u 3 >_{-1}^0.5 =  5.99757900641553e-11
Newton iteration   4 , energy = 8.650802991090188<A u 4 , A u 4 >_{-1}^0.5 =  5.326139270082297e-15
loadstep 37
Newton iteration   0 , energy = 8.647024844164546<A u 0 , A u 0 >_{-1}^0.5 =  0.004871909687068079
Newton iteration   1 , energy = 8.64701318965287<A u 1 , A u 1 >_{-1}^0.5 =  0.00032319656453480973
Newton iteration   2 , energy = 8.647013137386656<A u 2 , A u 2 >_{-1}^0.5 =  1.1182875717610878e-06
Newton iteration   3 , energy = 8.64701313738604<A u 3 , A u 3 >_{-1}^0.5 =  4.221410145221285e-11
Newton iteration   4 , energy = 8.647013137386041<A u 4 , A u 4 >_{-1}^0.5 =  6.1878333714899875e-15
loadstep 38
Newton iteration   0 , energy = 8.643211735038836<A u 0 , A u 0 >_{-1}^0.5 =  0.0047735296976276235
Newton iteration   1 , energy = 8.643200536368832<A u 1 , A u 1 >_{-1}^0.5 =  0.00029869230501240556
Newton iteration   2 , energy = 8.643200491729171<A u 2 , A u 2 >_{-1}^0.5 =  9.520444188391924e-07
Newton iteration   3 , energy = 8.64320049172872<A u 3 , A u 3 >_{-1}^0.5 =  2.9890963843612336e-11
Newton iteration   4 , energy = 8.643200491728713<A u 4 , A u 4 >_{-1}^0.5 =  5.927485176174408e-15
loadstep 39
Newton iteration   0 , energy = 8.639376750158188<A u 0 , A u 0 >_{-1}^0.5 =  0.004679969019268834
Newton iteration   1 , energy = 8.639365977353362<A u 1 , A u 1 >_{-1}^0.5 =  0.0002766296964333796
Newton iteration   2 , energy = 8.639365939065923<A u 2 , A u 2 >_{-1}^0.5 =  8.129098312091563e-07
Newton iteration   3 , energy = 8.639365939065597<A u 3 , A u 3 >_{-1}^0.5 =  2.129041460502372e-11
Newton iteration   4 , energy = 8.639365939065595<A u 4 , A u 4 >_{-1}^0.5 =  6.14764718223009e-15
loadstep 40
Newton iteration   0 , energy = 8.635520712911946<A u 0 , A u 0 >_{-1}^0.5 =  0.0045909054884695245
Newton iteration   1 , energy = 8.635510338490109<A u 1 , A u 1 >_{-1}^0.5 =  0.00025672269308442945
Newton iteration   2 , energy = 8.635510305515965<A u 2 , A u 2 >_{-1}^0.5 =  6.961190117130188e-07
Newton iteration   3 , energy = 8.63551030551572<A u 3 , A u 3 >_{-1}^0.5 =  1.524923937169174e-11
Newton iteration   4 , energy = 8.635510305515723<A u 4 , A u 4 >_{-1}^0.5 =  5.533620467872868e-15
loadstep 41
Newton iteration   0 , energy = 8.631644393221707<A u 0 , A u 0 >_{-1}^0.5 =  0.004506043165565104
Newton iteration   1 , energy = 8.631634391957725<A u 1 , A u 1 >_{-1}^0.5 =  0.0002387229941945253
Newton iteration   2 , energy = 8.631634363446206<A u 2 , A u 2 >_{-1}^0.5 =  5.977994014832271e-07
Newton iteration   3 , energy = 8.631634363446025<A u 3 , A u 3 >_{-1}^0.5 =  1.0983789132632781e-11
Newton iteration   4 , energy = 8.631634363446027<A u 4 , A u 4 >_{-1}^0.5 =  5.9886015152998185e-15
loadstep 42
Newton iteration   0 , energy = 8.627748512060565<A u 0 , A u 0 >_{-1}^0.5 =  0.004425109889385947
Newton iteration   1 , energy = 8.627738860772805<A u 1 , A u 1 >_{-1}^0.5 =  0.00022241454518084048
Newton iteration   2 , energy = 8.627738836024482<A u 2 , A u 2 >_{-1}^0.5 =  5.147943878355631e-07
Newton iteration   3 , energy = 8.62773883602435<A u 3 , A u 3 >_{-1}^0.5 =  7.954455532014165e-12
Newton iteration   4 , energy = 8.627738836024356<A u 4 , A u 4 >_{-1}^0.5 =  7.639383993849149e-15
loadstep 43
Newton iteration   0 , energy = 8.623833745515205<A u 0 , A u 0 >_{-1}^0.5 =  0.004347855081376776
Newton iteration   1 , energy = 8.623824422873449<A u 1 , A u 1 >_{-1}^0.5 =  0.00020760891187514776
Newton iteration   2 , energy = 8.623824401310939<A u 2 , A u 2 >_{-1}^0.5 =  4.4452343138069405e-07
Newton iteration   3 , energy = 8.623824401310843<A u 3 , A u 3 >_{-1}^0.5 =  5.790523533304125e-12
Newton iteration   4 , energy = 8.623824401310843<A u 4 , A u 4 >_{-1}^0.5 =  6.807306854157925e-15
loadstep 44
Newton iteration   0 , energy = 8.619900728443671<A u 0 , A u 0 >_{-1}^0.5 =  0.004274047772884702
Newton iteration   1 , energy = 8.619891714796632<A u 1 , A u 1 >_{-1}^0.5 =  0.00019414137852595735
Newton iteration   2 , energy = 8.619891695941396<A u 2 , A u 2 >_{-1}^0.5 =  3.8487082489926307e-07
Newton iteration   3 , energy = 8.619891695941323<A u 3 , A u 3 >_{-1}^0.5 =  4.237631258332885e-12
Newton iteration   4 , energy = 8.619891695941321<A u 4 , A u 4 >_{-1}^0.5 =  6.4558182569603805e-15
loadstep 45
Newton iteration   0 , energy = 8.615950057775086<A u 0 , A u 0 >_{-1}^0.5 =  0.004203474831744078
Newton iteration   1 , energy = 8.615941334995359<A u 1 , A u 1 >_{-1}^0.5 =  0.0001818676473661749
Newton iteration   2 , energy = 8.615941318449257<A u 2 , A u 2 >_{-1}^0.5 =  3.3409699005387563e-07
Newton iteration   3 , energy = 8.615941318449202<A u 3 , A u 3 >_{-1}^0.5 =  3.1150010060471828e-12
Newton iteration   4 , energy = 8.615941318449208<A u 4 , A u 4 >_{-1}^0.5 =  7.051750902255632e-15
loadstep 46
Newton iteration   0 , energy = 8.611982295491368<A u 0 , A u 0 >_{-1}^0.5 =  0.004135939366897804
Newton iteration   1 , energy = 8.611973846836175<A u 1 , A u 1 >_{-1}^0.5 =  0.00017066103967723
Newton iteration   2 , energy = 8.611973832266731<A u 2 , A u 2 >_{-1}^0.5 =  2.907675680107316e-07
Newton iteration   3 , energy = 8.611973832266687<A u 3 , A u 3 >_{-1}^0.5 =  2.3039250921449486e-12
Newton iteration   4 , energy = 8.611973832266692<A u 4 , A u 4 >_{-1}^0.5 =  6.120557491177614e-15
loadstep 47
Newton iteration   0 , energy = 8.607997971326375<A u 0 , A u 0 >_{-1}^0.5 =  0.004071259291904658
Newton iteration   1 , energy = 8.607989781312382<A u 1 , A u 1 >_{-1}^0.5 =  0.00016041011593615804
Newton iteration   2 , energy = 8.607989768440916<A u 2 , A u 2 >_{-1}^0.5 =  2.5369657297005814e-07
Newton iteration   3 , energy = 8.607989768440884<A u 3 , A u 3 >_{-1}^0.5 =  1.7107371502438864e-12
Newton iteration   4 , energy = 8.607989768440891<A u 4 , A u 4 >_{-1}^0.5 =  6.489989588000374e-15
loadstep 48
Newton iteration   0 , energy = 8.60399758521309<A u 0 , A u 0 >_{-1}^0.5 =  0.00400926603042159
Newton iteration   1 , energy = 8.60398963950384<A u 1 , A u 1 >_{-1}^0.5 =  0.00015101664724705597
Newton iteration   2 , energy = 8.603989628095974<A u 2 , A u 2 >_{-1}^0.5 =  2.2190070060807851e-07
Newton iteration   3 , energy = 8.603989628095954<A u 3 , A u 3 >_{-1}^0.5 =  1.2758448139147053e-12
Newton iteration   4 , energy = 8.603989628095952<A u 4 , A u 4 >_{-1}^0.5 =  6.429033961914888e-15
loadstep 49
Newton iteration   0 , energy = 8.59998160950617<A u 0 , A u 0 >_{-1}^0.5 =  0.003949803348439382
Newton iteration   1 , energy = 8.599973894810804<A u 1 , A u 1 >_{-1}^0.5 =  0.00014239388187217074
Newton iteration   2 , energy = 8.59997388466869<A u 2 , A u 2 >_{-1}^0.5 =  1.9456250103741827e-07
Newton iteration   3 , energy = 8.599973884668675<A u 3 , A u 3 >_{-1}^0.5 =  9.556867936503832e-13
Newton iteration   4 , energy = 8.599973884668675<A u 4 , A u 4 >_{-1}^0.5 =  6.420373503862344e-15

