Incompressible Navier-Stokes with heat conduction and convection

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Download mesh here

[1]:

# Video generated with Paraview from the output files
from IPython.display import Video
Video("visualizations/navier_stokes_temperature.mp4", embed=True, width=800, height=400)

[1]:

[ ]:

import jax
import jax.numpy as jnp
import meshio

from autopdex import SimState, dae, spaces

jax.config.update("jax_enable_x64", True)

Load mesh

[3]:

mesh = meshio.read("meshes/navier_stokes_mesh.msh")

Boundary conditions

  • Parabolic inflow at x = 0 and no-slip at the cylinder and top/bottom walls.

  • T = 0 at the inflow; the cylinder temperature ramps from 0 to 20 during the first 3.5 seconds.

  • p = 0 on the right outlet edge as pressure reference.

[4]:

def on_left(x):
    return jnp.isclose(x[0], 0.0)

def on_cylinder(x):
    return jnp.isclose((x[0] - 0.2) ** 2 + (x[1] - 0.2) ** 2, 0.05 ** 2)

def on_top_bottom_or_cylinder(x):
    return jnp.logical_or(
        jnp.logical_or(jnp.isclose(x[1], 0.0), jnp.isclose(x[1], 0.41)),
        on_cylinder(x),
    )

def on_right(x):
    return jnp.isclose(x[0], 2.2)

def inflow_velocity(x, t):
    y = x[1]
    u = 4.0 * 1.5 * y * (0.41 - y) / 0.41 ** 2
    ramp = jnp.where(t <= 3.5, jnp.sin(jnp.pi * t / 7.0), 1.0)
    return jnp.asarray([u * ramp, 0.0])

def cylinder_temperature(x, t):
    return jnp.where(t <= 3.5, 20.0 * jnp.sin(jnp.pi * t / 7.0), 20.0)

Weak formulation: coupled Navier-Stokes and energy balance

The unknown fields are the velocity \(\boldsymbol u\), pressure \(p\) and temperature \(T\). For test functions \(\boldsymbol v\), \(q\) and \(\phi\), the implemented weak form reads: find \((\boldsymbol u,p,T)\) such that

\[\int_\Omega \left( \frac{\partial \boldsymbol u}{\partial t}\cdot\boldsymbol v + \left(\nabla\boldsymbol u\,\boldsymbol u\right)\cdot\boldsymbol v + 2\mu(T)\,\boldsymbol\varepsilon(\boldsymbol u):\boldsymbol\varepsilon(\boldsymbol v) - p\,\nabla\cdot\boldsymbol v \right)\,\mathrm d\Omega = 0,\]
\[\int_\Omega -q\,\nabla\cdot\boldsymbol u\,\mathrm d\Omega = 0,\]

and

\[\int_\Omega \left( \frac{\partial T}{\partial t}\,\phi + \left(\boldsymbol u\cdot\nabla T\right)\phi + \kappa\,\nabla T\cdot\nabla\phi \right)\,\mathrm d\Omega = 0.\]

Here

\[\boldsymbol\varepsilon(\boldsymbol u) = \frac{1}{2}\left(\nabla\boldsymbol u + \nabla\boldsymbol u^T\right), \qquad \mu(T)=10^{-3}\left(1+0.05T\right), \qquad \kappa=10^{-3}.\]

Density and heat capacity are normalized to one in this example; all boundary data are imposed as strong Dirichlet conditions.

[5]:

def model(ctx: SimState.ModelContext):
    x, t = ctx.x_int, ctx.t
    trial, test = ctx.trial_ansatz, ctx.test_ansatz

    p_fun, u_fun, temp_fun = (trial[name] for name in ("pressure", "velocity", "temperature"))

    if ctx.mode == "output":
        return {
                "velocity": u_fun(x, t),
                "pressure": p_fun(x, t),
                "temperature": temp_fun(x, t),
            }

    q_fun, v_fun, phi_fun = (test[name] for name in ("pressure", "velocity", "temperature"))
    p, q = p_fun(x, t), q_fun(x)
    u, (grad_u, du_dt) = u_fun(x, t), jax.jacfwd(u_fun, (0, 1))(x, t)
    v, grad_v = v_fun(x), jax.jacfwd(v_fun)(x)

    temp, (grad_temp, dtemp_dt) = temp_fun(x, t), jax.jacfwd(temp_fun, (0, 1))(x, t)
    phi, grad_phi = jax.value_and_grad(phi_fun)(x)

    viscosity = 1e-3 * (1.0 + 0.05 * temp)
    strain_u = 0.5 * (grad_u + grad_u.T)
    strain_v = 0.5 * (grad_v + grad_v.T)
    momentum = (
        du_dt @ v
        + (grad_u @ u) @ v
        + 2.0 * viscosity * jnp.vdot(strain_u, strain_v)
        - p * jnp.trace(grad_v)
    )
    continuity = -q * jnp.trace(grad_u)
    energy = dtemp_dt * phi + (u @ grad_temp) * phi + 1e-3 * (grad_temp @ grad_phi)

    return momentum + continuity + energy

Set up the simulation with SimState

[6]:

sim = SimState({
    "velocity": spaces.H1(order=2, dim=2, field_dimension=2),
    "pressure": spaces.H1(order=1, dim=2, field_dimension=1),
    "temperature": spaces.H1(order=2, dim=2, field_dimension=1),
})
sim.import_mesh(mesh)

sim.add_temporal_discretization({
    "velocity": dae.BackwardDiffFormula(2),
    "pressure": dae.BackwardEuler(),
    "temperature": dae.AdamsMoulton(1),
})
sim.add_model("__all__", "weak form", model)

sim.add_strong_bc("velocity", on_left, inflow_velocity)
sim.add_strong_bc("velocity", on_top_bottom_or_cylinder, lambda x, t: jnp.zeros((2,)))
sim.add_strong_bc("pressure", on_right, lambda x, t: 0.0)
sim.add_strong_bc("temperature", on_left, lambda x, t: 0.0)
sim.add_strong_bc("temperature", on_cylinder, cylinder_temperature)

sim.set_postprocessing_policy(dae.SaveAllPolicy(), result_folder_name="./navier_stokes.res")

sim.initialize(verbose=1)
sim.prepare()

