Stokes integrators

stokes_integrators

This module provides a fine-tuned solver class for the Stokes system of conservation equations and a function to automatically set the associated null spaces. Users instantiate the StokesSolver class by providing relevant parameters and call the solve method to request a solver update.

iterative_stokes_solver_parameters = {'mat_type': 'matfree', 'ksp_type': 'preonly', 'pc_type': 'fieldsplit', 'pc_fieldsplit_type': 'schur', 'pc_fieldsplit_schur_type': 'full', 'fieldsplit_0': {'ksp_type': 'cg', 'ksp_rtol': 1e-05, 'ksp_max_it': 1000, 'pc_type': 'python', 'pc_python_type': 'gadopt.SPDAssembledPC', 'assembled_pc_type': 'gamg', 'assembled_mg_levels_pc_type': 'sor', 'assembled_pc_gamg_threshold': 0.01, 'assembled_pc_gamg_square_graph': 100, 'assembled_pc_gamg_coarse_eq_limit': 1000, 'assembled_pc_gamg_mis_k_minimum_degree_ordering': True}, 'fieldsplit_1': {'ksp_type': 'fgmres', 'ksp_rtol': 0.0001, 'ksp_max_it': 200, 'pc_type': 'python', 'pc_python_type': 'firedrake.MassInvPC', 'Mp_pc_type': 'ksp', 'Mp_ksp_ksp_rtol': 1e-05, 'Mp_ksp_ksp_type': 'cg', 'Mp_ksp_pc_type': 'sor'}} module-attribute

Default iterative solver parameters for solution of stokes system.

We configure the Schur complement approach as described in Section of 4.3 of Davies et al. (2022), using PETSc's fieldsplit preconditioner type, which provides a class of preconditioners for mixed problems that allows a user to apply different preconditioners to different blocks of the system.

The fieldsplit_0 entries configure solver options for the velocity block. The linear systems associated with this matrix are solved using a combination of the Conjugate Gradient (cg) method and an algebraic multigrid preconditioner (GAMG).

The fieldsplit_1 entries contain solver options for the Schur complement solve itself. For preconditioning, we approximate the Schur complement matrix with a mass matrix scaled by viscosity, with the viscosity provided through the optional mu keyword argument to Stokes solver. Since this preconditioner step involves an iterative solve, the Krylov method used for the Schur complement needs to be of flexible type, and we use FGMRES by default.

We note that our default solver parameters can be augmented or adjusted by accessing the solver_parameter dictionary, for example:

   stokes_solver.solver_parameters['fieldsplit_0']['ksp_converged_reason'] = None
   stokes_solver.solver_parameters['fieldsplit_0']['ksp_rtol'] = 1e-3
   stokes_solver.solver_parameters['fieldsplit_0']['assembled_pc_gamg_threshold'] = -1
   stokes_solver.solver_parameters['fieldsplit_1']['ksp_converged_reason'] = None
   stokes_solver.solver_parameters['fieldsplit_1']['ksp_rtol'] = 1e-2

Note

G-ADOPT defaults to iterative solvers in 3-D.

direct_stokes_solver_parameters = {'mat_type': 'aij', 'ksp_type': 'preonly', 'pc_type': 'lu', 'pc_factor_mat_solver_type': 'mumps'} module-attribute

Default direct solver parameters for solution of Stokes system.

Configured to use LU factorisation, using the MUMPS library.

Note

G-ADOPT defaults to direct solvers in 2-D.

newton_stokes_solver_parameters = {'snes_type': 'newtonls', 'snes_linesearch_type': 'l2', 'snes_max_it': 100, 'snes_atol': 1e-10, 'snes_rtol': 1e-05} module-attribute

Default solver parameters for non-linear systems.

We use a setup based on Newton's method (newtonls) with a secant line search over the L2-norm of the function.

StokesSolver(solution, T, approximation, bcs={}, quad_degree=6, solver_parameters=None, J=None, constant_jacobian=False, coupled_tstep=None, theta=0.5, nullspace=None, transpose_nullspace=None, near_nullspace=None)

Solves the Stokes system in place.

Note: Null space options can be generated using create_stokes_nullspace.

Parameters:

Name	Type	Description	Default
solution	Function	Firedrake function representing mixed Stokes system	required
T	Function	Firedrake function representing temperature	required
approximation	BaseApproximation	Approximation describing system of equations	required
bcs	dict[int, dict[str, Number]]	Dictionary of identifier-value pairs specifying boundary conditions	{}
quad_degree	int	Quadrature degree. Default value is 2p + 1, where p is the polynomial degree of the trial space	6
solver_parameters	Optional[dict[str, str | Number] | str]	Either a dictionary of PETSc solver parameters or a string specifying a default set of parameters defined in G-ADOPT	None
J	Optional[Function]	Firedrake function representing the Jacobian of the system	None
constant_jacobian	bool	Whether the Jacobian of the system is constant	False
coupled_tstep	Optional[float]	Timestep for advancing free surface equation	None
theta	float	Timestepping prefactor for free surface equation, where theta = 0: Forward Euler, theta = 0.5: Crank-Nicolson (default), or theta = 1: Backward Euler	0.5
nullspace	MixedVectorSpaceBasis	A MixedVectorSpaceBasis for the operator's kernel	None
transpose_nullspace	MixedVectorSpaceBasis	A MixedVectorSpaceBasis for the kernel of the operator's transpose	None
near_nullspace	MixedVectorSpaceBasis	A MixedVectorSpaceBasis for the operator's smallest eigenmodes (e.g. rigid body modes)	None

