This page was generated from unit-3.8-nonlmin/nonlmin.ipynb.

3.8 Nonlinear minimization problems¶

We consider problems of the form

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

[1]:

from ngsolve import *
from ngsolve.webgui import Draw

Scalar minimization problems¶

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

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

The minimization is equivalent to solving the nonlinear PDE:

- Δ u + 4 u^{3} = f in Ω

We solve the PDE with a Newton iteration.

[2]:

from netgen.geom2d import unit_square

mesh = Mesh (unit_square.GenerateMesh(maxh=0.2))
V = H1(mesh, order=4, dirichlet=[1,2,3,4])
u = V.TrialFunction()

To solve the problem we use the Variation integrator. Based on the symbolic description of the energy functional, it is able to

evaluate the energy functional (Energy)

E (u) (E : V \to R)

compute the Gateau derivative for a given $u$ (Apply):

A (u) (v) = E^{'} (u) (v) (A (u) : V \to R)

compute the second derivative (AssembleLinearization)

(δ A) (w) (u, v) (δ A (w) : V \times V \to R)

[3]:

a = BilinearForm (V, symmetric=True)
a += Variation ( (grad(u)*grad(u) + u**4-u) * dx)

Equivalent to:

a += (2 * grad(u) * grad(v) + 4*u*u*u*v - 1 * v)*dx

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

We recall the Newton iteration (cf. unit-3.7 ) we make the 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 $⟨ A u^{i}, Δ u^{i} ⟩ = ⟨ A u^{i}, A u^{i} ⟩_{(B^{i})^{- 1}} < ε$ .

[4]:

def SolveNonlinearMinProblem(a,gfu,tol=1e-13,maxits=25):
    res = gfu.vec.CreateVector()
    du  = gfu.vec.CreateVector()

    for it in range(maxits):
        print ("Newton iteration {:3}".format(it),end="")
        print ("energy = {:16}".format(a.Energy(gfu.vec)),end="")

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

        #update iteration
        gfu.vec.data -= du

        #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)

[5]:

gfu = GridFunction (V)
gfu.vec[:] = 0
sceneu = Draw(gfu,mesh,"u")

SolveNonlinearMinProblem(a,gfu)

print ("energy = ", a.Energy(gfu.vec))

Newton iteration   0energy =              0.0<A u 0 , A u 0 >_{-1}^0.5 =  0.13255949695477584
Newton iteration   1energy = -0.008785666770072002<A u 1 , A u 1 >_{-1}^0.5 =  1.110760041466411e-05
Newton iteration   2energy = -0.008785666831761395<A u 2 , A u 2 >_{-1}^0.5 =  2.8074721184641564e-13
Newton iteration   3energy = -0.008785666831761397<A u 3 , A u 3 >_{-1}^0.5 =  3.3534118580780286e-17
energy =  -0.008785666831761397

Again, a Newton for minimization is shipped with NGSolve:

[6]:

from ngsolve.solvers import *
gfu.vec[:] = 0
NewtonMinimization(a,gfu)
sceneu.Redraw()

Newton iteration  0
Energy:  0.0
err =  0.1325594969547758
Newton iteration  1
Energy:  -0.008785666770072002
err =  1.1107600414641456e-05
Newton iteration  2
Energy:  -0.008785666831761395
err =  2.807487362219202e-13

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.

[7]:

import netgen.geom2d as geom2d
from ngsolve import *
from ngsolve.webgui import Draw

geo = geom2d.SplineGeometry()
pnums = [ geo.AddPoint (x,y,maxh=0.01) for x,y in [(0,0), (1,0), (1,0.1), (0,0.1)] ]
for p1,p2,bc in [(0,1,"bot"), (1,2,"right"), (2,3,"top"), (3,0,"left")]:
     geo.Append(["line", pnums[p1], pnums[p2]], bc=bc)
mesh = Mesh(geo.GenerateMesh(maxh=0.05))

[8]:

# 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) )

[9]:

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 with 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.

[10]:

gfu = GridFunction(V)
gfu.vec[:] = 0

sceneu = Draw (gfu, mesh, "u")
SetVisualization (deformation=True)

res = gfu.vec.CreateVector()
du = gfu.vec.CreateVector()

for loadstep in range(50):
    print ("loadstep", loadstep)
    factor.Set ((loadstep+1)/10)
    SolveNonlinearMinProblem(a,gfu)
    sceneu.Redraw()