[13]:

Draw(gfu_l,mesh, interpolate_multidim=True, animate=True,
     deformation=True, min=0, max=1, autoscale=False)

[13]:

BaseWebGuiScene

Supplementary 1: Allen-Cahn equation¶

The Allen-Cahn equations describe the process of phase separation and is the ( $L^{2}$ ) gradient-flow equation to the energy

E (v) = \int_{Ω} ε | \nabla v |^{2} + \underset{Ψ (v)}{\underset{⏟}{v^{2} (1 - v^{2})}} d x

where $Ψ (v)$ is the so-called double-well potential with the minima $- 1, 0, 1$ (with $0$ being an unstable minima).

The solution to the Allen-Cahn equation solves

\partial_{t} u = \frac{δ E}{δ u}

The quantity $u$ is an indicator for a phase where $- 1$ refers to one phase and $1$ to another phase.

The equation has two driving forces:

$u$ is pulled into one of the two stable minima states ( $- 1$ and $1$ ) of the nonlinear term $u^{2} (1 - u^{2})$ (separation of the phases)
the diffusion term scaled with $ε$ enforces a smooth transition between the two phases. $ε$ determines the size of the transition layer

We use the Energy formulation for energy minimization combined with a simple time stepping with an implicit Euler discretization:

M u^{n + 1} - M u^{n} = Δ t \underset{=: A (u)}{\underset{⏟}{\frac{δ E}{δ u}}} (u^{n + 1})

which we can interprete as a nonlinear minimization problem again with the energy

E^{I E} (v) = \int_{Ω} \frac{ε}{2} | \nabla v |^{2} + v^{2} (1 - v^{2}) + \frac{1}{2 Δ t} | v - u^{n} |^{2} d x

To solve the nonlinear equation at every time step we again rely on Newton’s method. We first define the periodic geometry, setup the formulation and then apply Newton’s method in the next cells:

[14]:

# define periodic geometry and generate mesh
shape = Rectangle(1,1).Face()
right=shape.edges.Max(X)
right.name="right"
shape.edges.Min(X).Identify(right,name="left")
top=shape.edges.Max(Y)
top.name="top"
shape.edges.Min(Y).Identify(top,name="bottom")
geom = OCCGeometry(shape, dim=2)
mesh = Mesh(geom.GenerateMesh(maxh=0.1))

[15]:

#use a periodic fe space correspondingly
V = Periodic(H1(mesh, order=3))
u = V.TrialFunction()

eps = 4e-3
dt = Parameter(1e-1)
gfu = GridFunction(V)
gfuold = GridFunction(V)
a = BilinearForm (V, symmetric=False)
a += Variation( (eps/2*grad(u)*grad(u) + ((1-u*u)*(1-u*u))
                     + 0.5/dt*(u-gfuold)*(u-gfuold)) * dx)