Source code in g-adopt/gadopt/stokes_integrators.py

def __init__(
    self,
    solution: fd.Function,
    T: fd.Function,
    approximation: BaseApproximation,
    bcs: dict[int, dict[str, Number]] = {},
    quad_degree: int = 6,
    solver_parameters: Optional[dict[str, str | Number] | str] = None,
    J: Optional[fd.Function] = None,
    constant_jacobian: bool = False,
    coupled_tstep: Optional[float] = None,
    theta: float = 0.5,
    nullspace: fd.MixedVectorSpaceBasis = None,
    transpose_nullspace: fd.MixedVectorSpaceBasis = None,
    near_nullspace: fd.MixedVectorSpaceBasis = None,
):
    self.solution_space = solution.function_space()
    self.mesh = self.solution_space.mesh()
    self.tests = fd.TestFunctions(self.solution_space)
    self.solution = solution
    self.T = T
    self.approximation = approximation
    self.bcs = bcs
    self.quad_degree = quad_degree
    self.solver_parameters = solver_parameters
    self.coupled_tstep = coupled_tstep
    self.theta = theta
    self.nullspace = nullspace
    self.transpose_nullspace = transpose_nullspace
    self.near_nullspace = near_nullspace

    self.solution_old = solution.copy(deepcopy=True)
    self.solution_split = fd.split(solution)
    self.solution_old_split = fd.split(self.solution_old)
    self.solution_theta_split = [
        theta * sol + (1 - theta) * sol_old
        for sol, sol_old in zip(self.solution_split, self.solution_old_split)
    ]

    self.J = J
    self.constant_jacobian = constant_jacobian
    self.linear = not depends_on(self.approximation.mu, self.solution)

    self.k = upward_normal(self.mesh)

    # Setup boundary conditions
    self.weak_bcs = {}
    self.strong_bcs = []

    # Free surface parameters
    self.free_surface_map = {}
    self.eta_ind = 2
    self.buoyancy_fs = [None] * len(self.solution_split)

    self.set_boundary_conditions()

    # eta is a list of 0, 1 or multiple free surface fields
    self.u, self.p, *self.eta = fd.split(self.solution)

    self.rho_mass = self.approximation.rho_continuity()

    self.equations = []
    self.F = 0.0  # Weak form of the system

    self.set_equations()
    self.set_form()

    self.set_solver_options()

    # Solver object is set up later to permit editing default solver options.
    self._solver_ready = False

set_boundary_conditions()

Sets strong and weak boundary conditions.

Source code in g-adopt/gadopt/stokes_integrators.py

def set_boundary_conditions(self) -> None:
    """Sets strong and weak boundary conditions."""
    bc_map = {"u": self.solution_space.sub(0)}
    if self.mesh.cartesian:
        bc_map["ux"] = bc_map["u"].sub(0)
        bc_map["uy"] = bc_map["u"].sub(1)
        if self.mesh.geometric_dimension == 3:
            bc_map["uz"] = bc_map["u"].sub(2)

    for bc_id, bc in self.bcs.items():
        weak_bc = defaultdict(float)

        for bc_type, val in bc.items():
            match bc_type:
                case "u" | "ux" | "uy" | "uz":
                    self.strong_bcs.append(
                        fd.DirichletBC(bc_map[bc_type], val, bc_id)
                    )
                case "free_surface":
                    weak_bc["normal_stress"] += self.set_free_surface_boundary(
                        val, bc_id
                    )
                    self.eta_ind += 1
                case _:
                    weak_bc[bc_type] += val

        self.weak_bcs[bc_id] = weak_bc

set_solver_options()

Sets PETSc solver parameters.