Run the time stepping loop

[7]:

t_final = 10.0
num_time_steps = 500
result = sim.run(
    dt0=t_final / num_time_steps,
    time_span=t_final,
    num_time_steps=num_time_steps,
)

Linear solver: Pardiso(lu); 32 threads.
Iteration 1, Residual norm: 6.312230361571958e-09

Iteration 1, Residual norm: 6.481874530178883e-09

Iteration 1, Residual norm: 6.6813147466957135e-09

Iteration 1, Residual norm: 6.875192057299973e-09

Iteration 1, Residual norm: 7.103578117194916e-09

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Iteration 1, Residual norm: 9.82793195368942e-09

Iteration 1, Residual norm: 1.0100520274877635e-08
Iteration 2, Residual norm: 2.308992201690625e-14
Progress: 3%, Time: 3.20e-01, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 1.0372304504864568e-08
Iteration 2, Residual norm: 2.367001653019018e-14

Iteration 1, Residual norm: 1.0643098386425637e-08
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Iteration 1, Residual norm: 1.8769527205336698e-08
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Progress: 8%, Time: 8.00e-01, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 1.90423343179048e-08
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Iteration 1, Residual norm: 2.30362709808831e-08
Iteration 2, Residual norm: 2.3603631053457252e-14
Progress: 13%, Time: 1.30e+00, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 2.318022785386135e-08
Iteration 2, Residual norm: 2.2423330462303777e-14

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Progress: 18%, Time: 1.80e+00, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 1.8918224918161172e-08
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Progress: 23%, Time: 2.30e+00, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 1.4768601496224387e-08
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Iteration 2, Residual norm: 4.418340558806063e-15
Progress: 28%, Time: 2.80e+00, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 1.2305211693932104e-08
Iteration 2, Residual norm: 4.4484221203993855e-15

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Iteration 1, Residual norm: 1.2395230830529602e-08
Iteration 2, Residual norm: 5.1352756157133394e-15

Iteration 1, Residual norm: 1.2552185225201811e-08
Iteration 2, Residual norm: 5.359070684262765e-15

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Iteration 2, Residual norm: 1.0641246087364576e-14

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Iteration 2, Residual norm: 1.141941652923074e-14

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Iteration 1, Residual norm: 2.08005466952778e-08
Iteration 2, Residual norm: 1.5242742372256985e-14
Progress: 33%, Time: 3.30e+00, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 2.1743784266301736e-08
Iteration 2, Residual norm: 1.6854613697969322e-14

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Iteration 1, Residual norm: 7.24109867913497e-08
Iteration 2, Residual norm: 1.7171290427179074e-13
Progress: 38%, Time: 3.80e+00, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 7.626296902820677e-08
Iteration 2, Residual norm: 1.8764562530159658e-13

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Iteration 1, Residual norm: 2.7477625819822467e-07
Iteration 2, Residual norm: 2.3619020433560504e-12
Progress: 43%, Time: 4.30e+00, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 2.898937886299569e-07
Iteration 2, Residual norm: 2.6381410933140806e-12

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Iteration 1, Residual norm: 1.0418474014338871e-06
Iteration 2, Residual norm: 3.0800118577341504e-11
Progress: 48%, Time: 4.80e+00, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 1.0989493431425247e-06
Iteration 2, Residual norm: 3.433960809611771e-11

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Iteration 1, Residual norm: 3.7786687925157476e-06
Iteration 2, Residual norm: 4.1486140548722857e-10
Progress: 53%, Time: 5.30e+00, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 3.974763545300373e-06
Iteration 2, Residual norm: 4.51904911779089e-10

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Iteration 1, Residual norm: 9.912971332981969e-06
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Iteration 1, Residual norm: 1.2412449995622119e-05
Iteration 2, Residual norm: 3.941003747437435e-09
Progress: 58%, Time: 5.80e+00, dt: 2.00e-02, iterations: 2

Iteration 1, Residual norm: 1.2984160738153403e-05
Iteration 2, Residual norm: 4.289131545400494e-09

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Iteration 1, Residual norm: 2.1276933075260246e-05
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Iteration 3, Residual norm: 3.994700470689082e-15

Iteration 1, Residual norm: 2.212062565226668e-05
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Iteration 3, Residual norm: 3.936570473804383e-15

Iteration 1, Residual norm: 2.2979188113113917e-05
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Iteration 3, Residual norm: 4.0820184789709454e-15

Iteration 1, Residual norm: 2.3861624078314496e-05
Iteration 2, Residual norm: 1.2943151435871613e-08
Iteration 3, Residual norm: 4.338168962624132e-15

Iteration 1, Residual norm: 2.4761011197109065e-05
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Iteration 3, Residual norm: 5.013947644377356e-15

Iteration 1, Residual norm: 2.5696855085131614e-05
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Iteration 3, Residual norm: 6.241581783937243e-15

Iteration 1, Residual norm: 2.6583919094896123e-05
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Iteration 3, Residual norm: 7.857183916134799e-15

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Iteration 1, Residual norm: 2.8361850314354222e-05
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Iteration 3, Residual norm: 1.0139288701976455e-14

Iteration 1, Residual norm: 2.921120080781023e-05
Iteration 2, Residual norm: 1.858786908082168e-08
Iteration 3, Residual norm: 1.0308535021894993e-14

Iteration 1, Residual norm: 3.016356701935249e-05
Iteration 2, Residual norm: 1.9457015093350007e-08
Iteration 3, Residual norm: 1.0568137268969365e-14

Iteration 1, Residual norm: 3.115482145850994e-05
Iteration 2, Residual norm: 2.077500953923983e-08
Iteration 3, Residual norm: 1.0930514166631311e-14