[16]:

from math import pi
gfu = GridFunction(V)
#gfu.Set(sin(2*pi*x)) # regular initial values
gfu.Set(sin(1e7*(x+y*y))) #<- essentially a random function
gfu_t = GridFunction(V, multidim=0)
gfu_t.AddMultiDimComponent(0.1*gfu.vec)

[17]:

t = 0; tend = 5; cnt = 0; sample_rate = int(floor(0.5/dt.Get()))
while t < tend - 0.5 * dt.Get():
    gfuold.vec.data = gfu.vec
    SolveNonlinearMinProblem(a,gfu)
    if (cnt+1) % sample_rate == 0:
        gfu_t.AddMultiDimComponent(0.1*gfu.vec)
    t += dt.Get(); cnt += 1
    print("t = ", t)

Newton iteration   0 , energy = 11.155527591736018<A u 0 , A u 0 >_{-1}^0.5 =  3.9029676615100333
Newton iteration   1 , energy = 3.031758350166657<A u 1 , A u 1 >_{-1}^0.5 =  0.5872637690441912
Newton iteration   2 , energy = 2.851815320675236<A u 2 , A u 2 >_{-1}^0.5 =  0.051517175466848336
Newton iteration   3 , energy = 2.8504702558831227<A u 3 , A u 3 >_{-1}^0.5 =  0.0012747177081418896
Newton iteration   4 , energy = 2.8504694427694806<A u 4 , A u 4 >_{-1}^0.5 =  1.6950503329541726e-06
Newton iteration   5 , energy = 2.85046944276804<A u 5 , A u 5 >_{-1}^0.5 =  4.23230584300913e-12
Newton iteration   6 , energy = 2.850469442768041<A u 6 , A u 6 >_{-1}^0.5 =  3.19059330547849e-16
t =  0.1
Newton iteration   0 , energy = 1.4798260778065344<A u 0 , A u 0 >_{-1}^0.5 =  0.8225916484098298
Newton iteration   1 , energy = 1.1356965620675556<A u 1 , A u 1 >_{-1}^0.5 =  0.04100866266173253
Newton iteration   2 , energy = 1.134850460634233<A u 2 , A u 2 >_{-1}^0.5 =  0.0004808987015879975
Newton iteration   3 , energy = 1.1348503449763898<A u 3 , A u 3 >_{-1}^0.5 =  1.8404732682074978e-07
Newton iteration   4 , energy = 1.134850344976372<A u 4 , A u 4 >_{-1}^0.5 =  4.295610942078956e-14
t =  0.2
Newton iteration   0 , energy = 1.0118266141901342<A u 0 , A u 0 >_{-1}^0.5 =  0.34912788832806796
Newton iteration   1 , energy = 0.9512089197686354<A u 1 , A u 1 >_{-1}^0.5 =  0.012892609168279086
Newton iteration   2 , energy = 0.9511256001882656<A u 2 , A u 2 >_{-1}^0.5 =  5.5361844310412465e-05
Newton iteration   3 , energy = 0.9511255986557665<A u 3 , A u 3 >_{-1}^0.5 =  1.9056429087232417e-09
Newton iteration   4 , energy = 0.951125598655766<A u 4 , A u 4 >_{-1}^0.5 =  1.4630089922040322e-16
t =  0.30000000000000004
Newton iteration   0 , energy = 0.9110905743548463<A u 0 , A u 0 >_{-1}^0.5 =  0.25875410183657105
Newton iteration   1 , energy = 0.878369043151633<A u 1 , A u 1 >_{-1}^0.5 =  0.011877729217851326
Newton iteration   2 , energy = 0.8782983072320303<A u 2 , A u 2 >_{-1}^0.5 =  5.367126170405351e-05
Newton iteration   3 , energy =  0.8782983057917<A u 3 , A u 3 >_{-1}^0.5 =  1.6969039216074304e-09
Newton iteration   4 , energy = 0.8782983057917005<A u 4 , A u 4 >_{-1}^0.5 =  1.4893988049094296e-16
t =  0.4
Newton iteration   0 , energy = 0.8471297102003006<A u 0 , A u 0 >_{-1}^0.5 =  0.25377371432720536
Newton iteration   1 , energy = 0.8159840307174713<A u 1 , A u 1 >_{-1}^0.5 =  0.013776892966505897
Newton iteration   2 , energy = 0.8158888393927236<A u 2 , A u 2 >_{-1}^0.5 =  6.721093216992284e-05
Newton iteration   3 , energy = 0.8158888371340216<A u 3 , A u 3 >_{-1}^0.5 =  2.3080134756385272e-09
Newton iteration   4 , energy = 0.815888837134021<A u 4 , A u 4 >_{-1}^0.5 =  1.669690693889013e-16
t =  0.5
Newton iteration   0 , energy = 0.7833250288136523<A u 0 , A u 0 >_{-1}^0.5 =  0.26294142546776966
Newton iteration   1 , energy = 0.7500716429111843<A u 1 , A u 1 >_{-1}^0.5 =  0.0158999052371038
Newton iteration   2 , energy = 0.7499447952623312<A u 2 , A u 2 >_{-1}^0.5 =  8.98957905213857e-05
Newton iteration   3 , energy = 0.749944791221584<A u 3 , A u 3 >_{-1}^0.5 =  4.324126849446335e-09
Newton iteration   4 , energy = 0.7499447912215844<A u 4 , A u 4 >_{-1}^0.5 =  1.838504591635453e-16
t =  0.6
Newton iteration   0 , energy = 0.7154946862277995<A u 0 , A u 0 >_{-1}^0.5 =  0.2669123350109644
Newton iteration   1 , energy = 0.6812710902175816<A u 1 , A u 1 >_{-1}^0.5 =  0.016754748959607643
Newton iteration   2 , energy = 0.6811301930417182<A u 2 , A u 2 >_{-1}^0.5 =  0.00010545670330969519
Newton iteration   3 , energy = 0.6811301874809285<A u 3 , A u 3 >_{-1}^0.5 =  7.221056206460974e-09
Newton iteration   4 , energy = 0.6811301874809279<A u 4 , A u 4 >_{-1}^0.5 =  1.8814400909991558e-16
t =  0.7
Newton iteration   0 , energy = 0.6467836243973103<A u 0 , A u 0 >_{-1}^0.5 =  0.26232034334542514
Newton iteration   1 , energy = 0.6136408730671259<A u 1 , A u 1 >_{-1}^0.5 =  0.015828021070455677
Newton iteration   2 , energy = 0.6135151722367927<A u 2 , A u 2 >_{-1}^0.5 =  9.120067945796306e-05
Newton iteration   3 , energy = 0.6135151680778683<A u 3 , A u 3 >_{-1}^0.5 =  5.087267437641023e-09
Newton iteration   4 , energy = 0.6135151680778679<A u 4 , A u 4 >_{-1}^0.5 =  2.047607551591038e-16
t =  0.7999999999999999