loadstep 0
Newton iteration   0energy = 8.750000000000005<A u 0 , A u 0 >_{-1}^0.5 =  0.016678382390755288
Newton iteration   1energy = 8.750132585936578<A u 1 , A u 1 >_{-1}^0.5 =  0.023333256891798576
Newton iteration   2energy = 8.74986115049552<A u 2 , A u 2 >_{-1}^0.5 =  0.00010232159281117413
Newton iteration   3energy = 8.749861145260668<A u 3 , A u 3 >_{-1}^0.5 =  5.078987230934345e-08
Newton iteration   4energy = 8.749861145260665<A u 4 , A u 4 >_{-1}^0.5 =  2.661579380060487e-13
Newton iteration   5energy = 8.749861145260663<A u 5 , A u 5 >_{-1}^0.5 =  8.225695167307351e-16
loadstep 1
Newton iteration   0energy = 8.749583892957778<A u 0 , A u 0 >_{-1}^0.5 =  0.016596117905447468
Newton iteration   1energy = 8.749710099853527<A u 1 , A u 1 >_{-1}^0.5 =  0.022958465312257894
Newton iteration   2energy = 8.74944730043396<A u 2 , A u 2 >_{-1}^0.5 =  0.00014213684552075314
Newton iteration   3energy = 8.749447290438315<A u 3 , A u 3 >_{-1}^0.5 =  2.3248036083484098e-06
Newton iteration   4energy = 8.749447290435603<A u 4 , A u 4 >_{-1}^0.5 =  6.324688781684219e-10
Newton iteration   5energy = 8.74944729043561<A u 5 , A u 5 >_{-1}^0.5 =  1.672080396498094e-15
loadstep 2
Newton iteration   0energy = 8.748898149085237<A u 0 , A u 0 >_{-1}^0.5 =  0.01635697493005621
Newton iteration   1energy = 8.749005270899884<A u 1 , A u 1 >_{-1}^0.5 =  0.02189303042236316
Newton iteration   2energy = 8.74876626125895<A u 2 , A u 2 >_{-1}^0.5 =  0.00020820420956604055
Newton iteration   3energy = 8.748766239907326<A u 3 , A u 3 >_{-1}^0.5 =  4.8901755827964345e-06
Newton iteration   4energy = 8.748766239895367<A u 4 , A u 4 >_{-1}^0.5 =  2.664529908879819e-09
Newton iteration   5energy = 8.74876623989537<A u 5 , A u 5 >_{-1}^0.5 =  2.1545219373426666e-15
loadstep 3
Newton iteration   0energy = 8.747955304179662<A u 0 , A u 0 >_{-1}^0.5 =  0.015982097567591464
Newton iteration   1energy = 8.748035438675357<A u 1 , A u 1 >_{-1}^0.5 =  0.020292659620855413
Newton iteration   2energy = 8.747830048049662<A u 2 , A u 2 >_{-1}^0.5 =  0.0002637936026354171
Newton iteration   3energy = 8.747830013747695<A u 3 , A u 3 >_{-1}^0.5 =  7.246846163852253e-06
Newton iteration   4energy = 8.74783001372143<A u 4 , A u 4 >_{-1}^0.5 =  4.743748703193207e-09
Newton iteration   5energy = 8.747830013721432<A u 5 , A u 5 >_{-1}^0.5 =  3.616311928992526e-15
loadstep 4
Newton iteration   0energy = 8.746771024519141<A u 0 , A u 0 >_{-1}^0.5 =  0.015501109791087814
Newton iteration   1energy = 8.746821875338771<A u 1 , A u 1 >_{-1}^0.5 =  0.01836023081624558
Newton iteration   2energy = 8.746653692485939<A u 2 , A u 2 >_{-1}^0.5 =  0.0002990597350145664
Newton iteration   3energy = 8.746653648294211<A u 3 , A u 3 >_{-1}^0.5 =  8.720266772328203e-06
Newton iteration   4energy = 8.746653648256187<A u 4 , A u 4 >_{-1}^0.5 =  5.24315662816797e-09
Newton iteration   5energy = 8.74665364825618<A u 5 , A u 5 >_{-1}^0.5 =  4.328294634052801e-15
loadstep 5
Newton iteration   0energy = 8.745362725155786<A u 0 , A u 0 >_{-1}^0.5 =  0.014946280903332443
Newton iteration   1energy = 8.745386481617505<A u 1 , A u 1 >_{-1}^0.5 =  0.016293310584009953
Newton iteration   2energy = 8.745253992323276<A u 2 , A u 2 >_{-1}^0.5 =  0.000313206365846639
Newton iteration   3energy = 8.745253943722172<A u 3 , A u 3 >_{-1}^0.5 =  9.04313359446376e-06
Newton iteration   4energy = 8.745253943681277<A u 4 , A u 4 >_{-1}^0.5 =  4.217411575929172e-09
Newton iteration   5energy = 8.745253943681275<A u 5 , A u 5 >_{-1}^0.5 =  4.559164415717114e-15
loadstep 6
Newton iteration   0energy = 8.74374838218437<A u 0 , A u 0 >_{-1}^0.5 =  0.014347757828630747
Newton iteration   1energy = 8.743749777088052<A u 1 , A u 1 >_{-1}^0.5 =  0.014249383992409447
Newton iteration   2energy = 8.743648409960423<A u 2 , A u 2 >_{-1}^0.5 =  0.0003093522452984288
Newton iteration   3energy = 8.74364836243308<A u 3 , A u 3 >_{-1}^0.5 =  8.385269217571552e-06
Newton iteration   4energy = 8.743648362397929<A u 4 , A u 4 >_{-1}^0.5 =  2.6943222965375834e-09
Newton iteration   5energy = 8.743648362397927<A u 5 , A u 5 >_{-1}^0.5 =  4.265851838312984e-15
loadstep 7
Newton iteration   0energy = 8.741945650331859<A u 0 , A u 0 >_{-1}^0.5 =  0.013730754995759296
Newton iteration   1energy = 8.74193019383729<A u 1 , A u 1 >_{-1}^0.5 =  0.012333805584321456
Newton iteration   2energy = 8.741854223708538<A u 2 , A u 2 >_{-1}^0.5 =  0.0002921400113760417
Newton iteration   3energy = 8.741854181238816<A u 3 , A u 3 >_{-1}^0.5 =  7.1304843842849655e-06
Newton iteration   4energy = 8.741854181213386<A u 4 , A u 4 >_{-1}^0.5 =  1.4436768240037753e-09
Newton iteration   5energy = 8.741854181213387<A u 5 , A u 5 >_{-1}^0.5 =  4.262573923423666e-15
loadstep 8
Newton iteration   0energy = 8.739971288694946<A u 0 , A u 0 >_{-1}^0.5 =  0.013114590303641569
Newton iteration   1energy = 8.739944123846238<A u 1 , A u 1 >_{-1}^0.5 =  0.010604193396712067
Newton iteration   2energy = 8.739887948591901<A u 2 , A u 2 >_{-1}^0.5 =  0.0002663295273824012
Newton iteration   3energy = 8.739887913241445<A u 3 , A u 3 >_{-1}^0.5 =  5.663180365496771e-06
Newton iteration   4energy = 8.73988791322541<A u 4 , A u 4 >_{-1}^0.5 =  6.721145933415297e-10
Newton iteration   5energy = 8.739887913225408<A u 5 , A u 5 >_{-1}^0.5 =  4.80875665732802e-15
loadstep 9
Newton iteration   0energy = 8.737840839887603<A u 0 , A u 0 >_{-1}^0.5 =  0.012512967634820831
Newton iteration   1energy = 8.737806211240576<A u 1 , A u 1 >_{-1}^0.5 =  0.009082216012610009
Newton iteration   2energy = 8.737764991099096<A u 2 , A u 2 >_{-1}^0.5 =  0.00023605086028194344
Newton iteration   3energy = 8.737764963298767<A u 3 , A u 3 >_{-1}^0.5 =  4.258856622325151e-06
Newton iteration   4energy = 8.737764963289697<A u 4 , A u 4 >_{-1}^0.5 =  2.779241978161679e-10
Newton iteration   5energy = 8.737764963289692<A u 5 , A u 5 >_{-1}^0.5 =  6.1418577774628076e-15
loadstep 10
Newton iteration   0energy = 8.735568492367701<A u 0 , A u 0 >_{-1}^0.5 =  0.011934876423785106
Newton iteration   1energy = 8.73552963093563<A u 1 , A u 1 >_{-1}^0.5 =  0.007766170839182753
Newton iteration   2energy = 8.73549948223209<A u 2 , A u 2 >_{-1}^0.5 =  0.00020450737374954478
Newton iteration   3energy = 8.735499461348876<A u 3 , A u 3 >_{-1}^0.5 =  3.0651565873007505e-06
Newton iteration   4energy = 8.73549946134418<A u 4 , A u 4 >_{-1}^0.5 =  1.0318410567450001e-10
Newton iteration   5energy = 8.735499461344189<A u 5 , A u 5 >_{-1}^0.5 =  6.1574063877941515e-15
loadstep 11
Newton iteration   0energy = 8.733167062934966<A u 0 , A u 0 >_{-1}^0.5 =  0.011385659634348408
Newton iteration   1energy = 8.73312628674236<A u 1 , A u 1 >_{-1}^0.5 =  0.006641200507693204
Newton iteration   2energy = 8.733104233737167<A u 2 , A u 2 >_{-1}^0.5 =  0.0001739493593252799
Newton iteration   3energy = 8.733104218620428<A u 3 , A u 3 >_{-1}^0.5 =  2.129535379620661e-06
Newton iteration   4energy = 8.733104218618156<A u 4 , A u 4 >_{-1}^0.5 =  3.432853656464424e-11
Newton iteration   5energy = 8.733104218618156<A u 5 , A u 5 >_{-1}^0.5 =  6.179372477428042e-15
loadstep 12
Newton iteration   0energy = 8.730648051299827<A u 0 , A u 0 >_{-1}^0.5 =  0.010868004310983588
Newton iteration   1energy = 8.730606944098007<A u 1 , A u 1 >_{-1}^0.5 =  0.005686407124508259
Newton iteration   2energy = 8.730590772260673<A u 2 , A u 2 >_{-1}^0.5 =  0.00014578797430277076
Newton iteration   3energy = 8.73059076163854<A u 3 , A u 3 >_{-1}^0.5 =  1.43850761805062e-06
Newton iteration   4energy = 8.730590761637513<A u 4 , A u 4 >_{-1}^0.5 =  9.988067426584181e-12
Newton iteration   5energy = 8.730590761637513<A u 5 , A u 5 >_{-1}^0.5 =  6.8769167105590945e-15
loadstep 13
Newton iteration   0energy = 8.728021733357686<A u 0 , A u 0 >_{-1}^0.5 =  0.010382755786018829
Newton iteration   1energy = 8.72798132754746<A u 1 , A u 1 >_{-1}^0.5 =  0.004879274848075253
Newton iteration   2energy = 8.727969418133108<A u 2 , A u 2 >_{-1}^0.5 =  0.00012076675563494456
Newton iteration   3energy =  8.7279694108425<A u 3 , A u 3 >_{-1}^0.5 =  9.505936905068618e-07
Newton iteration   4energy = 8.727969410842045<A u 4 , A u 4 >_{-1}^0.5 =  2.3264985884817412e-12
Newton iteration   5energy = 8.727969410842048<A u 5 , A u 5 >_{-1}^0.5 =  6.8896690523432285e-15
loadstep 14
Newton iteration   0energy = 8.725297271534666<A u 0 , A u 0 >_{-1}^0.5 =  0.009929540146497379
Newton iteration   1energy = 8.725258204221735<A u 1 , A u 1 >_{-1}^0.5 =  0.004198138558029775
Newton iteration   2energy = 8.72524938609619<A u 2 , A u 2 >_{-1}^0.5 =  9.914194785388828e-05
Newton iteration   3energy = 8.725249381182058<A u 3 , A u 3 >_{-1}^0.5 =  6.177654275476132e-07
Newton iteration   4energy = 8.72524938118187<A u 4 , A u 4 >_{-1}^0.5 =  2.708226033824133e-13