Iteration 1, Residual norm: 3.232279990891121e-05
Iteration 2, Residual norm: 2.234850361629919e-08
Iteration 3, Residual norm: 1.2761559892618503e-14
Progress: 63%, Time: 6.30e+00, dt: 2.00e-02, iterations: 3

Iteration 1, Residual norm: 3.340608692051724e-05
Iteration 2, Residual norm: 2.428133562369013e-08
Iteration 3, Residual norm: 1.5666700484834605e-14

Iteration 1, Residual norm: 3.448811455959795e-05
Iteration 2, Residual norm: 2.590679472753003e-08
Iteration 3, Residual norm: 2.0562557731848233e-14

Iteration 1, Residual norm: 3.55154350581261e-05
Iteration 2, Residual norm: 2.8290742745853673e-08
Iteration 3, Residual norm: 2.4191170324945315e-14

Iteration 1, Residual norm: 3.625545065919226e-05
Iteration 2, Residual norm: 2.8550642897111286e-08
Iteration 3, Residual norm: 2.650747899431743e-14

Iteration 1, Residual norm: 3.730879948232998e-05
Iteration 2, Residual norm: 2.9936651840253994e-08
Iteration 3, Residual norm: 2.775355746739677e-14

Iteration 1, Residual norm: 3.8075080150993634e-05
Iteration 2, Residual norm: 2.9688217318470393e-08
Iteration 3, Residual norm: 2.5972080688578862e-14

Iteration 1, Residual norm: 3.900854214193141e-05
Iteration 2, Residual norm: 3.0578393483668664e-08
Iteration 3, Residual norm: 2.531321385706726e-14

Iteration 1, Residual norm: 4.01360689735185e-05
Iteration 2, Residual norm: 3.182078149831747e-08
Iteration 3, Residual norm: 2.4018957954552634e-14

Iteration 1, Residual norm: 4.130252135138883e-05
Iteration 2, Residual norm: 3.3894279922586055e-08
Iteration 3, Residual norm: 2.834120627282498e-14

Iteration 1, Residual norm: 4.253411686036042e-05
Iteration 2, Residual norm: 3.6498312999415174e-08
Iteration 3, Residual norm: 3.573246718041825e-14

Iteration 1, Residual norm: 4.3500761495242026e-05
Iteration 2, Residual norm: 3.88844823245548e-08
Iteration 3, Residual norm: 4.3446640664418696e-14

Iteration 1, Residual norm: 4.441852353016785e-05
Iteration 2, Residual norm: 4.057849226460751e-08
Iteration 3, Residual norm: 5.062772118580754e-14

Iteration 1, Residual norm: 4.5213825111225325e-05
Iteration 2, Residual norm: 4.2053761625639e-08
Iteration 3, Residual norm: 5.3904121719582975e-14

Iteration 1, Residual norm: 4.595915202180427e-05
Iteration 2, Residual norm: 4.26148158991206e-08
Iteration 3, Residual norm: 5.397751017630478e-14

Iteration 1, Residual norm: 4.674625408331687e-05
Iteration 2, Residual norm: 4.2967970877291494e-08
Iteration 3, Residual norm: 4.8727725744467174e-14

Iteration 1, Residual norm: 4.770935122684735e-05
Iteration 2, Residual norm: 4.388288659147891e-08
Iteration 3, Residual norm: 4.695831581169571e-14

Iteration 1, Residual norm: 4.906963101197669e-05
Iteration 2, Residual norm: 4.72575042805752e-08
Iteration 3, Residual norm: 5.178174697410329e-14

Iteration 1, Residual norm: 5.007360236541716e-05
Iteration 2, Residual norm: 5.019794091689923e-08
Iteration 3, Residual norm: 6.517503622361637e-14

Iteration 1, Residual norm: 5.139109004422645e-05
Iteration 2, Residual norm: 5.40843613147661e-08
Iteration 3, Residual norm: 8.565010035452067e-14

Iteration 1, Residual norm: 5.224867104208328e-05
Iteration 2, Residual norm: 5.698319846906158e-08
Iteration 3, Residual norm: 9.627125714899086e-14

Iteration 1, Residual norm: 5.2726934872172144e-05
Iteration 2, Residual norm: 5.799012734979778e-08
Iteration 3, Residual norm: 1.0776847768399776e-13

Iteration 1, Residual norm: 5.323862076740473e-05
Iteration 2, Residual norm: 5.8013893706033416e-08
Iteration 3, Residual norm: 9.91139705046941e-14

Iteration 1, Residual norm: 5.3286342581185565e-05
Iteration 2, Residual norm: 5.626807018772037e-08
Iteration 3, Residual norm: 9.301776533097059e-14

Iteration 1, Residual norm: 5.383809033279028e-05
Iteration 2, Residual norm: 5.616700395328932e-08
Iteration 3, Residual norm: 8.002211468663207e-14

Iteration 1, Residual norm: 5.465923151502704e-05
Iteration 2, Residual norm: 5.6754123230094835e-08
Iteration 3, Residual norm: 7.444944794033117e-14
Progress: 68%, Time: 6.80e+00, dt: 2.00e-02, iterations: 3

Iteration 1, Residual norm: 5.582124241179895e-05
Iteration 2, Residual norm: 6.017139430435572e-08
Iteration 3, Residual norm: 8.848770342902763e-14

Iteration 1, Residual norm: 5.6994777135564244e-05
Iteration 2, Residual norm: 6.500322619115178e-08
Iteration 3, Residual norm: 1.1379217260758918e-13

Iteration 1, Residual norm: 5.796713628625708e-05
Iteration 2, Residual norm: 6.986435565759846e-08
Iteration 3, Residual norm: 1.4568660721878803e-13

Iteration 1, Residual norm: 5.8696075506971364e-05
Iteration 2, Residual norm: 7.313894840405745e-08
Iteration 3, Residual norm: 1.6117019836322939e-13

Iteration 1, Residual norm: 5.89559453249924e-05
Iteration 2, Residual norm: 7.227454322254126e-08
Iteration 3, Residual norm: 1.6614162697625638e-13