Newton iteration   0 , energy = 0.5810325754854718<A u 0 , A u 0 >_{-1}^0.5 =  0.2530974031842429
Newton iteration   1 , energy = 0.5501189255035114<A u 1 , A u 1 >_{-1}^0.5 =  0.014529399171055368
Newton iteration   2 , energy = 0.5500130418792271<A u 2 , A u 2 >_{-1}^0.5 =  7.363869717001023e-05
Newton iteration   3 , energy = 0.5500130391678325<A u 3 , A u 3 >_{-1}^0.5 =  2.943680711071955e-09
Newton iteration   4 , energy = 0.550013039167832<A u 4 , A u 4 >_{-1}^0.5 =  2.406068616154831e-16
t =  0.8999999999999999
Newton iteration   0 , energy = 0.5200191278732138<A u 0 , A u 0 >_{-1}^0.5 =  0.24195821487970726
Newton iteration   1 , energy = 0.49195055544356614<A u 1 , A u 1 >_{-1}^0.5 =  0.016199909231168572
Newton iteration   2 , energy = 0.49181865857024165<A u 2 , A u 2 >_{-1}^0.5 =  0.00013877000575511985
Newton iteration   3 , energy = 0.49181864894098165<A u 3 , A u 3 >_{-1}^0.5 =  1.6162235030107774e-08
Newton iteration   4 , energy = 0.49181864894098165<A u 4 , A u 4 >_{-1}^0.5 =  3.813020755027571e-16
t =  0.9999999999999999
Newton iteration   0 , energy = 0.46529832304361385<A u 0 , A u 0 >_{-1}^0.5 =  0.22254165742733853
Newton iteration   1 , energy = 0.441713819748227<A u 1 , A u 1 >_{-1}^0.5 =  0.017254434644772075
Newton iteration   2 , energy = 0.4415638855590664<A u 2 , A u 2 >_{-1}^0.5 =  0.0002033177059838874
Newton iteration   3 , energy = 0.4415638648875152<A u 3 , A u 3 >_{-1}^0.5 =  3.792703612968718e-08
Newton iteration   4 , energy = 0.4415638648875143<A u 4 , A u 4 >_{-1}^0.5 =  1.4696311126787105e-15
t =  1.0999999999999999
Newton iteration   0 , energy = 0.42053725198497327<A u 0 , A u 0 >_{-1}^0.5 =  0.19203379287773834
Newton iteration   1 , energy = 0.40290011095002537<A u 1 , A u 1 >_{-1}^0.5 =  0.013500635959563585
Newton iteration   2 , energy = 0.402808476230338<A u 2 , A u 2 >_{-1}^0.5 =  0.00011906321559774835
Newton iteration   3 , energy = 0.40280846914184915<A u 3 , A u 3 >_{-1}^0.5 =  1.2055948466063786e-08
Newton iteration   4 , energy = 0.4028084691418493<A u 4 , A u 4 >_{-1}^0.5 =  2.8602016512036763e-16
t =  1.2
Newton iteration   0 , energy = 0.3878905829158591<A u 0 , A u 0 >_{-1}^0.5 =  0.15838607130918503
Newton iteration   1 , energy = 0.37581637454258304<A u 1 , A u 1 >_{-1}^0.5 =  0.010038128682433001
Newton iteration   2 , energy = 0.37576571894758526<A u 2 , A u 2 >_{-1}^0.5 =  9.086309166066276e-05
Newton iteration   3 , energy = 0.37576571481921284<A u 3 , A u 3 >_{-1}^0.5 =  1.091465958214028e-08
Newton iteration   4 , energy = 0.3757657148192129<A u 4 , A u 4 >_{-1}^0.5 =  3.0995325086740605e-16
t =  1.3
Newton iteration   0 , energy = 0.36584485571508024<A u 0 , A u 0 >_{-1}^0.5 =  0.12785885968187724
Newton iteration   1 , energy = 0.3579010752659412<A u 1 , A u 1 >_{-1}^0.5 =  0.00649395571299005
Newton iteration   2 , energy = 0.3578799083983545<A u 2 , A u 2 >_{-1}^0.5 =  4.1328065151927325e-05
Newton iteration   3 , energy = 0.35787990754431814<A u 3 , A u 3 >_{-1}^0.5 =  2.3114042845319445e-09
Newton iteration   4 , energy = 0.35787990754431853<A u 4 , A u 4 >_{-1}^0.5 =  2.6550694787597516e-16
t =  1.4000000000000001
Newton iteration   0 , energy = 0.3513724391475177<A u 0 , A u 0 >_{-1}^0.5 =  0.10411973287732425
Newton iteration   1 , energy = 0.34605187071092036<A u 1 , A u 1 >_{-1}^0.5 =  0.0036799679998081843
Newton iteration   2 , energy = 0.3460450873066709<A u 2 , A u 2 >_{-1}^0.5 =  1.1245501639158333e-05
Newton iteration   3 , energy = 0.3460450872434394<A u 3 , A u 3 >_{-1}^0.5 =  1.5959157826850064e-10
Newton iteration   4 , energy = 0.34604508724343935<A u 4 , A u 4 >_{-1}^0.5 =  2.7371527835833993e-16
t =  1.5000000000000002
Newton iteration   0 , energy = 0.34155788317473273<A u 0 , A u 0 >_{-1}^0.5 =  0.08812837881644642
Newton iteration   1 , energy = 0.3377216170972494<A u 1 , A u 1 >_{-1}^0.5 =  0.002247896160198926
Newton iteration   2 , energy = 0.33771908855831195<A u 2 , A u 2 >_{-1}^0.5 =  3.0487967124063097e-06
Newton iteration   3 , energy = 0.3377190885536645<A u 3 , A u 3 >_{-1}^0.5 =  9.322382820504892e-12
Newton iteration   4 , energy = 0.3377190885536643<A u 4 , A u 4 >_{-1}^0.5 =  2.5769569761306345e-16
t =  1.6000000000000003
Newton iteration   0 , energy = 0.33433895200512403<A u 0 , A u 0 >_{-1}^0.5 =  0.0780663778491266
Newton iteration   1 , energy = 0.33131861677086527<A u 1 , A u 1 >_{-1}^0.5 =  0.0016725179237095929
Newton iteration   2 , energy = 0.3313172174355442<A u 2 , A u 2 >_{-1}^0.5 =  1.37504159055911e-06
Newton iteration   3 , energy = 0.33131721743459874<A u 3 , A u 3 >_{-1}^0.5 =  1.6855282733345524e-12
Newton iteration   4 , energy = 0.33131721743459885<A u 4 , A u 4 >_{-1}^0.5 =  2.650720703774998e-16
t =  1.7000000000000004
Newton iteration   0 , energy = 0.3285582359249102<A u 0 , A u 0 >_{-1}^0.5 =  0.07154071008401247