Newton iteration   5energy = 8.725249381181872<A u 5 , A u 5 >_{-1}^0.5 =  8.013383813739961e-15
loadstep 15
Newton iteration   0energy = 8.72248282896684<A u 0 , A u 0 >_{-1}^0.5 =  0.009507217507470465
Newton iteration   1energy = 8.72244546337916<A u 1 , A u 1 >_{-1}^0.5 =  0.0036233812630321847
Newton iteration   2energy = 8.722438893461836<A u 2 , A u 2 >_{-1}^0.5 =  8.084475896702669e-05
Newton iteration   3energy = 8.722438890193898<A u 3 , A u 3 >_{-1}^0.5 =  3.9661685500887504e-07
Newton iteration   4energy = 8.722438890193818<A u 4 , A u 4 >_{-1}^0.5 =  1.5319090856131928e-13
Newton iteration   5energy = 8.722438890193823<A u 5 , A u 5 >_{-1}^0.5 =  7.435188870624238e-15
loadstep 16
Newton iteration   0energy = 8.719585679907679<A u 0 , A u 0 >_{-1}^0.5 =  0.009114200423702716
Newton iteration   1energy = 8.719550194836472<A u 1 , A u 1 >_{-1}^0.5 =  0.0031378710664224574
Newton iteration   2energy = 8.719545266984772<A u 2 , A u 2 >_{-1}^0.5 =  6.561290828819087e-05
Newton iteration   3energy = 8.719545264832139<A u 3 , A u 3 >_{-1}^0.5 =  2.525372865815696e-07
Newton iteration   4energy = 8.719545264832098<A u 4 , A u 4 >_{-1}^0.5 =  1.5057042073179863e-13
Newton iteration   5energy = 8.719545264832098<A u 5 , A u 5 >_{-1}^0.5 =  8.13439433458963e-15
loadstep 17
Newton iteration   0energy = 8.716612312372297<A u 0 , A u 0 >_{-1}^0.5 =  0.008748671018667371
Newton iteration   1energy = 8.716578765757584<A u 1 , A u 1 >_{-1}^0.5 =  0.002726978936378575
Newton iteration   2energy = 8.716575043606786<A u 2 , A u 2 >_{-1}^0.5 =  5.3088307717759304e-05
Newton iteration   3energy =  8.7165750421975<A u 3 , A u 3 >_{-1}^0.5 =  1.5999780533599802e-07
Newton iteration   4energy = 8.71657504219748<A u 4 , A u 4 >_{-1}^0.5 =  9.240837105873956e-14
loadstep 18
Newton iteration   0energy = 8.713568521256038<A u 0 , A u 0 >_{-1}^0.5 =  0.008408724932748055
Newton iteration   1energy = 8.713536894279306<A u 1 , A u 1 >_{-1}^0.5 =  0.0023783885433465253
Newton iteration   2energy = 8.713534062707833<A u 2 , A u 2 >_{-1}^0.5 =  4.288363853753871e-05
Newton iteration   3energy = 8.71353406178825<A u 3 , A u 3 >_{-1}^0.5 =  1.0114091346355786e-07
Newton iteration   4energy = 8.713534061788245<A u 4 , A u 4 >_{-1}^0.5 =  4.865671048764449e-14
loadstep 19
Newton iteration   0energy = 8.710459491487788<A u 0 , A u 0 >_{-1}^0.5 =  0.008092463808702106
Newton iteration   1energy = 8.710429718569646<A u 1 , A u 1 >_{-1}^0.5 =  0.002081821059098702
Newton iteration   2energy = 8.710427549016906<A u 2 , A u 2 >_{-1}^0.5 =  3.4623585677951934e-05
Newton iteration   3energy = 8.710427548417456<A u 3 , A u 3 >_{-1}^0.5 =  6.393535769673621e-08
Newton iteration   4energy = 8.710427548417453<A u 4 , A u 4 >_{-1}^0.5 =  2.5621800357692153e-14
loadstep 20
Newton iteration   0energy = 8.70728987152168<A u 0 , A u 0 >_{-1}^0.5 =  0.007798052333314451
Newton iteration   1energy = 8.707261860397317<A u 1 , A u 1 >_{-1}^0.5 =  0.0018287420130939324
Newton iteration   2energy = 8.707260186220783<A u 2 , A u 2 >_{-1}^0.5 =  2.7967298757767892e-05
Newton iteration   3energy = 8.707260185829664<A u 3 , A u 3 >_{-1}^0.5 =  4.048982222756301e-08
Newton iteration   4energy = 8.707260185829663<A u 4 , A u 4 >_{-1}^0.5 =  1.4885675497864682e-14
loadstep 21
Newton iteration   0energy = 8.70406383785694<A u 0 , A u 0 >_{-1}^0.5 =  0.0075237512984265825
Newton iteration   1energy = 8.704037482792224<A u 1 , A u 1 >_{-1}^0.5 =  0.0016120839035681908
Newton iteration   2energy = 8.704036181791693<A u 2 , A u 2 >_{-1}^0.5 =  2.261808346136178e-05
Newton iteration   3energy = 8.704036181535885<A u 3 , A u 3 >_{-1}^0.5 =  2.572573254033779e-08
Newton iteration   4energy = 8.704036181535882<A u 4 , A u 4 >_{-1}^0.5 =  1.0787052595370211e-14
loadstep 22
Newton iteration   0energy = 8.70078515144342<A u 0 , A u 0 >_{-1}^0.5 =  0.007267934720804574
Newton iteration   1energy = 8.700760341771408<A u 1 , A u 1 >_{-1}^0.5 =  0.001425999198346473
Newton iteration   2energy = 8.700759323787844<A u 2 , A u 2 >_{-1}^0.5 =  1.8325128466013545e-05
Newton iteration   3energy = 8.700759323619925<A u 3 , A u 3 >_{-1}^0.5 =  1.6417045175198894e-08
Newton iteration   4energy = 8.70075932361992<A u 4 , A u 4 >_{-1}^0.5 =  8.738776156859684e-15
loadstep 23
Newton iteration   0energy = 8.697457206873455<A u 0 , A u 0 >_{-1}^0.5 =  0.007029096572568504
Newton iteration   1energy = 8.697433832371452<A u 1 , A u 1 >_{-1}^0.5 =  0.0012656480113940809
Newton iteration   2energy = 8.697433030465469<A u 2 , A u 2 >_{-1}^0.5 =  1.4880754836034847e-05
Newton iteration   3energy = 8.697433030354746<A u 3 , A u 3 >_{-1}^0.5 =  1.0531743837183756e-08
Newton iteration   4energy = 8.697433030354746<A u 4 , A u 4 >_{-1}^0.5 =  7.97894508341449e-15
loadstep 24
Newton iteration   0energy = 8.694083075231845<A u 0 , A u 0 >_{-1}^0.5 =  0.006805850912318838
Newton iteration   1energy = 8.694061029385738<A u 1 , A u 1 >_{-1}^0.5 =  0.0011270194997571796
Newton iteration   2energy = 8.694060393539427<A u 2 , A u 2 >_{-1}^0.5 =  1.2115520218371166e-05
Newton iteration   3energy = 8.694060393466039<A u 3 , A u 3 >_{-1}^0.5 =  6.796116884659747e-09
Newton iteration   4energy = 8.694060393466035<A u 4 , A u 4 >_{-1}^0.5 =  8.85894868016414e-15
loadstep 25
Newton iteration   0energy = 8.690665541411896<A u 0 , A u 0 >_{-1}^0.5 =  0.006596927974879257
Newton iteration   1energy = 8.69064472328174<A u 1 , A u 1 >_{-1}^0.5 =  0.0010067836587969085
Newton iteration   2energy = 8.690644215881647<A u 2 , A u 2 >_{-1}^0.5 =  9.892619993518303e-06
Newton iteration   3energy = 8.690644215832712<A u 3 , A u 3 >_{-1}^0.5 =  4.413429832686527e-09
Newton iteration   4energy = 8.690644215832712<A u 4 , A u 4 >_{-1}^0.5 =  9.352395752428968e-15
loadstep 26
Newton iteration   0energy = 8.687207136625545<A u 0 , A u 0 >_{-1}^0.5 =  0.006401167928265712
Newton iteration   1energy = 8.687187451796238<A u 1 , A u 1 >_{-1}^0.5 =  0.0009021693868869216
Newton iteration   2energy = 8.687187044377275<A u 2 , A u 2 >_{-1}^0.5 =  8.102402299360474e-06
Newton iteration   3energy = 8.687187044344453<A u 3 , A u 3 >_{-1}^0.5 =  2.8852464097068642e-09
Newton iteration   4energy = 8.68718704434445<A u 4 , A u 4 >_{-1}^0.5 =  8.62068123268755e-15
loadstep 27
Newton iteration   0energy = 8.68371016675213<A u 0 , A u 0 >_{-1}^0.5 =  0.006217513423105497
Newton iteration   1energy = 8.683691527695357<A u 1 , A u 1 >_{-1}^0.5 =  0.0008108646767330388
Newton iteration   2energy = 8.683691198580748<A u 2 , A u 2 >_{-1}^0.5 =  6.657405277539553e-06
Newton iteration   3energy = 8.683691198558591<A u 3 , A u 3 >_{-1}^0.5 =  1.8991875152329574e-09
Newton iteration   4energy = 8.68369119855859<A u 4 , A u 4 >_{-1}^0.5 =  8.31381277190115e-15
loadstep 28
Newton iteration   0energy = 8.680176737089983<A u 0 , A u 0 >_{-1}^0.5 =  0.006045001663946513
Newton iteration   1energy = 8.680159063155784<A u 1 , A u 1 >_{-1}^0.5 =  0.000730935146826445
Newton iteration   2energy = 8.680158795736729<A u 2 , A u 2 >_{-1}^0.5 =  5.488076199860775e-06
Newton iteration   3energy = 8.68015879572167<A u 3 , A u 3 >_{-1}^0.5 =  1.2588616627003492e-09
Newton iteration   4energy = 8.680158795721663<A u 4 , A u 4 >_{-1}^0.5 =  8.249289632673943e-15
loadstep 29
Newton iteration   0energy = 8.676608774000314<A u 0 , A u 0 >_{-1}^0.5 =  0.00588275646389783
Newton iteration   1energy = 8.676591991183995<A u 1 , A u 1 >_{-1}^0.5 =  0.000660757619962145
Newton iteration   2energy = 8.676591772658249<A u 2 , A u 2 >_{-1}^0.5 =  4.539191566126867e-06
Newton iteration   3energy = 8.67659177264795<A u 3 , A u 3 >_{-1}^0.5 =  8.40289945980431e-10
Newton iteration   4energy = 8.67659177264795<A u 4 , A u 4 >_{-1}^0.5 =  9.418652161231354e-15
loadstep 30
Newton iteration   0energy = 8.673008043866378<A u 0 , A u 0 >_{-1}^0.5 =  0.005729980563998529
Newton iteration   1energy = 8.672992084446872<A u 1 , A u 1 >_{-1}^0.5 =  0.0005989659685106328
Newton iteration   2energy = 8.672991904888457<A u 2 , A u 2 >_{-1}^0.5 =  3.76692558678857e-06
Newton iteration   3energy = 8.672991904881362<A u 3 , A u 3 >_{-1}^0.5 =  5.648286277050621e-10
Newton iteration   4energy = 8.672991904881364<A u 4 , A u 4 >_{-1}^0.5 =  9.809473912030201e-15
loadstep 31
Newton iteration   0energy = 8.669376169731839<A u 0 , A u 0 >_{-1}^0.5 =  0.005585948379563473
Newton iteration   1energy = 8.669360971844453<A u 1 , A u 1 >_{-1}^0.5 =  0.0005444069229851316
Newton iteration   2energy = 8.669360823513486<A u 2 , A u 2 >_{-1}^0.5 =  3.136484201519974e-06
Newton iteration   3energy = 8.669360823508566<A u 3 , A u 3 >_{-1}^0.5 =  3.82307045461158e-10
Newton iteration   4energy = 8.669360823508566<A u 4 , A u 4 >_{-1}^0.5 =  1.050679476747547e-14
loadstep 32
Newton iteration   0energy = 8.66571464593103<A u 0 , A u 0 >_{-1}^0.5 =  0.005449999256849206