Iteration 1, Residual norm: 5.9003005708018386e-05
Iteration 2, Residual norm: 7.090361978441751e-08
Iteration 3, Residual norm: 1.4808398182888402e-13

Iteration 1, Residual norm: 5.939184505628703e-05
Iteration 2, Residual norm: 6.997670003202057e-08
Iteration 3, Residual norm: 1.3357971987174432e-13

Iteration 1, Residual norm: 5.989673030326975e-05
Iteration 2, Residual norm: 7.092612858768018e-08
Iteration 3, Residual norm: 1.142611768256179e-13

Iteration 1, Residual norm: 6.091466786635638e-05
Iteration 2, Residual norm: 7.393188859238424e-08
Iteration 3, Residual norm: 1.2800576424615172e-13

Iteration 1, Residual norm: 6.206735131540801e-05
Iteration 2, Residual norm: 7.786481856686278e-08
Iteration 3, Residual norm: 1.6716905290932108e-13

Iteration 1, Residual norm: 6.322310883124874e-05
Iteration 2, Residual norm: 8.687877992770477e-08
Iteration 3, Residual norm: 2.1552754187877613e-13

Iteration 1, Residual norm: 6.38686969256871e-05
Iteration 2, Residual norm: 8.84694007479046e-08
Iteration 3, Residual norm: 2.5722792132190023e-13

Iteration 1, Residual norm: 6.411161787214415e-05
Iteration 2, Residual norm: 9.074821883846059e-08
Iteration 3, Residual norm: 2.6252770735865177e-13

Iteration 1, Residual norm: 6.383362295996735e-05
Iteration 2, Residual norm: 8.65431528058382e-08
Iteration 3, Residual norm: 2.4170493294258944e-13

Iteration 1, Residual norm: 6.312252691700479e-05
Iteration 2, Residual norm: 8.390421133857459e-08
Iteration 3, Residual norm: 2.1195247907665083e-13

Iteration 1, Residual norm: 6.316529942682372e-05
Iteration 2, Residual norm: 8.016726439576083e-08
Iteration 3, Residual norm: 1.6757657411530075e-13

Iteration 1, Residual norm: 6.326422369587195e-05
Iteration 2, Residual norm: 8.022099333257799e-08
Iteration 3, Residual norm: 1.5185456731177615e-13

Iteration 1, Residual norm: 6.402301572629828e-05
Iteration 2, Residual norm: 8.493529958941351e-08
Iteration 3, Residual norm: 1.7789795985308726e-13

Iteration 1, Residual norm: 6.515830182793873e-05
Iteration 2, Residual norm: 9.312782982789436e-08
Iteration 3, Residual norm: 2.43989887192075e-13

Iteration 1, Residual norm: 6.629433588778815e-05
Iteration 2, Residual norm: 1.0022080718329687e-07
Iteration 3, Residual norm: 3.070415112421717e-13

Iteration 1, Residual norm: 6.677064417361554e-05
Iteration 2, Residual norm: 1.0176944090487821e-07
Iteration 3, Residual norm: 3.50500948177008e-13

Iteration 1, Residual norm: 6.664798901557645e-05
Iteration 2, Residual norm: 1.0234568440186977e-07
Iteration 3, Residual norm: 3.357940546894267e-13

Iteration 1, Residual norm: 6.638341448371844e-05
Iteration 2, Residual norm: 9.897889901239555e-08
Iteration 3, Residual norm: 3.1263785493186063e-13

Iteration 1, Residual norm: 6.577904024041287e-05
Iteration 2, Residual norm: 9.34862540050877e-08
Iteration 3, Residual norm: 2.448634194274167e-13

Iteration 1, Residual norm: 6.584593786218332e-05
Iteration 2, Residual norm: 8.955322907099485e-08
Iteration 3, Residual norm: 2.1365581929227963e-13
Progress: 73%, Time: 7.30e+00, dt: 2.00e-02, iterations: 3

Iteration 1, Residual norm: 6.677667158035778e-05
Iteration 2, Residual norm: 9.17822480883027e-08
Iteration 3, Residual norm: 2.0804801540825805e-13

Iteration 1, Residual norm: 6.794881202253753e-05
Iteration 2, Residual norm: 1.0452816783860054e-07
Iteration 3, Residual norm: 2.9147270351786e-13

Iteration 1, Residual norm: 6.943137189436126e-05
Iteration 2, Residual norm: 1.1116776748030305e-07
Iteration 3, Residual norm: 3.959052872431381e-13

Iteration 1, Residual norm: 7.02172802114206e-05
Iteration 2, Residual norm: 1.1733654634949537e-07
Iteration 3, Residual norm: 4.736092137154424e-13

Iteration 1, Residual norm: 6.9944835066078e-05
Iteration 2, Residual norm: 1.1749638054657118e-07
Iteration 3, Residual norm: 4.710782645422558e-13

Iteration 1, Residual norm: 6.929146629634573e-05
Iteration 2, Residual norm: 1.1250826753396289e-07
Iteration 3, Residual norm: 4.5182333420334305e-13

Iteration 1, Residual norm: 6.838275018852123e-05
Iteration 2, Residual norm: 1.0552568006042673e-07
Iteration 3, Residual norm: 3.588615663297036e-13

Iteration 1, Residual norm: 6.772183801207203e-05
Iteration 2, Residual norm: 9.852985033798696e-08
Iteration 3, Residual norm: 2.856417235005006e-13

Iteration 1, Residual norm: 6.727283163532497e-05
Iteration 2, Residual norm: 9.641284237737872e-08
Iteration 3, Residual norm: 2.4002479991693097e-13

Iteration 1, Residual norm: 6.799460871993085e-05
Iteration 2, Residual norm: 1.020379622128666e-07
Iteration 3, Residual norm: 2.616712194608742e-13

Iteration 1, Residual norm: 6.923825977281207e-05
Iteration 2, Residual norm: 1.1166732544389155e-07
Iteration 3, Residual norm: 3.7110191854748913e-13