Newton iteration   1 , energy = 0.3260170661772191<A u 1 , A u 1 >_{-1}^0.5 =  0.0014250703219029172
Newton iteration   2 , energy = 0.32601605036749176<A u 2 , A u 2 >_{-1}^0.5 =  9.518034225752939e-07
Newton iteration   3 , energy = 0.3260160503670384<A u 3 , A u 3 >_{-1}^0.5 =  8.3714311978967e-13
Newton iteration   4 , energy = 0.3260160503670385<A u 4 , A u 4 >_{-1}^0.5 =  2.4549035015296827e-16
t =  1.8000000000000005
Newton iteration   0 , energy = 0.3236418456543297<A u 0 , A u 0 >_{-1}^0.5 =  0.0669205426878862
Newton iteration   1 , energy = 0.3214155565031594<A u 1 , A u 1 >_{-1}^0.5 =  0.0012774943359804476
Newton iteration   2 , energy = 0.3214147402288587<A u 2 , A u 2 >_{-1}^0.5 =  7.395919751555945e-07
Newton iteration   3 , energy = 0.321414740228585<A u 3 , A u 3 >_{-1}^0.5 =  4.774291796573343e-13
Newton iteration   4 , energy = 0.32141474022858496<A u 4 , A u 4 >_{-1}^0.5 =  2.571560115085417e-16
t =  1.9000000000000006
Newton iteration   0 , energy = 0.319307200330189<A u 0 , A u 0 >_{-1}^0.5 =  0.06336334534033089
Newton iteration   1 , energy = 0.317309482637797<A u 1 , A u 1 >_{-1}^0.5 =  0.0011636519432441383
Newton iteration   2 , energy = 0.31730880539045125<A u 2 , A u 2 >_{-1}^0.5 =  5.881579096364029e-07
Newton iteration   3 , energy = 0.3173088053902784<A u 3 , A u 3 >_{-1}^0.5 =  2.7551746653838786e-13
Newton iteration   4 , energy = 0.31730880539027817<A u 4 , A u 4 >_{-1}^0.5 =  2.454894382228644e-16
t =  2.0000000000000004
Newton iteration   0 , energy = 0.3154019446744455<A u 0 , A u 0 >_{-1}^0.5 =  0.06046988423639982
Newton iteration   1 , energy = 0.3135813462842381<A u 1 , A u 1 >_{-1}^0.5 =  0.0010696980913600478
Newton iteration   2 , energy = 0.3135807740019491<A u 2 , A u 2 >_{-1}^0.5 =  4.782193869677695e-07
Newton iteration   3 , energy = 0.31358077400183504<A u 3 , A u 3 >_{-1}^0.5 =  1.6249661865693085e-13
Newton iteration   4 , energy = 0.31358077400183504<A u 4 , A u 4 >_{-1}^0.5 =  2.7138089177844877e-16
t =  2.1000000000000005
Newton iteration   0 , energy = 0.31183294221715574<A u 0 , A u 0 >_{-1}^0.5 =  0.058030929719797515
Newton iteration   1 , energy = 0.3101555890654213<A u 1 , A u 1 >_{-1}^0.5 =  0.0009939083502081252
Newton iteration   2 , energy = 0.3101550950154471<A u 2 , A u 2 >_{-1}^0.5 =  4.0262913456839673e-07
Newton iteration   3 , energy = 0.31015509501536603<A u 3 , A u 3 >_{-1}^0.5 =  1.0406296807034973e-13
Newton iteration   4 , energy = 0.3101550950153661<A u 4 , A u 4 >_{-1}^0.5 =  2.6194439518922196e-16
t =  2.2000000000000006
Newton iteration   0 , energy = 0.3085381470982799<A u 0 , A u 0 >_{-1}^0.5 =  0.0559095082983912
Newton iteration   1 , energy = 0.30698089151783714<A u 1 , A u 1 >_{-1}^0.5 =  0.0009350737925028049
Newton iteration   2 , energy = 0.30698045423489756<A u 2 , A u 2 >_{-1}^0.5 =  3.5099162841240115e-07
Newton iteration   3 , energy = 0.30698045423483605<A u 3 , A u 3 >_{-1}^0.5 =  7.518928999533018e-14
t =  2.3000000000000007
Newton iteration   0 , energy = 0.3054751642681964<A u 0 , A u 0 >_{-1}^0.5 =  0.05400150209874556
Newton iteration   1 , energy = 0.30402228122943686<A u 1 , A u 1 >_{-1}^0.5 =  0.0008896044308022321
Newton iteration   2 , energy = 0.3040218854452222<A u 2 , A u 2 >_{-1}^0.5 =  3.138328664212689e-07
Newton iteration   3 , energy = 0.3040218854451729<A u 3 , A u 3 >_{-1}^0.5 =  5.988362983976485e-14
t =  2.400000000000001
Newton iteration   0 , energy = 0.3026153122254497<A u 0 , A u 0 >_{-1}^0.5 =  0.05222818323717892
Newton iteration   1 , energy = 0.30125624096623144<A u 1 , A u 1 >_{-1}^0.5 =  0.0008515491921994277
Newton iteration   2 , energy = 0.3012558783232956<A u 2 , A u 2 >_{-1}^0.5 =  2.8565170095095356e-07
Newton iteration   3 , energy = 0.3012558783232549<A u 3 , A u 3 >_{-1}^0.5 =  5.029095399199219e-14
t =  2.500000000000001
Newton iteration   0 , energy = 0.29993930200181346<A u 0 , A u 0 >_{-1}^0.5 =  0.05053687668854502
Newton iteration   1 , energy = 0.2986667580279501<A u 1 , A u 1 >_{-1}^0.5 =  0.0008146689318661688
Newton iteration   2 , energy = 0.29866642611941413<A u 2 , A u 2 >_{-1}^0.5 =  2.62874132326621e-07
Newton iteration   3 , energy = 0.29866642611937977<A u 3 , A u 3 >_{-1}^0.5 =  4.334231603157561e-14
t =  2.600000000000001
Newton iteration   0 , energy = 0.29743360125599233<A u 0 , A u 0 >_{-1}^0.5 =  0.0488998411208707
Newton iteration   1 , energy = 0.29624203636160634<A u 1 , A u 1 >_{-1}^0.5 =  0.0007749917838309931
Newton iteration   2 , energy = 0.2962417359981101<A u 2 , A u 2 >_{-1}^0.5 =  2.415012946682288e-07
Newton iteration   3 , energy = 0.29624173599808096<A u 3 , A u 3 >_{-1}^0.5 =  3.746480378823313e-14
t =  2.700000000000001
Newton iteration   0 , energy = 0.29508755528322356<A u 0 , A u 0 >_{-1}^0.5 =  0.04730895089365253