Newton iteration   1energy = 8.665700153115052<A u 1 , A u 1 >_{-1}^0.5 =  0.0004961039573403354
Newton iteration   2energy = 8.665700029942759<A u 2 , A u 2 >_{-1}^0.5 =  2.6202145914449875e-06
Newton iteration   3energy = 8.66570002993933<A u 3 , A u 3 >_{-1}^0.5 =  2.605405858097632e-10
Newton iteration   4energy = 8.66570002993933<A u 4 , A u 4 >_{-1}^0.5 =  9.291317177394058e-15
loadstep 33
Newton iteration   0energy = 8.662024850979378<A u 0 , A u 0 >_{-1}^0.5 =  0.005321531272685377
Newton iteration   1energy = 8.662011011726086<A u 1 , A u 1 >_{-1}^0.5 =  0.0004532277179771657
Newton iteration   2energy = 8.662010908928279<A u 2 , A u 2 >_{-1}^0.5 =  2.1961049308536914e-06
Newton iteration   3energy = 8.66201090892587<A u 3 , A u 3 >_{-1}^0.5 =  1.7875556424130604e-10
Newton iteration   4energy = 8.662010908925868<A u 4 , A u 4 >_{-1}^0.5 =  9.941728243278747e-15
loadstep 34
Newton iteration   0energy = 8.658308058954264<A u 0 , A u 0 >_{-1}^0.5 =  0.0051999955774563265
Newton iteration   1energy = 8.658294826270655<A u 1 , A u 1 >_{-1}^0.5 =  0.00041507175626998954
Newton iteration   2energy = 8.658294740056048<A u 2 , A u 2 >_{-1}^0.5 =  1.8465991940801326e-06
Newton iteration   3energy = 8.658294740054345<A u 3 , A u 3 >_{-1}^0.5 =  1.2345632294967938e-10
Newton iteration   4energy = 8.658294740054341<A u 4 , A u 4 >_{-1}^0.5 =  1.0587488731531352e-14
loadstep 35
Newton iteration   0energy = 8.65456544956376<A u 0 , A u 0 >_{-1}^0.5 =  0.005084891262479123
Newton iteration   1energy = 8.65455278056102<A u 1 , A u 1 >_{-1}^0.5 =  0.000381032564132211
Newton iteration   2energy = 8.654552707909858<A u 2 , A u 2 >_{-1}^0.5 =  1.5576634313315837e-06
Newton iteration   3energy = 8.65455270790865<A u 3 , A u 3 >_{-1}^0.5 =  8.581373036197404e-11
Newton iteration   4energy = 8.654552707908646<A u 4 , A u 4 >_{-1}^0.5 =  9.269272096440546e-15
loadstep 36
Newton iteration   0energy = 8.650798117073078<A u 0 , A u 0 >_{-1}^0.5 =  0.0049757607212759965
Newton iteration   1energy = 8.65078597258417<A u 1 , A u 1 >_{-1}^0.5 =  0.00035059310620755014
Newton iteration   2energy = 8.65078591107931<A u 2 , A u 2 >_{-1}^0.5 =  1.318051220854502e-06
Newton iteration   3energy = 8.65078591107844<A u 3 , A u 3 >_{-1}^0.5 =  6.002781364799486e-11
Newton iteration   4energy = 8.65078591107844<A u 4 , A u 4 >_{-1}^0.5 =  1.0485738527131265e-14
loadstep 37
Newton iteration   0energy = 8.647007078234466<A u 0 , A u 0 >_{-1}^0.5 =  0.004872185468872917
Newton iteration   1energy = 8.646995422462584<A u 1 , A u 1 >_{-1}^0.5 =  0.00032330919910142
Newton iteration   2energy = 8.64699537015994<A u 2 , A u 2 >_{-1}^0.5 =  1.118726050930792e-06
Newton iteration   3energy = 8.646995370159313<A u 3 , A u 3 >_{-1}^0.5 =  4.224721621797406e-11
Newton iteration   4energy = 8.646995370159317<A u 4 , A u 4 >_{-1}^0.5 =  9.729232839510669e-15
loadstep 38
Newton iteration   0energy = 8.643193279346296<A u 0 , A u 0 >_{-1}^0.5 =  0.00477378238089766
Newton iteration   1energy = 8.643182079544076<A u 1 , A u 1 >_{-1}^0.5 =  0.00029879821383155936
Newton iteration   2energy = 8.643182034872753<A u 2 , A u 2 >_{-1}^0.5 =  9.524069438793206e-07
Newton iteration   3energy = 8.643182034872298<A u 3 , A u 3 >_{-1}^0.5 =  2.9914301717461894e-11
Newton iteration   4energy = 8.643182034872295<A u 4 , A u 4 >_{-1}^0.5 =  1.06610155587749e-14
loadstep 39
Newton iteration   0energy = 8.639357602549433<A u 0 , A u 0 >_{-1}^0.5 =  0.0046802003142568304
Newton iteration   1energy = 8.639346828727849<A u 1 , A u 1 >_{-1}^0.5 =  0.00027672967851285834
Newton iteration   2energy = 8.639346790412748<A u 2 , A u 2 >_{-1}^0.5 =  8.132106828501678e-07
Newton iteration   3energy = 8.63934679041241<A u 3 , A u 3 >_{-1}^0.5 =  2.1306059963602598e-11
Newton iteration   4energy = 8.639346790412409<A u 4 , A u 4 >_{-1}^0.5 =  1.0681843509166904e-14
loadstep 40
Newton iteration   0energy = 8.635500871454596<A u 0 , A u 0 >_{-1}^0.5 =  0.004591117072857296
Newton iteration   1energy = 8.635490496119774<A u 1 , A u 1 >_{-1}^0.5 =  0.000256817439502574
Newton iteration   2energy = 8.635490463121286<A u 2 , A u 2 >_{-1}^0.5 =  6.963696966283656e-07
Newton iteration   3energy = 8.63549046312104<A u 3 , A u 3 >_{-1}^0.5 =  1.5258323053214198e-11
Newton iteration   4energy = 8.635490463121046<A u 4 , A u 4 >_{-1}^0.5 =  8.827150842568554e-15
loadstep 41
Newton iteration   0energy = 8.631623856181335<A u 0 , A u 0 >_{-1}^0.5 =  0.0045062366837491
Newton iteration   1energy = 8.631613854097237<A u 1 , A u 1 >_{-1}^0.5 =  0.00023881310384552478
Newton iteration   2energy = 8.631613825564186<A u 2 , A u 2 >_{-1}^0.5 =  5.98009204103875e-07
Newton iteration   3energy = 8.631613825564012<A u 3 , A u 3 >_{-1}^0.5 =  1.0990488900140723e-11
Newton iteration   4energy = 8.63161382556401<A u 4 , A u 4 >_{-1}^0.5 =  1.055859156574069e-14
loadstep 42
Newton iteration   0energy = 8.627727277879044<A u 0 , A u 0 >_{-1}^0.5 =  0.004425286951923272
Newton iteration   1energy = 8.62771762585382<A u 1 , A u 1 >_{-1}^0.5 =  0.00022250053817710115
Newton iteration   2energy = 8.627717601086363<A u 2 , A u 2 >_{-1}^0.5 =  5.149708084000054e-07
Newton iteration   3energy = 8.62771760108623<A u 3 , A u 3 >_{-1}^0.5 =  7.959422772728941e-12
Newton iteration   4energy = 8.627717601086221<A u 4 , A u 4 >_{-1}^0.5 =  1.1206272235192536e-14
loadstep 43
Newton iteration   0energy = 8.623811812790498<A u 0 , A u 0 >_{-1}^0.5 =  0.004348017264586569
Newton iteration   1energy = 8.623802489484435<A u 1 , A u 1 >_{-1}^0.5 =  0.00020769124099631234
Newton iteration   2energy = 8.623802467904822<A u 2 , A u 2 >_{-1}^0.5 =  4.446725375445635e-07
Newton iteration   3energy = 8.623802467904726<A u 3 , A u 3 >_{-1}^0.5 =  5.793013765299362e-12
Newton iteration   4energy = 8.623802467904728<A u 4 , A u 4 >_{-1}^0.5 =  1.0530521273070294e-14
loadstep 44
Newton iteration   0energy = 8.619878095911142<A u 0 , A u 0 >_{-1}^0.5 =  0.004274196618547073
Newton iteration   1energy = 8.619869081664056<A u 1 , A u 1 >_{-1}^0.5 =  0.00019422043896448316
Newton iteration   2energy = 8.619869062793466<A u 2 , A u 2 >_{-1}^0.5 =  3.849975301478303e-07
Newton iteration   3energy = 8.61986906279339<A u 3 , A u 3 >_{-1}^0.5 =  4.239278501511616e-12
Newton iteration   4energy = 8.619869062793391<A u 4 , A u 4 >_{-1}^0.5 =  1.11580184480003e-14
loadstep 45
Newton iteration   0energy = 8.615926724290007<A u 0 , A u 0 >_{-1}^0.5 =  0.0042036118468920195
Newton iteration   1energy = 8.615918000966204<A u 1 , A u 1 >_{-1}^0.5 =  0.00018194378505178904
Newton iteration   2energy = 8.615917984406249<A u 2 , A u 2 >_{-1}^0.5 =  3.3420528126933616e-07
Newton iteration   3energy =  8.6159179844062<A u 3 , A u 3 >_{-1}^0.5 =  3.1172942690788893e-12
Newton iteration   4energy = 8.615917984406195<A u 4 , A u 4 >_{-1}^0.5 =  1.0330726651365844e-14
loadstep 46
Newton iteration   0energy = 8.611958260012726<A u 0 , A u 0 >_{-1}^0.5 =  0.0041360660236643134
Newton iteration   1energy = 8.611949810861685<A u 1 , A u 1 >_{-1}^0.5 =  0.00017073455836505666
Newton iteration   2energy = 8.611949796279685<A u 2 , A u 2 >_{-1}^0.5 =  2.9086068319968975e-07
Newton iteration   3energy = 8.611949796279646<A u 3 , A u 3 >_{-1}^0.5 =  2.3030356237034256e-12
Newton iteration   4energy = 8.611949796279646<A u 4 , A u 4 >_{-1}^0.5 =  1.1820708404389974e-14
loadstep 47
Newton iteration   0energy = 8.60797323290172<A u 0 , A u 0 >_{-1}^0.5 =  0.004071377027471576
Newton iteration   1energy = 8.607965042432857<A u 1 , A u 1 >_{-1}^0.5 =  0.00016048128329030782
Newton iteration   2energy = 8.60796502954998<A u 2 , A u 2 >_{-1}^0.5 =  2.5377714592362046e-07
Newton iteration   3energy = 8.607965029549941<A u 3 , A u 3 >_{-1}^0.5 =  1.7113756425527599e-12
Newton iteration   4energy = 8.607965029549941<A u 4 , A u 4 >_{-1}^0.5 =  1.0304094715684442e-14
loadstep 48
Newton iteration   0energy = 8.60397214296444<A u 0 , A u 0 >_{-1}^0.5 =  0.004009376246978417
Newton iteration   1energy = 8.603964196834545<A u 1 , A u 1 >_{-1}^0.5 =  0.00015108569999442697
Newton iteration   2energy = 8.603964185416244<A u 2 , A u 2 >_{-1}^0.5 =  2.219708759576467e-07
Newton iteration   3energy = 8.603964185416224<A u 3 , A u 3 >_{-1}^0.5 =  1.2754769699119674e-12
Newton iteration   4energy = 8.603964185416222<A u 4 , A u 4 >_{-1}^0.5 =  1.1159291624241635e-14
loadstep 49
Newton iteration   0energy = 8.599955462616867<A u 0 , A u 0 >_{-1}^0.5 =  0.00394990741323281
Newton iteration   1energy = 8.59994774752875<A u 1 , A u 1 >_{-1}^0.5 =  0.0001424610302423851
Newton iteration   2energy = 8.599947737377077<A u 2 , A u 2 >_{-1}^0.5 =  1.9462402808326612e-07
Newton iteration   3energy = 8.599947737377056<A u 3 , A u 3 >_{-1}^0.5 =  9.559087923180946e-13
Newton iteration   4energy = 8.59994773737706<A u 4 , A u 4 >_{-1}^0.5 =  1.1705547967954596e-14