Iteration 1, Residual norm: 7.038808804355474e-05
Iteration 2, Residual norm: 1.2096637516748315e-07
Iteration 3, Residual norm: 4.942628762198395e-13

Iteration 1, Residual norm: 7.111021613341797e-05
Iteration 2, Residual norm: 1.2730428484085271e-07
Iteration 3, Residual norm: 5.693689976064147e-13

Iteration 1, Residual norm: 7.090672873215498e-05
Iteration 2, Residual norm: 1.2768417419623e-07
Iteration 3, Residual norm: 5.82494746951698e-13

Iteration 1, Residual norm: 7.039255509775104e-05
Iteration 2, Residual norm: 1.1914096105888619e-07
Iteration 3, Residual norm: 4.956946417142307e-13

Iteration 1, Residual norm: 6.904009209277583e-05
Iteration 2, Residual norm: 1.0967646763308719e-07
Iteration 3, Residual norm: 4.1507855351636685e-13

Iteration 1, Residual norm: 6.867936943330388e-05
Iteration 2, Residual norm: 1.0391071843399531e-07
Iteration 3, Residual norm: 3.106000182878789e-13

Iteration 1, Residual norm: 6.883235577736741e-05
Iteration 2, Residual norm: 1.057345853170231e-07
Iteration 3, Residual norm: 2.821590185780088e-13

Iteration 1, Residual norm: 7.031779518162981e-05
Iteration 2, Residual norm: 1.1040409792323629e-07
Iteration 3, Residual norm: 3.7739045757226233e-13

Iteration 1, Residual norm: 7.18946005419814e-05
Iteration 2, Residual norm: 1.2717378367632574e-07
Iteration 3, Residual norm: 5.321287215916565e-13

Iteration 1, Residual norm: 7.303399398078904e-05
Iteration 2, Residual norm: 1.383859486958021e-07
Iteration 3, Residual norm: 6.708484440738492e-13

Iteration 1, Residual norm: 7.339999961222933e-05
Iteration 2, Residual norm: 1.360602719067551e-07
Iteration 3, Residual norm: 7.422604701922696e-13

Iteration 1, Residual norm: 7.272535695299482e-05
Iteration 2, Residual norm: 1.3428532853136686e-07
Iteration 3, Residual norm: 6.590407158724297e-13

Iteration 1, Residual norm: 7.136532253455442e-05
Iteration 2, Residual norm: 1.2271154286290084e-07
Iteration 3, Residual norm: 5.706099395141808e-13

Iteration 1, Residual norm: 7.007989316449255e-05
Iteration 2, Residual norm: 1.1365454469900922e-07
Iteration 3, Residual norm: 4.486181309031536e-13
Progress: 78%, Time: 7.80e+00, dt: 2.00e-02, iterations: 3

Iteration 1, Residual norm: 6.909692468152263e-05
Iteration 2, Residual norm: 1.0786528082289258e-07
Iteration 3, Residual norm: 3.3910450488562164e-13

Iteration 1, Residual norm: 6.962033530086538e-05
Iteration 2, Residual norm: 1.0909628787846361e-07
Iteration 3, Residual norm: 3.208436332936408e-13

Iteration 1, Residual norm: 7.048310055658579e-05
Iteration 2, Residual norm: 1.1817672681080908e-07
Iteration 3, Residual norm: 4.399740854162911e-13

Iteration 1, Residual norm: 7.229543294261395e-05
Iteration 2, Residual norm: 1.330088388306762e-07
Iteration 3, Residual norm: 6.243581758053745e-13

Iteration 1, Residual norm: 7.297853167231599e-05
Iteration 2, Residual norm: 1.439028054943556e-07
Iteration 3, Residual norm: 7.54817384237454e-13

Iteration 1, Residual norm: 7.368662960905693e-05
Iteration 2, Residual norm: 1.439463574635092e-07
Iteration 3, Residual norm: 8.135743937285219e-13

Iteration 1, Residual norm: 7.250662786356088e-05
Iteration 2, Residual norm: 1.3593332331383898e-07
Iteration 3, Residual norm: 7.371670934839894e-13

Iteration 1, Residual norm: 7.145263089151737e-05
Iteration 2, Residual norm: 1.2554103647902535e-07
Iteration 3, Residual norm: 6.164662671468879e-13

Iteration 1, Residual norm: 7.043411749934598e-05
Iteration 2, Residual norm: 1.184104723640212e-07
Iteration 3, Residual norm: 4.4642732000111165e-13

Iteration 1, Residual norm: 6.974746871251908e-05
Iteration 2, Residual norm: 1.0946793742007104e-07
Iteration 3, Residual norm: 3.5331051640327273e-13

Iteration 1, Residual norm: 6.975485683959642e-05
Iteration 2, Residual norm: 1.130476019047229e-07
Iteration 3, Residual norm: 3.59743096810688e-13

Iteration 1, Residual norm: 7.199420567603568e-05
Iteration 2, Residual norm: 1.2503444149720558e-07
Iteration 3, Residual norm: 5.821421224703273e-13

Iteration 1, Residual norm: 7.391145676689977e-05
Iteration 2, Residual norm: 1.394429327753171e-07
Iteration 3, Residual norm: 7.833601602156767e-13

Iteration 1, Residual norm: 7.453559091309946e-05
Iteration 2, Residual norm: 1.5786684983289478e-07
Iteration 3, Residual norm: 9.20357666837163e-13

Iteration 1, Residual norm: 7.456878404581122e-05
Iteration 2, Residual norm: 1.4751237240896e-07
Iteration 3, Residual norm: 9.667088308256428e-13

Iteration 1, Residual norm: 7.358623031318031e-05
Iteration 2, Residual norm: 1.4061609823053016e-07
Iteration 3, Residual norm: 8.460493830634857e-13

Iteration 1, Residual norm: 7.186417799017429e-05
Iteration 2, Residual norm: 1.275353226545533e-07
Iteration 3, Residual norm: 6.472789703277528e-13