Newton iteration   1 , energy = 0.29397208180268336<A u 1 , A u 1 >_{-1}^0.5 =  0.0007318010389492004
Newton iteration   2 , energy = 0.29397181398729666<A u 2 , A u 2 >_{-1}^0.5 =  2.1864078851395085e-07
Newton iteration   3 , energy = 0.2939718139872729<A u 3 , A u 3 >_{-1}^0.5 =  3.1290687652296695e-14
t =  2.800000000000001
Newton iteration   0 , energy = 0.29289147485304795<A u 0 , A u 0 >_{-1}^0.5 =  0.045767942282800914
Newton iteration   1 , energy = 0.2918472820798932<A u 1 , A u 1 >_{-1}^0.5 =  0.0006867421186889949
Newton iteration   2 , energy = 0.29184704623157903<A u 2 , A u 2 >_{-1}^0.5 =  1.942287090991803e-07
Newton iteration   3 , energy = 0.29184704623155994<A u 3 , A u 3 >_{-1}^0.5 =  2.47096197176073e-14
t =  2.9000000000000012
Newton iteration   0 , energy = 0.29083567869225335<A u 0 , A u 0 >_{-1}^0.5 =  0.044285259334220964
Newton iteration   1 , energy = 0.28985783725122105<A u 1 , A u 1 >_{-1}^0.5 =  0.0006422196211125165
Newton iteration   2 , energy = 0.28985763099455336<A u 2 , A u 2 >_{-1}^0.5 =  1.6998740391686497e-07
Newton iteration   3 , energy = 0.289857630994539<A u 3 , A u 3 >_{-1}^0.5 =  1.8595142136157353e-14
t =  3.0000000000000013
Newton iteration   0 , energy = 0.2889102362214215<A u 0 , A u 0 >_{-1}^0.5 =  0.04286943153866994
Newton iteration   1 , energy = 0.28799372375048027<A u 1 , A u 1 >_{-1}^0.5 =  0.0006001764701857079
Newton iteration   2 , energy = 0.2879935436171827<A u 2 , A u 2 >_{-1}^0.5 =  1.4760228587673114e-07
Newton iteration   3 , energy = 0.2879935436171718<A u 3 , A u 3 >_{-1}^0.5 =  1.3698576360179294e-14
t =  3.1000000000000014
Newton iteration   0 , energy = 0.28710507639904415<A u 0 , A u 0 >_{-1}^0.5 =  0.041527106603070976
Newton iteration   1 , energy = 0.2862448814946366<A u 1 , A u 1 >_{-1}^0.5 =  0.000561604270828211
Newton iteration   2 , energy = 0.28624472377270044<A u 2 , A u 2 >_{-1}^0.5 =  1.279317359759126e-07
Newton iteration   3 , energy = 0.28624472377269233<A u 3 , A u 3 >_{-1}^0.5 =  1.0164918291750335e-14
t =  3.2000000000000015
Newton iteration   0 , energy = 0.28541020400251493<A u 0 , A u 0 >_{-1}^0.5 =  0.04026276584023924
Newton iteration   1 , energy = 0.28460142852637166<A u 1 , A u 1 >_{-1}^0.5 =  0.0005266329945374004
Newton iteration   2 , energy = 0.2846012898370938<A u 2 , A u 2 >_{-1}^0.5 =  1.1111894431712491e-07
Newton iteration   3 , energy = 0.28460128983708777<A u 3 , A u 3 >_{-1}^0.5 =  7.684063018423214e-15
t =  3.3000000000000016
Newton iteration   0 , energy = 0.28381589335594626<A u 0 , A u 0 >_{-1}^0.5 =  0.03907903897497687
Newton iteration   1 , energy = 0.2830538286715773<A u 1 , A u 1 >_{-1}^0.5 =  0.0004949004002671524
Newton iteration   2 , energy = 0.2830537061935375<A u 2 , A u 2 >_{-1}^0.5 =  9.693706411425455e-08
Newton iteration   3 , energy = 0.28305370619353276<A u 3 , A u 3 >_{-1}^0.5 =  5.8939831335759276e-15
t =  3.4000000000000017
Newton iteration   0 , energy = 0.282312828894207<A u 0 , A u 0 >_{-1}^0.5 =  0.03797696388689092
Newton iteration   1 , energy = 0.2815930104000623<A u 1 , A u 1 >_{-1}^0.5 =  0.00046593220451703656
Newton iteration   2 , energy = 0.2815929018414163<A u 2 , A u 2 >_{-1}^0.5 =  8.50303809171891e-08
Newton iteration   3 , energy = 0.2815929018414129<A u 3 , A u 3 >_{-1}^0.5 =  4.572447136960331e-15
t =  3.5000000000000018
Newton iteration   0 , energy = 0.28089220711467927<A u 0 , A u 0 >_{-1}^0.5 =  0.03695602264201758
Newton iteration   1 , energy = 0.280210457290666<A u 1 , A u 1 >_{-1}^0.5 =  0.00043937399829195974
Newton iteration   2 , energy = 0.28021036075573674<A u 2 , A u 2 >_{-1}^0.5 =  7.502751144713063e-08
Newton iteration   3 , energy = 0.280210360755734<A u 3 , A u 3 >_{-1}^0.5 =  3.58716939972748e-15
t =  3.600000000000002
Newton iteration   0 , energy = 0.2795458189068637<A u 0 , A u 0 >_{-1}^0.5 =  0.03601407754910985
Newton iteration   1 , energy = 0.27889828380875215<A u 1 , A u 1 >_{-1}^0.5 =  0.00041504802390297487
Newton iteration   2 , energy = 0.27889819766779134<A u 2 , A u 2 >_{-1}^0.5 =  6.659068183711379e-08
Newton iteration   3 , energy = 0.27889819766778906<A u 3 , A u 3 >_{-1}^0.5 =  2.841909661310525e-15
t =  3.700000000000002
Newton iteration   0 , energy = 0.2782661188952641<A u 0 , A u 0 >_{-1}^0.5 =  0.035147391376697125
Newton iteration   1 , energy = 0.2776492967301608<A u 1 , A u 1 >_{-1}^0.5 =  0.0003928971030734926
Newton iteration   2 , energy = 0.27764921953888283<A u 2 , A u 2 >_{-1}^0.5 =  5.9437738601283895e-08
Newton iteration   3 , energy = 0.277649219538881<A u 3 , A u 3 >_{-1}^0.5 =  2.2665850853678274e-15
t =  3.800000000000002
Newton iteration   0 , energy = 0.27704627721572056<A u 0 , A u 0 >_{-1}^0.5 =  0.034350828975997935
Newton iteration   1 , energy = 0.276457035117315<A u 1 , A u 1 >_{-1}^0.5 =  0.00037289594840102773
Newton iteration   2 , energy = 0.2764569655854736<A u 2 , A u 2 >_{-1}^0.5 =  5.3346152356448625e-08