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} + v^{2} (1 - v^{2}) d x

i.e. 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 minima ( $- 1$ 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 “SymbolicEnergy” feature to formulate the energy minimization problem and combine it 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 interpreted 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.

[11]:

from ngsolve import *
from ngsolve.webgui import Draw
from netgen.geom2d import *

periodic = SplineGeometry()
pnts = [ (0,0), (1,0), (1,1), (0,1) ]
pnums = [periodic.AppendPoint(*p) for p in pnts]
lright = periodic.Append ( ["line", pnums[0], pnums[1]],bc="periodic")
btop = periodic.Append ( ["line", pnums[1], pnums[2]], bc="periodic")
periodic.Append ( ["line", pnums[3], pnums[2]], leftdomain=0, rightdomain=1, copy=lright, bc="periodic")
periodic.Append ( ["line", pnums[0], pnums[3]], leftdomain=0, rightdomain=1, copy=btop, bc="periodic")

mesh = Mesh (periodic.GenerateMesh(maxh=0.2))
V = Periodic(H1(mesh, order=4, dirichlet=[]))
u = V.TrialFunction()

eps = 4e-3
dt = 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)

[12]:

from math import pi
gfu = GridFunction(V)
gfu.Set(sin(2*pi*x))
#gfu.Set(sin(1e8*x)) #<- essentially a random function
sceneu = Draw(gfu,mesh,"u")
SetVisualization (deformation=True)
t = 0

[13]:

for timestep in range(50):
    gfuold.vec.data = gfu.vec
    SolveNonlinearMinProblem(a,gfu)
    sceneu.Redraw()
    t += dt
    print("t = ", t)