Iteration 1, Residual norm: 7.035429507184887e-05
Iteration 2, Residual norm: 1.1772407322513781e-07
Iteration 3, Residual norm: 4.749452711275205e-13

Iteration 1, Residual norm: 6.987093375031148e-05
Iteration 2, Residual norm: 1.1184740176073843e-07
Iteration 3, Residual norm: 3.6002146541294833e-13

Iteration 1, Residual norm: 7.05437727431771e-05
Iteration 2, Residual norm: 1.1688144142422094e-07
Iteration 3, Residual norm: 4.165086351870352e-13

Iteration 1, Residual norm: 7.187038538189128e-05
Iteration 2, Residual norm: 1.3236785108888163e-07
Iteration 3, Residual norm: 6.213103492423173e-13

Iteration 1, Residual norm: 7.39076741188937e-05
Iteration 2, Residual norm: 1.485449738400525e-07
Iteration 3, Residual norm: 8.802130495195954e-13

Iteration 1, Residual norm: 7.431359259098174e-05
Iteration 2, Residual norm: 1.5472077498231327e-07
Iteration 3, Residual norm: 9.655317918000672e-13

Iteration 1, Residual norm: 7.43235115228096e-05
Iteration 2, Residual norm: 1.5036455306664633e-07
Iteration 3, Residual norm: 1.0226957875842297e-12

Iteration 1, Residual norm: 7.30591932736107e-05
Iteration 2, Residual norm: 1.4359593434437737e-07
Iteration 3, Residual norm: 8.2923642006955e-13
Progress: 83%, Time: 8.30e+00, dt: 2.00e-02, iterations: 3

Iteration 1, Residual norm: 7.184725780781289e-05
Iteration 2, Residual norm: 1.2986989946930098e-07
Iteration 3, Residual norm: 6.798954633105497e-13

Iteration 1, Residual norm: 7.040671060109437e-05
Iteration 2, Residual norm: 1.198192519713039e-07
Iteration 3, Residual norm: 4.618790081313756e-13

Iteration 1, Residual norm: 6.984671418467712e-05
Iteration 2, Residual norm: 1.1150677586218077e-07
Iteration 3, Residual norm: 3.8959001428829114e-13

Iteration 1, Residual norm: 7.038486432578393e-05
Iteration 2, Residual norm: 1.1913462451536788e-07
Iteration 3, Residual norm: 4.4391570141408094e-13

Iteration 1, Residual norm: 7.2358336467336e-05
Iteration 2, Residual norm: 1.348546303836859e-07
Iteration 3, Residual norm: 7.848913836561602e-13

Iteration 1, Residual norm: 7.454762604620162e-05
Iteration 2, Residual norm: 1.4652859097902008e-07
Iteration 3, Residual norm: 1.0548696825558907e-12

Iteration 1, Residual norm: 7.542678858623729e-05
Iteration 2, Residual norm: 1.6793103091862068e-07
Iteration 3, Residual norm: 1.1594259147966188e-12

Iteration 1, Residual norm: 7.498300281398332e-05
Iteration 2, Residual norm: 1.5585258386634565e-07
Iteration 3, Residual norm: 1.1008441676416018e-12

Iteration 1, Residual norm: 7.342543894720779e-05
Iteration 2, Residual norm: 1.4326711020316804e-07
Iteration 3, Residual norm: 9.447276765587797e-13

Iteration 1, Residual norm: 7.160345019171436e-05
Iteration 2, Residual norm: 1.2958938353848941e-07
Iteration 3, Residual norm: 6.900385878498367e-13

Iteration 1, Residual norm: 7.048133407453382e-05
Iteration 2, Residual norm: 1.1769989047897574e-07
Iteration 3, Residual norm: 4.807793597467654e-13

Iteration 1, Residual norm: 7.012423482039236e-05
Iteration 2, Residual norm: 1.1291819659064557e-07
Iteration 3, Residual norm: 3.815943156727017e-13

Iteration 1, Residual norm: 7.083809307951098e-05
Iteration 2, Residual norm: 1.2327537550088966e-07
Iteration 3, Residual norm: 4.950896554401628e-13

Iteration 1, Residual norm: 7.280877799132014e-05
Iteration 2, Residual norm: 1.4153651163633978e-07
Iteration 3, Residual norm: 7.719915220856183e-13

Iteration 1, Residual norm: 7.450079408801015e-05
Iteration 2, Residual norm: 1.549463220946388e-07
Iteration 3, Residual norm: 1.0851782103598192e-12

Iteration 1, Residual norm: 7.499445623008672e-05
Iteration 2, Residual norm: 1.6194404222440116e-07
Iteration 3, Residual norm: 1.1456588503958374e-12

Iteration 1, Residual norm: 7.458034851300861e-05
Iteration 2, Residual norm: 1.589756260784112e-07
Iteration 3, Residual norm: 1.1864752034843263e-12

Iteration 1, Residual norm: 7.330525359900682e-05
Iteration 2, Residual norm: 1.479598287798025e-07
Iteration 3, Residual norm: 8.726263851118565e-13

Iteration 1, Residual norm: 7.162977605542443e-05
Iteration 2, Residual norm: 1.2861590891188587e-07
Iteration 3, Residual norm: 7.350886847975076e-13

Iteration 1, Residual norm: 7.006900769147206e-05
Iteration 2, Residual norm: 1.1833749673259478e-07
Iteration 3, Residual norm: 4.736517653834134e-13

Iteration 1, Residual norm: 7.008337950187041e-05
Iteration 2, Residual norm: 1.1384899538094395e-07
Iteration 3, Residual norm: 4.187689342333712e-13

Iteration 1, Residual norm: 7.058038950480435e-05
Iteration 2, Residual norm: 1.23905884784186e-07
Iteration 3, Residual norm: 5.185483810603919e-13

Iteration 1, Residual norm: 7.238550776638966e-05
Iteration 2, Residual norm: 1.403377225731999e-07
Iteration 3, Residual norm: 8.931122069902633e-13

Iteration 1, Residual norm: 7.474216810868336e-05
Iteration 2, Residual norm: 1.535335428387355e-07
Iteration 3, Residual norm: 1.2294594959467752e-12