Newton iteration   3 , energy = 0.27645696558547234<A u 3 , A u 3 >_{-1}^0.5 =  1.7942224496011876e-15
t =  3.900000000000002
Newton iteration   0 , energy = 0.2758802061476704<A u 0 , A u 0 >_{-1}^0.5 =  0.0336182194953775
Newton iteration   1 , energy = 0.275315782621881<A u 1 , A u 1 >_{-1}^0.5 =  0.00035498199342934766
Newton iteration   2 , energy = 0.27531571961049045<A u 2 , A u 2 >_{-1}^0.5 =  4.814509709626193e-08
Newton iteration   3 , energy = 0.27531571961048923<A u 3 , A u 3 >_{-1}^0.5 =  1.4517456559532703e-15
t =  4.000000000000002
Newton iteration   0 , energy = 0.27476255790752957<A u 0 , A u 0 >_{-1}^0.5 =  0.03294278655417879
Newton iteration   1 , energy = 0.27422055148050817<A u 1 , A u 1 >_{-1}^0.5 =  0.00033902324952341515
Newton iteration   2 , energy = 0.27422049400754955<A u 2 , A u 2 >_{-1}^0.5 =  4.370317901966756e-08
Newton iteration   3 , energy = 0.2742204940075488<A u 3 , A u 3 >_{-1}^0.5 =  1.2029743955843305e-15
t =  4.100000000000001
Newton iteration   0 , energy = 0.273688696101748<A u 0 , A u 0 >_{-1}^0.5 =  0.03231754514948758
Newton iteration   1 , energy = 0.2731670433823861<A u 1 , A u 1 >_{-1}^0.5 =  0.0003248196932807901
Newton iteration   2 , energy = 0.2731669906244711<A u 2 , A u 2 >_{-1}^0.5 =  3.9916977538799526e-08
Newton iteration   3 , energy = 0.27316699062447036<A u 3 , A u 3 >_{-1}^0.5 =  9.888253872196682e-16
t =  4.200000000000001
Newton iteration   0 , energy = 0.27265464810833245<A u 0 , A u 0 >_{-1}^0.5 =  0.031735596888519346
Newton iteration   1 , energy = 0.2721515957630506<A u 1 , A u 1 >_{-1}^0.5 =  0.0003121269731680887
Newton iteration   2 , energy = 0.27215154704790234<A u 2 , A u 2 >_{-1}^0.5 =  3.6701916338598945e-08
Newton iteration   3 , energy = 0.2721515470479015<A u 3 , A u 3 >_{-1}^0.5 =  8.433717951412317e-16
t =  4.300000000000001
Newton iteration   0 , energy = 0.2716570474966338<A u 0 , A u 0 >_{-1}^0.5 =  0.031190300855575344
Newton iteration   1 , energy = 0.27117112247149877<A u 1 , A u 1 >_{-1}^0.5 =  0.0003006897733736882
Newton iteration   2 , energy = 0.27117107726119166<A u 2 , A u 2 >_{-1}^0.5 =  3.3984647977790945e-08
Newton iteration   3 , energy = 0.27117107726119133<A u 3 , A u 3 >_{-1}^0.5 =  7.582284547611633e-16
t =  4.4
Newton iteration   0 , energy = 0.2706930746904997<A u 0 , A u 0 >_{-1}^0.5 =  0.03067533807183733
Newton iteration   1 , energy = 0.2702230558015484<A u 1 , A u 1 >_{-1}^0.5 =  0.0002902727509576759
Newton iteration   2 , energy = 0.27022301366959317<A u 2 , A u 2 >_{-1}^0.5 =  3.169618372578609e-08
Newton iteration   3 , energy = 0.2702230136695925<A u 3 , A u 3 >_{-1}^0.5 =  6.378743535051877e-16
t =  4.5
Newton iteration   0 , energy = 0.2697604013033479<A u 0 , A u 0 >_{-1}^0.5 =  0.03018471586581177
Newton iteration   1 , energy = 0.26930529353765226<A u 1 , A u 1 >_{-1}^0.5 =  0.0002806792254648955
Newton iteration   2 , energy = 0.26930525414467854<A u 2 , A u 2 >_{-1}^0.5 =  2.9766651102240373e-08
Newton iteration   3 , energy = 0.2693052541446782<A u 3 , A u 3 >_{-1}^0.5 =  5.95494848352045e-16
t =  4.6
Newton iteration   0 , energy = 0.2688571399005529<A u 0 , A u 0 >_{-1}^0.5 =  0.029712771605002766
Newton iteration   1 , energy = 0.2684161509060477<A u 1 , A u 1 >_{-1}^0.5 =  0.0002717525330428682
Newton iteration   2 , energy = 0.2684161139789862<A u 2 , A u 2 >_{-1}^0.5 =  2.812317708103957e-08
Newton iteration   3 , energy = 0.26841611397898574<A u 3 , A u 3 >_{-1}^0.5 =  5.419853565871418e-16
t =  4.699999999999999
Newton iteration   0 , energy = 0.2679817973356853<A u 0 , A u 0 >_{-1}^0.5 =  0.029254230523451902
Newton iteration   1 , energy = 0.2675543141138422<A u 1 , A u 1 >_{-1}^0.5 =  0.0002633618275664335
Newton iteration   2 , energy = 0.26755427943196064<A u 2 , A u 2 >_{-1}^0.5 =  2.669090173722351e-08
Newton iteration   3 , energy = 0.2675542794319601<A u 3 , A u 3 >_{-1}^0.5 =  5.28868572186883e-16
t =  4.799999999999999
Newton iteration   0 , energy = 0.26713322723971317<A u 0 , A u 0 >_{-1}^0.5 =  0.028804348630074375
Newton iteration   1 , energy = 0.2667187904558303<A u 1 , A u 1 >_{-1}^0.5 =  0.0002553810925794131
Newton iteration   2 , energy = 0.2667187578440901<A u 2 , A u 2 >_{-1}^0.5 =  2.5395720057489295e-08
Newton iteration   3 , energy = 0.2667187578440898<A u 3 , A u 3 >_{-1}^0.5 =  5.0782478585141e-16
t =  4.899999999999999
Newton iteration   0 , energy = 0.266310576590937<A u 0 , A u 0 >_{-1}^0.5 =  0.02835913299393717
Newton iteration   1 , energy = 0.2659088504789751<A u 1 , A u 1 >_{-1}^0.5 =  0.00024767399372416736
Newton iteration   2 , energy = 0.26590881980593656<A u 2 , A u 2 >_{-1}^0.5 =  2.4167900537430136e-08
Newton iteration   3 , energy = 0.265908819805936<A u 3 , A u 3 >_{-1}^0.5 =  4.372225479169036e-16
t =  4.999999999999998