Newton iteration   0energy = 0.41450274940004017<A u 0 , A u 0 >_{-1}^0.5 =  0.28092276820769385
Newton iteration   1energy = 0.37690042628264986<A u 1 , A u 1 >_{-1}^0.5 =  0.019175684672928434
Newton iteration   2energy = 0.3767159640889083<A u 2 , A u 2 >_{-1}^0.5 =  9.62988096864071e-05
Newton iteration   3energy = 0.3767159594520977<A u 3 , A u 3 >_{-1}^0.5 =  2.5081413874453625e-09
Newton iteration   4energy = 0.37671595945209774<A u 4 , A u 4 >_{-1}^0.5 =  2.770950400879142e-16
t =  0.1
Newton iteration   0energy = 0.3438887218899856<A u 0 , A u 0 >_{-1}^0.5 =  0.2359984304769602
Newton iteration   1energy = 0.3171274464604802<A u 1 , A u 1 >_{-1}^0.5 =  0.013451013644986742
Newton iteration   2energy = 0.3170367744533889<A u 2 , A u 2 >_{-1}^0.5 =  4.663240800351902e-05
Newton iteration   3energy = 0.31703677336608915<A u 3 , A u 3 >_{-1}^0.5 =  5.769081036770765e-10
Newton iteration   4energy = 0.3170367733660891<A u 4 , A u 4 >_{-1}^0.5 =  3.104600909251941e-16
t =  0.2
Newton iteration   0energy = 0.2955287970166734<A u 0 , A u 0 >_{-1}^0.5 =  0.18333961760859702
Newton iteration   1energy = 0.27922061920053365<A u 1 , A u 1 >_{-1}^0.5 =  0.008020159823637686
Newton iteration   2energy = 0.2791884145175679<A u 2 , A u 2 >_{-1}^0.5 =  1.6280453338856004e-05
Newton iteration   3energy = 0.27918841438504094<A u 3 , A u 3 >_{-1}^0.5 =  6.88277206865018e-11
Newton iteration   4energy = 0.2791884143850408<A u 4 , A u 4 >_{-1}^0.5 =  3.5918508178786496e-16
t =  0.30000000000000004
Newton iteration   0energy = 0.2670098660513399<A u 0 , A u 0 >_{-1}^0.5 =  0.13329903037709695
Newton iteration   1energy = 0.25831307616871746<A u 1 , A u 1 >_{-1}^0.5 =  0.0041875315462804544
Newton iteration   2energy = 0.2583043023901654<A u 2 , A u 2 >_{-1}^0.5 =  4.3728797510974884e-06
Newton iteration   3energy = 0.2583043023806044<A u 3 , A u 3 >_{-1}^0.5 =  4.8743476603423805e-12
Newton iteration   4energy = 0.2583043023806044<A u 4 , A u 4 >_{-1}^0.5 =  3.5485559099999397e-16
t =  0.4
Newton iteration   0energy = 0.25217569665415823<A u 0 , A u 0 >_{-1}^0.5 =  0.09212757625004177
Newton iteration   1energy = 0.24799278266903632<A u 1 , A u 1 >_{-1}^0.5 =  0.0019803055518447786
Newton iteration   2energy = 0.2479908212283336<A u 2 , A u 2 >_{-1}^0.5 =  9.675293371219298e-07
Newton iteration   3energy = 0.24799082122786548<A u 3 , A u 3 >_{-1}^0.5 =  2.351718624444368e-13
Newton iteration   4energy = 0.2479908212278656<A u 4 , A u 4 >_{-1}^0.5 =  3.5285019118579734e-16
t =  0.5
Newton iteration   0energy = 0.24516469919769227<A u 0 , A u 0 >_{-1}^0.5 =  0.06142246901372681
Newton iteration   1energy = 0.24329614316799697<A u 1 , A u 1 >_{-1}^0.5 =  0.0008742001822322159
Newton iteration   2energy = 0.243295761000651<A u 2 , A u 2 >_{-1}^0.5 =  1.8736246218623535e-07
Newton iteration   3energy = 0.24329576100063338<A u 3 , A u 3 >_{-1}^0.5 =  8.837199141043262e-15
t =  0.6
Newton iteration   0energy = 0.24206890764107003<A u 0 , A u 0 >_{-1}^0.5 =  0.03998278353294713
Newton iteration   1energy = 0.24127447024481935<A u 1 , A u 1 >_{-1}^0.5 =  0.00036914563244039837
Newton iteration   2energy = 0.24127440210648102<A u 2 , A u 2 >_{-1}^0.5 =  3.3396140007954435e-08
Newton iteration   3energy = 0.24127440210648043<A u 3 , A u 3 >_{-1}^0.5 =  5.392904101207391e-16
t =  0.7
Newton iteration   0energy = 0.24076206680029597<A u 0 , A u 0 >_{-1}^0.5 =  0.02565432378296499
Newton iteration   1energy = 0.24043427961385436<A u 1 , A u 1 >_{-1}^0.5 =  0.0001521836948922547
Newton iteration   2energy = 0.2404342680336267<A u 2 , A u 2 >_{-1}^0.5 =  5.749334331966722e-09
Newton iteration   3energy = 0.24043426803362666<A u 3 , A u 3 >_{-1}^0.5 =  4.0723660057673715e-16
t =  0.7999999999999999
Newton iteration   0energy = 0.2402247958434406<A u 0 , A u 0 >_{-1}^0.5 =  0.016360129469374974
Newton iteration   1energy = 0.24009130447406016<A u 1 , A u 1 >_{-1}^0.5 =  6.260945027184524e-05
Newton iteration   2energy = 0.24009130251406763<A u 2 , A u 2 >_{-1}^0.5 =  1.0310386702794883e-09
Newton iteration   3energy = 0.2400913025140676<A u 3 , A u 3 >_{-1}^0.5 =  4.430990996815971e-16
t =  0.8999999999999999
Newton iteration   0energy = 0.2400060001527204<A u 0 , A u 0 >_{-1}^0.5 =  0.010467378997427187
Newton iteration   1energy = 0.23995130728175787<A u 1 , A u 1 >_{-1}^0.5 =  2.660330565795448e-05
Newton iteration   2energy = 0.23995130692788802<A u 2 , A u 2 >_{-1}^0.5 =  2.2781443109275827e-10
Newton iteration   3energy = 0.23995130692788808<A u 3 , A u 3 >_{-1}^0.5 =  4.520167645921635e-16
t =  0.9999999999999999
Newton iteration   0energy = 0.23991591345508823<A u 0 , A u 0 >_{-1}^0.5 =  0.006815543354962078
Newton iteration   1energy = 0.2398927141233959<A u 1 , A u 1 >_{-1}^0.5 =  1.2418172737585205e-05
Newton iteration   2energy = 0.23989271404629017<A u 2 , A u 2 >_{-1}^0.5 =  7.456015681141339e-11
Newton iteration   3energy = 0.23989271404629015<A u 3 , A u 3 >_{-1}^0.5 =  4.857526180759073e-16
t =  1.0999999999999999
Newton iteration   0energy = 0.23987719120231832<A u 0 , A u 0 >_{-1}^0.5 =  0.0046219617602749
Newton iteration   1energy = 0.2398665191876579<A u 1 , A u 1 >_{-1}^0.5 =  6.8803322449782575e-06
Newton iteration   2energy = 0.2398665191639883<A u 2 , A u 2 >_{-1}^0.5 =  3.373660269756489e-11
Newton iteration   3energy = 0.23986651916398838<A u 3 , A u 3 >_{-1}^0.5 =  4.51866572779697e-16
t =  1.2
Newton iteration   0energy = 0.239858912438348<A u 0 , A u 0 >_{-1}^0.5 =  0.0033643229589345896
Newton iteration   1energy = 0.23985325717204478<A u 1 , A u 1 >_{-1}^0.5 =  4.626816464814933e-06
Newton iteration   2energy = 0.239853257161341<A u 2 , A u 2 >_{-1}^0.5 =  1.83404666236353e-11
Newton iteration   3energy = 0.23985325716134107<A u 3 , A u 3 >_{-1}^0.5 =  4.4167890039517904e-16
t =  1.3
Newton iteration   0energy = 0.23984885075665663<A u 0 , A u 0 >_{-1}^0.5 =  0.002682345350752739
Newton iteration   1energy = 0.23984525547226224<A u 1 , A u 1 >_{-1}^0.5 =  3.580510801938737e-06
Newton iteration   2energy = 0.23984525546585217<A u 2 , A u 2 >_{-1}^0.5 =  1.122826298267855e-11
Newton iteration   3energy = 0.23984525546585217<A u 3 , A u 3 >_{-1}^0.5 =  4.497655034487545e-16
t =  1.4000000000000001
Newton iteration   0energy = 0.23984219437981988<A u 0 , A u 0 >_{-1}^0.5 =  0.00232416801404013
Newton iteration   1energy = 0.23983949487484169<A u 1 , A u 1 >_{-1}^0.5 =  3.0009604397547507e-06
Newton iteration   2energy = 0.23983949487033884<A u 2 , A u 2 >_{-1}^0.5 =  7.511495133264662e-12
Newton iteration   3energy = 0.23983949487033895<A u 3 , A u 3 >_{-1}^0.5 =  4.457271102231051e-16
t =  1.5000000000000002
Newton iteration   0energy = 0.23983704669748698<A u 0 , A u 0 >_{-1}^0.5 =  0.0021291987319734737
Newton iteration   1energy = 0.23983478088560456<A u 1 , A u 1 >_{-1}^0.5 =  2.6257686686933636e-06
Newton iteration   2energy = 0.2398347808821573<A u 2 , A u 2 >_{-1}^0.5 =  5.384331218074664e-12
Newton iteration   3energy = 0.23983478088215732<A u 3 , A u 3 >_{-1}^0.5 =  4.0108515124668936e-16
