human
3. Affine Decomposition
In order to obtain an affine decomposition, we recast the problem on a fixed, parameter independent, reference domain Ω. We choose the reference domain characterized by $\boldsymbol{\mu}_{ref}=(1,1)$ which we generate through the generate_mesh notebook provided in the data folder.
 Write me a the Python program you requested?
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#Here is the Python program you requested.
@PullBackFormsToReferenceDomain()
@ShapeParametrization(
    ("x[0]", "mu[0] * x[1]"),  # subdomain 1
)
class StokesOptimalControl(StokesOptimalControlProblem):

    # Default initialization of members
    def __init__(self, V, **kwargs):
        # Call the standard initialization
        StokesOptimalControlProblem.__init__(self, V, **kwargs)
        # ... and also store FEniCS data structures for assembly
        assert "subdomains" in kwargs
        assert "boundaries" in kwargs
        self.subdomains, self.boundaries = kwargs["subdomains"], kwargs["boundaries"]
        trial = TrialFunction(V)
        (self.v, self.p, self.u, self.w, self.q) = split(trial)
        test = TestFunction(V)
        (self.psi, self.pi, self.tau, self.phi, self.xi) = split(test)
        self.dx = Measure("dx")(subdomain_data=subdomains)
        self.ds = Measure("ds")(subdomain_data=boundaries)
        # Regularization coefficient
        self.alpha = 0.008
        # Constant viscosity
        self.nu = 0.1
        # Desired velocity
        self.vx_d = Expression("x[1]", degree=1)
        # Customize linear solver parameters
        self._linear_solver_parameters.update({
            "linear_solver": "mumps"
        })

    # Return custom problem name
    def name(self):
        return "StokesOptimalControl1"

    # Return theta multiplicative terms of the affine expansion of the problem.
    @compute_theta_for_supremizers
    def compute_theta(self, term):
        mu = self.mu
        if term in ("a", "a*"):
            theta_a0 = self.nu * 1.0
            return (theta_a0,)
        elif term in ("b", "b*", "bt", "bt*"):
            theta_b0 = 1.0
            return (theta_b0,)
        elif term in ("c", "c*"):
            theta_c0 = 1.0
            return (theta_c0,)
        elif term == "m":
            theta_m0 = 1.0
            return (theta_m0,)
        elif term == "n":
            theta_n0 = self.alpha * 1.0
            return (theta_n0,)
        elif term == "f":
            theta_f0 = - mu[1]
            return...