[18]:

Draw(gfu_t, mesh, interpolate_multidim=True, animate=True,
     min=-0.1, max=0.1, autoscale=False, deformation=True)

[18]:

BaseWebGuiScene

Supplementary 2: Minimal energy extension (postscript in unit-2.1.3 )¶

In unit-2.1.3 we discussed the BDDC preconditioner and characterized the coarse grid solution as the condensed problem with the continuity only w.r.t. the coarse grid dofs.

We can characterize this as a minimization problem involving

u \in V^{h o, d i s c}, u^{l o, c o n t} \in V^{l o, c o n t}, λ \in V^{l o, d i s c},

Here $u$ is the solution to the coarse-grid (decoupled) problem where $u^{l o, c o n t}$ represents the coarse (continuity-)dofs and $λ$ is the corresponding Lagrange multiplier to enforce continuity of $u$ w.r.t. the lo,cont-dofs.

[19]:

mesh = Mesh(geom.GenerateMesh(maxh=0.1))
fes_ho = Discontinuous(H1(mesh, order=10))
fes_lo = H1(mesh, order=1, dirichlet=".*")
fes_lam = Discontinuous(H1(mesh, order=1))
fes = fes_ho*fes_lo*fes_lam
uho, ulo, lam = fes.TrialFunction()

The energy that is to be minimized is:

\int_{Ω} \frac{1}{2} ‖ \nabla u ‖^{2} - u + \sum_{T} \sum_{V \in V (T)} ((u - u^{l o}) \cdot λ) |_{V} ⟶ \min!

[20]:

a = BilinearForm(fes)
a += Variation(0.5 * grad(uho)*grad(uho)*dx
               - 1*uho*dx
               + (uho-ulo)*lam*dx(element_vb=BBND))
gfu = GridFunction(fes)
NewtonMinimization(a=a, u=gfu)
Draw(gfu.components[0],mesh,deformation=True)

Newton iteration  0
Energy:  0.0
err =  0.3930405243750326
Newton iteration  1
Energy:  -0.07724042690050034
err =  5.5293793565357545e-15

[20]:

BaseWebGuiScene

The minimization problem is solved by the solution of the PDE:

\int_{Ω} \nabla u \cdot \nabla v = \int_{Ω} 1 \cdot v \forall v \in V^{h o, d i s c}

under the constraint

u (v) = u^{l o} (v) for all vertices v \in V (T) for all T .