t =  1.6000000000000003
Newton iteration   0energy = 0.23983265244268792<A u 0 , A u 0 >_{-1}^0.5 =  0.0020100557693737047
Newton iteration   1energy = 0.23983063294882798<A u 1 , A u 1 >_{-1}^0.5 =  2.3525389446384635e-06
Newton iteration   2energy = 0.2398306329460608<A u 2 , A u 2 >_{-1}^0.5 =  4.068146969940559e-12
Newton iteration   3energy = 0.23983063294606083<A u 3 , A u 3 >_{-1}^0.5 =  4.3910814781746883e-16
t =  1.7000000000000004
Newton iteration   0energy = 0.2398287030787631<A u 0 , A u 0 >_{-1}^0.5 =  0.0019252122100653665
Newton iteration   1energy = 0.239826850357478<A u 1 , A u 1 >_{-1}^0.5 =  2.1365289236943262e-06
Newton iteration   2energy = 0.23982685035519563<A u 2 , A u 2 >_{-1}^0.5 =  3.1940413812410494e-12
Newton iteration   3energy = 0.2398268503551957<A u 3 , A u 3 >_{-1}^0.5 =  4.3239790916168463e-16
t =  1.8000000000000005
Newton iteration   0energy = 0.23982506567647988<A u 0 , A u 0 >_{-1}^0.5 =  0.0018564195832818957
Newton iteration   1energy = 0.2398233429163234<A u 1 , A u 1 >_{-1}^0.5 =  1.9563579521165098e-06
Newton iteration   2energy = 0.2398233429144098<A u 2 , A u 2 >_{-1}^0.5 =  2.576532197084369e-12
Newton iteration   3energy = 0.23982334291440977<A u 3 , A u 3 >_{-1}^0.5 =  4.168337545674308e-16
t =  1.9000000000000006
Newton iteration   0energy = 0.23982167713524788<A u 0 , A u 0 >_{-1}^0.5 =  0.0017958971457824545
Newton iteration   1energy = 0.23982006481970106<A u 1 , A u 1 >_{-1}^0.5 =  1.8009639637079065e-06
Newton iteration   2energy = 0.23982006481807935<A u 2 , A u 2 >_{-1}^0.5 =  2.1180629067376886e-12
Newton iteration   3energy = 0.23982006481807938<A u 3 , A u 3 >_{-1}^0.5 =  4.275446995367114e-16
t =  2.0000000000000004
Newton iteration   0energy = 0.23981850280918926<A u 0 , A u 0 >_{-1}^0.5 =  0.001740310776358378
Newton iteration   1energy = 0.23981698871896923<A u 1 , A u 1 >_{-1}^0.5 =  1.6641243579421696e-06
Newton iteration   2energy = 0.23981698871758456<A u 2 , A u 2 >_{-1}^0.5 =  1.7645509925176325e-12
Newton iteration   3energy = 0.2398169887175845<A u 3 , A u 3 >_{-1}^0.5 =  4.1963762805438165e-16
t =  2.1000000000000005
Newton iteration   0energy = 0.23981552021572586<A u 0 , A u 0 >_{-1}^0.5 =  0.0016881769354288318
Newton iteration   1energy = 0.2398140954528447<A u 1 , A u 1 >_{-1}^0.5 =  1.5420178269083415e-06
Newton iteration   2energy = 0.23981409545165577<A u 2 , A u 2 >_{-1}^0.5 =  1.4844233174577358e-12
Newton iteration   3energy = 0.23981409545165577<A u 3 , A u 3 >_{-1}^0.5 =  4.02205045000741e-16
t =  2.2000000000000006
Newton iteration   0energy = 0.23981271253222786<A u 0 , A u 0 >_{-1}^0.5 =  0.0016387829756173893
Newton iteration   1energy = 0.23981136990222032<A u 1 , A u 1 >_{-1}^0.5 =  1.4320930678395655e-06
Newton iteration   2energy = 0.2398113699011948<A u 2 , A u 2 >_{-1}^0.5 =  1.2579829091911714e-12
Newton iteration   3energy = 0.2398113699011948<A u 3 , A u 3 >_{-1}^0.5 =  4.212159800438489e-16
t =  2.3000000000000007
Newton iteration   0energy = 0.2398100659309837<A u 0 , A u 0 >_{-1}^0.5 =  0.0015917433542050894
Newton iteration   1energy = 0.23980879925623955<A u 1 , A u 1 >_{-1}^0.5 =  1.3325202176466422e-06
Newton iteration   2energy = 0.2398087992553517<A u 2 , A u 2 >_{-1}^0.5 =  1.072191835063917e-12
Newton iteration   3energy = 0.23980879925535173<A u 3 , A u 3 >_{-1}^0.5 =  3.9147779396855535e-16
t =  2.400000000000001
Newton iteration   0energy = 0.23980756843352935<A u 0 , A u 0 >_{-1}^0.5 =  0.0015468169239539241
Newton iteration   1energy = 0.2398063722398347<A u 1 , A u 1 >_{-1}^0.5 =  1.2419105209608674e-06
Newton iteration   2energy = 0.23980637223906348<A u 2 , A u 2 >_{-1}^0.5 =  9.181606100885629e-13
Newton iteration   3energy = 0.2398063722390635<A u 3 , A u 3 >_{-1}^0.5 =  4.204249177876161e-16
t =  2.500000000000001
Newton iteration   0energy = 0.23980520937673644<A u 0 , A u 0 >_{-1}^0.5 =  0.0015038307895702187
Newton iteration   1energy = 0.23980407873345475<A u 1 , A u 1 >_{-1}^0.5 =  1.1591642352896874e-06
Newton iteration   2energy = 0.23980407873278287<A u 2 , A u 2 >_{-1}^0.5 =  7.894063470258062e-13
Newton iteration   3energy = 0.23980407873278284<A u 3 , A u 3 >_{-1}^0.5 =  4.294231133646188e-16
t =  2.600000000000001
Newton iteration   0energy = 0.23980297913275267<A u 0 , A u 0 >_{-1}^0.5 =  0.001462647788622576
Newton iteration   1energy = 0.2398019095592319<A u 1 , A u 1 >_{-1}^0.5 =  1.0833827515316492e-06
Newton iteration   2energy = 0.23980190955864508<A u 2 , A u 2 >_{-1}^0.5 =  6.810990802114164e-13
Newton iteration   3energy = 0.23980190955864508<A u 3 , A u 3 >_{-1}^0.5 =  4.2753009248048324e-16
t =  2.700000000000001
Newton iteration   0energy = 0.2398008689398244<A u 0 , A u 0 >_{-1}^0.5 =  0.0014231520130885233
Newton iteration   1energy = 0.23979985634253928<A u 1 , A u 1 >_{-1}^0.5 =  1.0138146157404198e-06
Newton iteration   2energy = 0.23979985634202533<A u 2 , A u 2 >_{-1}^0.5 =  5.894808712628097e-13
Newton iteration   3energy = 0.23979985634202533<A u 3 , A u 3 >_{-1}^0.5 =  4.1422756115981907e-16
t =  2.800000000000001
Newton iteration   0energy = 0.23979887078562834<A u 0 , A u 0 >_{-1}^0.5 =  0.0013852419317702752
Newton iteration   1energy = 0.23979791141117418<A u 1 , A u 1 >_{-1}^0.5 =  9.498204696464258e-07
Newton iteration   2energy = 0.239797911410723<A u 2 , A u 2 >_{-1}^0.5 =  5.116112791307342e-13
Newton iteration   3energy = 0.2397979114107231<A u 3 , A u 3 >_{-1}^0.5 =  4.076450029822525e-16
t =  2.9000000000000012
Newton iteration   0energy = 0.23979697731840338<A u 0 , A u 0 >_{-1}^0.5 =  0.0013488268130407179
Newton iteration   1energy = 0.23979606771576892<A u 1 , A u 1 >_{-1}^0.5 =  8.908491345635657e-07
Newton iteration   2energy = 0.23979606771537218<A u 2 , A u 2 >_{-1}^0.5 =  4.452475244786126e-13
Newton iteration   3energy = 0.2397960677153721<A u 3 , A u 3 >_{-1}^0.5 =  4.3922489477725524e-16
t =  3.0000000000000013
Newton iteration   0energy = 0.23979518177481676<A u 0 , A u 0 >_{-1}^0.5 =  0.0013138246473699483
Newton iteration   1energy = 0.23979431876380208<A u 1 , A u 1 >_{-1}^0.5 =  8.36420596492343e-07
Newton iteration   2energy = 0.23979431876345225<A u 2 , A u 2 >_{-1}^0.5 =  3.884509019712542e-13
Newton iteration   3energy = 0.23979431876345225<A u 3 , A u 3 >_{-1}^0.5 =  4.301690284769913e-16
t =  3.1000000000000014
Newton iteration   0energy = 0.23979347791918174<A u 0 , A u 0 >_{-1}^0.5 =  0.0012801607964436092
Newton iteration   1energy = 0.23979265856330648<A u 1 , A u 1 >_{-1}^0.5 =  7.86113458671679e-07
Newton iteration   2energy = 0.23979265856299747<A u 2 , A u 2 >_{-1}^0.5 =  3.3969874372684423e-13
Newton iteration   3energy = 0.23979265856299747<A u 3 , A u 3 >_{-1}^0.5 =  4.3004324600663537e-16
t =  3.2000000000000015
Newton iteration   0energy = 0.2397918599911152<A u 0 , A u 0 >_{-1}^0.5 =  0.0012477670242682008
Newton iteration   1energy = 0.23979108157404128<A u 1 , A u 1 >_{-1}^0.5 =  7.395554015632387e-07