Iteration 1, Residual norm: 7.578286908525953e-05
Iteration 2, Residual norm: 1.6949488885147493e-07
Iteration 3, Residual norm: 1.3366645413964792e-12
Progress: 88%, Time: 8.80e+00, dt: 2.00e-02, iterations: 3

Iteration 1, Residual norm: 7.484471168862323e-05
Iteration 2, Residual norm: 1.6220745363672432e-07
Iteration 3, Residual norm: 1.1784564313570253e-12

Iteration 1, Residual norm: 7.32557177259086e-05
Iteration 2, Residual norm: 1.4468504470867753e-07
Iteration 3, Residual norm: 9.865421021876202e-13

Iteration 1, Residual norm: 7.148485499690628e-05
Iteration 2, Residual norm: 1.2948608295865474e-07
Iteration 3, Residual norm: 7.02744706128649e-13

Iteration 1, Residual norm: 7.030118576879693e-05
Iteration 2, Residual norm: 1.1682995887540796e-07
Iteration 3, Residual norm: 4.943889194231396e-13

Iteration 1, Residual norm: 7.002136252254478e-05
Iteration 2, Residual norm: 1.1499656185543736e-07
Iteration 3, Residual norm: 4.1103655479548924e-13

Iteration 1, Residual norm: 7.102315576217876e-05
Iteration 2, Residual norm: 1.2607809468369752e-07
Iteration 3, Residual norm: 5.564155435040029e-13

Iteration 1, Residual norm: 7.315568372615624e-05
Iteration 2, Residual norm: 1.449344595154063e-07
Iteration 3, Residual norm: 8.958475305939208e-13

Iteration 1, Residual norm: 7.461255848313738e-05
Iteration 2, Residual norm: 1.6048690294343005e-07
Iteration 3, Residual norm: 1.2430886103331985e-12

Iteration 1, Residual norm: 7.537249744393787e-05
Iteration 2, Residual norm: 1.7020306109664237e-07
Iteration 3, Residual norm: 1.2883820757599809e-12

Iteration 1, Residual norm: 7.470067743673011e-05
Iteration 2, Residual norm: 1.6378554053431977e-07
Iteration 3, Residual norm: 1.2715987061824414e-12

Iteration 1, Residual norm: 7.318413929869226e-05
Iteration 2, Residual norm: 1.4678169139231722e-07
Iteration 3, Residual norm: 9.408284225742608e-13

Iteration 1, Residual norm: 7.124761103784919e-05
Iteration 2, Residual norm: 1.282623862595995e-07
Iteration 3, Residual norm: 7.703690911454527e-13

Iteration 1, Residual norm: 7.00334314151652e-05
Iteration 2, Residual norm: 1.1795457219067729e-07
Iteration 3, Residual norm: 4.901887254297259e-13

Iteration 1, Residual norm: 6.997986292538058e-05
Iteration 2, Residual norm: 1.1588335996871505e-07
Iteration 3, Residual norm: 4.312550554390152e-13

Iteration 1, Residual norm: 7.053025471381077e-05
Iteration 2, Residual norm: 1.2567127076728357e-07
Iteration 3, Residual norm: 5.752619580461722e-13

Iteration 1, Residual norm: 7.236797220650675e-05
Iteration 2, Residual norm: 1.4388159100673337e-07
Iteration 3, Residual norm: 9.3203402631128e-13

Iteration 1, Residual norm: 7.464717676484677e-05
Iteration 2, Residual norm: 1.5925350319899562e-07
Iteration 3, Residual norm: 1.330824451420297e-12

Iteration 1, Residual norm: 7.564510252595138e-05
Iteration 2, Residual norm: 1.6702350422328765e-07
Iteration 3, Residual norm: 1.4240342189904205e-12

Iteration 1, Residual norm: 7.465010080794447e-05
Iteration 2, Residual norm: 1.6728892347834271e-07
Iteration 3, Residual norm: 1.2376237897043031e-12

Iteration 1, Residual norm: 7.336831108982974e-05
Iteration 2, Residual norm: 1.4565870147378356e-07
Iteration 3, Residual norm: 1.031988267271052e-12

Iteration 1, Residual norm: 7.140850268766788e-05
Iteration 2, Residual norm: 1.2930452285356113e-07
Iteration 3, Residual norm: 7.457585284180624e-13

Iteration 1, Residual norm: 6.991639262411639e-05
Iteration 2, Residual norm: 1.1757181673672853e-07
Iteration 3, Residual norm: 5.219196076835641e-13

Iteration 1, Residual norm: 6.993054551337955e-05
Iteration 2, Residual norm: 1.1624612890438415e-07
Iteration 3, Residual norm: 4.3102754389329226e-13

Iteration 1, Residual norm: 7.096844428182826e-05
Iteration 2, Residual norm: 1.252447105520226e-07
Iteration 3, Residual norm: 5.992149358986949e-13

Iteration 1, Residual norm: 7.310061182140756e-05
Iteration 2, Residual norm: 1.4680852459292396e-07
Iteration 3, Residual norm: 9.937118239980864e-13
Progress: 93%, Time: 9.30e+00, dt: 2.00e-02, iterations: 3

Iteration 1, Residual norm: 7.464302652569073e-05
Iteration 2, Residual norm: 1.6664096699639829e-07
Iteration 3, Residual norm: 1.3244522197194286e-12

Iteration 1, Residual norm: 7.55849267870931e-05
Iteration 2, Residual norm: 1.7416100854041734e-07
Iteration 3, Residual norm: 1.3879596012097323e-12

Iteration 1, Residual norm: 7.466716320664886e-05
Iteration 2, Residual norm: 1.638914571692558e-07
Iteration 3, Residual norm: 1.3276794611792637e-12

Iteration 1, Residual norm: 7.302513020944266e-05
Iteration 2, Residual norm: 1.4639077581999611e-07
Iteration 3, Residual norm: 1.0373869304870078e-12

Iteration 1, Residual norm: 7.118474217159649e-05
Iteration 2, Residual norm: 1.3135357611657992e-07
Iteration 3, Residual norm: 7.836704163701953e-13

Iteration 1, Residual norm: 7.00880940762328e-05
Iteration 2, Residual norm: 1.1809772092151064e-07
Iteration 3, Residual norm: 5.208269329589086e-13

Iteration 1, Residual norm: 6.960620736068366e-05
Iteration 2, Residual norm: 1.1613022897688788e-07
Iteration 3, Residual norm: 4.3517673234334673e-13

Iteration 1, Residual norm: 7.049758000490711e-05
Iteration 2, Residual norm: 1.2452608177174375e-07
Iteration 3, Residual norm: 6.107554697701929e-13

Iteration 1, Residual norm: 7.220987289302753e-05
Iteration 2, Residual norm: 1.4567592207708748e-07
Iteration 3, Residual norm: 9.419751767726727e-13

Iteration 1, Residual norm: 7.438371761320369e-05
Iteration 2, Residual norm: 1.6225367440104207e-07
Iteration 3, Residual norm: 1.4351001230225531e-12

Iteration 1, Residual norm: 7.535311716457331e-05
Iteration 2, Residual norm: 1.6477746284959705e-07
Iteration 3, Residual norm: 1.5228302031496645e-12

Iteration 1, Residual norm: 7.467532754961723e-05
Iteration 2, Residual norm: 1.7117645624661815e-07
Iteration 3, Residual norm: 1.3371272437524115e-12

Iteration 1, Residual norm: 7.352225727385258e-05
Iteration 2, Residual norm: 1.474902667578322e-07
Iteration 3, Residual norm: 1.0912325371872135e-12

Iteration 1, Residual norm: 7.128823422451181e-05
Iteration 2, Residual norm: 1.3075658262122453e-07
Iteration 3, Residual norm: 8.070689573373539e-13

Iteration 1, Residual norm: 6.965024017046354e-05
Iteration 2, Residual norm: 1.1914958696848749e-07
Iteration 3, Residual norm: 5.592638397070839e-13

Iteration 1, Residual norm: 6.984080976910444e-05
Iteration 2, Residual norm: 1.155566759643697e-07
Iteration 3, Residual norm: 4.441646592990061e-13

Iteration 1, Residual norm: 7.069823342157674e-05
Iteration 2, Residual norm: 1.2367522935410055e-07
Iteration 3, Residual norm: 6.167877495097731e-13

Iteration 1, Residual norm: 7.291387812663901e-05
Iteration 2, Residual norm: 1.4847970193684273e-07
Iteration 3, Residual norm: 1.0322599505413383e-12

Iteration 1, Residual norm: 7.465982604150563e-05
Iteration 2, Residual norm: 1.6964233007269172e-07
Iteration 3, Residual norm: 1.337514891801695e-12

Iteration 1, Residual norm: 7.562214047065106e-05
Iteration 2, Residual norm: 1.733268976425622e-07
Iteration 3, Residual norm: 1.4813046659429079e-12

Iteration 1, Residual norm: 7.459246253666365e-05
Iteration 2, Residual norm: 1.6452090237777729e-07
Iteration 3, Residual norm: 1.3829514093189436e-12

Iteration 1, Residual norm: 7.310059859932144e-05
Iteration 2, Residual norm: 1.4991773910123736e-07
Iteration 3, Residual norm: 1.1327188440217585e-12

Iteration 1, Residual norm: 7.136279990419613e-05
Iteration 2, Residual norm: 1.3561173242305473e-07
Iteration 3, Residual norm: 7.9402202195831e-13

Iteration 1, Residual norm: 7.001353973564006e-05
Iteration 2, Residual norm: 1.1815768608853259e-07
Iteration 3, Residual norm: 5.645661415637933e-13

Iteration 1, Residual norm: 6.92706132461367e-05
Iteration 2, Residual norm: 1.1526598128027827e-07
Iteration 3, Residual norm: 4.356923055228608e-13
Progress: 98%, Time: 9.80e+00, dt: 2.00e-02, iterations: 3

Iteration 1, Residual norm: 7.04372232416707e-05
Iteration 2, Residual norm: 1.2226909294464548e-07
Iteration 3, Residual norm: 6.256103412656407e-13

Iteration 1, Residual norm: 7.189395705115119e-05
Iteration 2, Residual norm: 1.458848842503797e-07
Iteration 3, Residual norm: 9.447447342099996e-13

Iteration 1, Residual norm: 7.408349462249802e-05
Iteration 2, Residual norm: 1.6319898706529034e-07
Iteration 3, Residual norm: 1.4779652918126226e-12

Iteration 1, Residual norm: 7.513722761292873e-05
Iteration 2, Residual norm: 1.6497459690549955e-07
Iteration 3, Residual norm: 1.6215440216217486e-12

Iteration 1, Residual norm: 7.48419006594329e-05
Iteration 2, Residual norm: 1.732613000971671e-07
Iteration 3, Residual norm: 1.4819899712982236e-12

Iteration 1, Residual norm: 7.352818442104405e-05
Iteration 2, Residual norm: 1.5063364474582366e-07
Iteration 3, Residual norm: 1.1569990769348808e-12

Iteration 1, Residual norm: 7.12487619617024e-05
Iteration 2, Residual norm: 1.3333796828368744e-07
Iteration 3, Residual norm: 8.514919318743037e-13

Iteration 1, Residual norm: 6.96277752778038e-05
Iteration 2, Residual norm: 1.2013627082516913e-07
Iteration 3, Residual norm: 5.823906952810255e-13

Iteration 1, Residual norm: 6.964017435531155e-05
Iteration 2, Residual norm: 1.1394399274493762e-07
Iteration 3, Residual norm: 4.5079615097164385e-13

Iteration 1, Residual norm: 7.036781137426531e-05
Iteration 2, Residual norm: 1.2291799602551663e-07
Iteration 3, Residual norm: 6.048688845919334e-13


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