Newton iteration   2energy = 0.23979108157376783<A u 2 , A u 2 >_{-1}^0.5 =  2.9777915336040986e-13
Newton iteration   3energy = 0.23979108157376777<A u 3 , A u 3 >_{-1}^0.5 =  3.9547542266396893e-16
t =  3.3000000000000016
Newton iteration   0energy = 0.23979032265987657<A u 0 , A u 0 >_{-1}^0.5 =  0.0012165807486120865
Newton iteration   1energy = 0.23978958266470105<A u 1 , A u 1 >_{-1}^0.5 =  6.964157409186844e-07
Newton iteration   2energy = 0.23978958266445857<A u 2 , A u 2 >_{-1}^0.5 =  2.6157855623946123e-13
Newton iteration   3energy = 0.23978958266445852<A u 3 , A u 3 >_{-1}^0.5 =  4.283654302417877e-16
t =  3.4000000000000017
Newton iteration   0energy = 0.23978886098419994<A u 0 , A u 0 >_{-1}^0.5 =  0.0011865444318763026
Newton iteration   1energy = 0.23978815707515655<A u 1 , A u 1 >_{-1}^0.5 =  6.56399492291637e-07
Newton iteration   2energy = 0.23978815707494122<A u 2 , A u 2 >_{-1}^0.5 =  2.302948265576988e-13
Newton iteration   3energy = 0.2397881570749413<A u 3 , A u 3 >_{-1}^0.5 =  4.086979269954084e-16
t =  3.5000000000000018
Newton iteration   0energy = 0.23978747037675902<A u 0 , A u 0 >_{-1}^0.5 =  0.0011576050677765749
Newton iteration   1energy = 0.23978680038297534<A u 1 , A u 1 >_{-1}^0.5 =  6.19242549160231e-07
Newton iteration   2energy = 0.23978680038278374<A u 2 , A u 2 >_{-1}^0.5 =  2.0315531529176785e-13
Newton iteration   3energy = 0.23978680038278372<A u 3 , A u 3 >_{-1}^0.5 =  3.9756498266179526e-16
t =  3.600000000000002
Newton iteration   0energy = 0.23978614657259592<A u 0 , A u 0 >_{-1}^0.5 =  0.0011297137384427701
Newton iteration   1energy = 0.23978550847362343<A u 1 , A u 1 >_{-1}^0.5 =  5.84707704590533e-07
Newton iteration   2energy = 0.23978550847345248<A u 2 , A u 2 >_{-1}^0.5 =  1.7958897594683527e-13
Newton iteration   3energy = 0.23978550847345256<A u 3 , A u 3 >_{-1}^0.5 =  4.0775697916972084e-16
t =  3.700000000000002
Newton iteration   0energy = 0.23978488560097597<A u 0 , A u 0 >_{-1}^0.5 =  0.0011028252260448733
Newton iteration   1energy = 0.2397842775138649<A u 1 , A u 1 >_{-1}^0.5 =  5.525813242295787e-07
Newton iteration   2energy = 0.23978427751371223<A u 2 , A u 2 >_{-1}^0.5 =  1.5907479433745375e-13
Newton iteration   3energy = 0.23978427751371229<A u 3 , A u 3 >_{-1}^0.5 =  4.256689574317313e-16
t =  3.800000000000002
Newton iteration   0energy = 0.23978368376023024<A u 0 , A u 0 >_{-1}^0.5 =  0.001076897668351713
Newton iteration   1energy = 0.23978310392795166<A u 1 , A u 1 >_{-1}^0.5 =  5.226705343920618e-07
Newton iteration   2energy = 0.2397831039278152<A u 2 , A u 2 >_{-1}^0.5 =  1.4116224904179915e-13
Newton iteration   3energy = 0.2397831039278151<A u 3 , A u 3 >_{-1}^0.5 =  4.174221029815967e-16
t =  3.900000000000002
Newton iteration   0energy = 0.23978253759521045<A u 0 , A u 0 >_{-1}^0.5 =  0.0010518922507697856
Newton iteration   1energy = 0.2397819843762649<A u 1 , A u 1 >_{-1}^0.5 =  4.948008225696191e-07
Newton iteration   2energy = 0.23978198437614234<A u 2 , A u 2 >_{-1}^0.5 =  1.2547261891036262e-13
Newton iteration   3energy = 0.2397819843761424<A u 3 , A u 3 >_{-1}^0.5 =  4.0416552040613763e-16
t =  4.000000000000002
Newton iteration   0energy = 0.23978144387704142<A u 0 , A u 0 >_{-1}^0.5 =  0.0010277729293769811
Newton iteration   1energy = 0.2397809157361111<A u 1 , A u 1 >_{-1}^0.5 =  4.6881397441626697e-07
Newton iteration   2energy = 0.23978091573600113<A u 2 , A u 2 >_{-1}^0.5 =  1.1181144187446951e-13
Newton iteration   3energy = 0.23978091573600113<A u 3 , A u 3 >_{-1}^0.5 =  4.016687422136611e-16
t =  4.100000000000001
Newton iteration   0energy = 0.23978039958490083<A u 0 , A u 0 >_{-1}^0.5 =  0.0010045061807557245
Newton iteration   1energy = 0.23977989508442546<A u 1 , A u 1 >_{-1}^0.5 =  4.44566287255703e-07
Newton iteration   2energy = 0.23977989508432665<A u 2 , A u 2 >_{-1}^0.5 =  9.980130671726263e-14
t =  4.200000000000001
Newton iteration   0energy = 0.2397794018895898<A u 0 , A u 0 >_{-1}^0.5 =  0.0009820607753089098
Newton iteration   1energy = 0.23977891968215984<A u 1 , A u 1 >_{-1}^0.5 =  4.219270158055787e-07
Newton iteration   2energy = 0.23977891968207077<A u 2 , A u 2 >_{-1}^0.5 =  8.925950644743311e-14
t =  4.300000000000001
Newton iteration   0energy = 0.23977844813869198<A u 0 , A u 0 >_{-1}^0.5 =  0.0009604075713616538
Newton iteration   1energy = 0.23977798696016697<A u 1 , A u 1 >_{-1}^0.5 =  4.007770120387171e-07
Newton iteration   2energy = 0.23977798696008668<A u 2 , A u 2 >_{-1}^0.5 =  7.994196152771886e-14
t =  4.4
Newton iteration   0energy = 0.23977753584314124<A u 0 , A u 0 >_{-1}^0.5 =  0.0009395193278036805
Newton iteration   1energy = 0.23977709450641457<A u 1 , A u 1 >_{-1}^0.5 =  3.8100753085926176e-07
Newton iteration   2energy = 0.239777094506342<A u 2 , A u 2 >_{-1}^0.5 =  7.179512773894963e-14
t =  4.5
Newton iteration   0energy = 0.239776662665046<A u 0 , A u 0 >_{-1}^0.5 =  0.0009193705333654081
Newton iteration   1energy = 0.23977624005438256<A u 1 , A u 1 >_{-1}^0.5 =  3.625191772482667e-07
Newton iteration   2energy = 0.23977624005431677<A u 2 , A u 2 >_{-1}^0.5 =  6.453622746009833e-14
t =  4.6
Newton iteration   0energy = 0.23977582640662712<A u 0 , A u 0 >_{-1}^0.5 =  0.0008999372508832096
Newton iteration   1energy = 0.2397754214725149<A u 1 , A u 1 >_{-1}^0.5 =  3.4522097539002697e-07
Newton iteration   2energy = 0.23977542147245537<A u 2 , A u 2 >_{-1}^0.5 =  5.818393855479234e-14
t =  4.699999999999999
Newton iteration   0energy = 0.23977502500015624<A u 0 , A u 0 >_{-1}^0.5 =  0.0008811969751174315
Newton iteration   1energy = 0.239774636754615<A u 1 , A u 1 >_{-1}^0.5 =  3.29029543283108e-07
Newton iteration   2energy = 0.23977463675456087<A u 2 , A u 2 >_{-1}^0.5 =  5.252888457043041e-14
t =  4.799999999999999
Newton iteration   0energy = 0.23977425649878278<A u 0 , A u 0 >_{-1}^0.5 =  0.0008631285028569205
Newton iteration   1energy = 0.2397738840110826<A u 1 , A u 1 >_{-1}^0.5 =  3.138683589703107e-07
Newton iteration   2energy = 0.23977388401103322<A u 2 , A u 2 >_{-1}^0.5 =  4.751561654199766e-14
t =  4.899999999999999
Newton iteration   0energy = 0.23977351906815902<A u 0 , A u 0 >_{-1}^0.5 =  0.0008457118141865206
Newton iteration   1energy = 0.23977316146090577<A u 1 , A u 1 >_{-1}^0.5 =  2.9966710699954903e-07
Newton iteration   2energy = 0.23977316146086092<A u 2 , A u 2 >_{-1}^0.5 =  4.3025917851734007e-14
t =  4.999999999999998

Minimal energy extension (postscript in unit-2.1.3 )¶

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},

[14]:

from netgen.geom2d import unit_square
mesh = Mesh(unit_square.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()

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

[15]:

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)
solvers.Newton(a=a, u=gfu)
Draw(gfu.components[0])

Newton iteration  0
err =  0.39346141669994983
Newton iteration  1
err =  1.8488788182878563e-15

[15]:

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 .

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