Source code in g-adopt/gadopt/stokes_integrators.py

def set_solver_options(self) -> None:
    """Sets PETSc solver parameters."""
    # Application context for the inverse mass matrix preconditioner
    self.appctx = {
        "mu": self.approximation.mu / self.approximation.rho_continuity()
    }

    if isinstance(solver_preset := self.solver_parameters, dict):
        return

    if not depends_on(self.approximation.mu, self.solution):
        self.solver_parameters = {"snes_type": "ksponly"}
    else:
        self.solver_parameters = newton_stokes_solver_parameters.copy()

    if INFO >= log_level:
        self.solver_parameters["snes_monitor"] = None

    if solver_preset is not None:
        match solver_preset:
            case "direct":
                self.solver_parameters.update(direct_stokes_solver_parameters)
            case "iterative":
                self.solver_parameters.update(iterative_stokes_solver_parameters)
            case _:
                raise ValueError("Solver type must be 'direct' or 'iterative'.")
    elif self.mesh.topological_dimension() == 2 and self.mesh.cartesian:
        self.solver_parameters.update(direct_stokes_solver_parameters)
    else:
        self.solver_parameters.update(iterative_stokes_solver_parameters)

    # Extra options for iterative solvers
    if self.solver_parameters.get("pc_type") == "fieldsplit":
        if DEBUG >= log_level:
            self.solver_parameters["fieldsplit_0"]["ksp_converged_reason"] = None
            self.solver_parameters["fieldsplit_1"]["ksp_monitor"] = None
        elif INFO >= log_level:
            self.solver_parameters["fieldsplit_1"]["ksp_converged_reason"] = None

            if self.free_surface_map:
                # Gather pressure and free surface fields for Schur complement solve
                fields_ind = ",".join(map(str, range(1, len(self.solution_split))))
                self.solver_parameters.update(
                    {
                        "pc_fieldsplit_0_fields": "0",
                        "pc_fieldsplit_1_fields": fields_ind,
                    }
                )
                # Update mass inverse preconditioner
                self.solver_parameters["fieldsplit_1"].update(
                    {"pc_python_type": "gadopt.FreeSurfaceMassInvPC"}
                )

set_form()

Sets the weak form including linear and bilinear terms.

Source code in g-adopt/gadopt/stokes_integrators.py

def set_form(self) -> None:
    """Sets the weak form including linear and bilinear terms."""
    for equation, solution, solution_old in zip(
        self.equations, self.solution_split, self.solution_old_split
    ):
        if equation.mass_term:
            assert equation.scaling_factor == -self.theta
            self.F += equation.mass((solution - solution_old) / self.coupled_tstep)
        self.F -= equation.residual(solution)

setup_solver()

Sets up the solver.

Source code in g-adopt/gadopt/stokes_integrators.py

def setup_solver(self) -> None:
    """Sets up the solver."""
    if self.free_surface_map:
        self.appctx["free_surface"] = self.free_surface_map
        self.appctx["ds"] = self.equations[-1].ds

    if self.constant_jacobian:
        trial = fd.TrialFunction(self.solution_space)
        F = fd.replace(self.F, {self.solution: trial})
        a, L = fd.lhs(F), fd.rhs(F)

        self.problem = fd.LinearVariationalProblem(
            a, L, self.solution, bcs=self.strong_bcs, constant_jacobian=True
        )
        self.solver = fd.LinearVariationalSolver(
            self.problem,
            solver_parameters=self.solver_parameters,
            nullspace=self.nullspace,
            transpose_nullspace=self.transpose_nullspace,
            near_nullspace=self.near_nullspace,
            appctx=self.appctx,
            options_prefix=self.name,
        )
    else:
        self.problem = fd.NonlinearVariationalProblem(
            self.F, self.solution, bcs=self.strong_bcs, J=self.J
        )
        self.solver = fd.NonlinearVariationalSolver(
            self.problem,
            solver_parameters=self.solver_parameters,
            nullspace=self.nullspace,
            transpose_nullspace=self.transpose_nullspace,
            near_nullspace=self.near_nullspace,
            appctx=self.appctx,
            options_prefix=self.name,
        )

    self._solver_ready = True

solve()

Solves the system.

Source code in g-adopt/gadopt/stokes_integrators.py

def solve(self):
    """Solves the system."""
    if not self._solver_ready:
        self.setup_solver()

    self.solver.solve()

    self.solution_old.assign(self.solution)

force_on_boundary(subdomain_id, **kwargs)

Computes the force acting on a boundary.

Parameters:

Name	Type	Description	Default
subdomain_id	int | str	The subdomain ID of a physical boundary.	required

Returns:

Type	Description
Function	The force acting on the boundary.

Source code in g-adopt/gadopt/stokes_integrators.py

def force_on_boundary(self, subdomain_id: int | str, **kwargs) -> fd.Function:
    """Computes the force acting on a boundary.

    Arguments:
      subdomain_id: The subdomain ID of a physical boundary.

    Returns:
      The force acting on the boundary.

    """
    if not hasattr(self, 'BoundaryNormalStressSolvers'):
        self.BoundaryNormalStressSolvers = {}

    if subdomain_id not in self.BoundaryNormalStressSolvers:
        self.BoundaryNormalStressSolvers[subdomain_id] = BoundaryNormalStressSolver(
            self, subdomain_id, **kwargs
        )

    return self.BoundaryNormalStressSolvers[subdomain_id].solve()

ViscoelasticStokesSolver(z, stress_old, displacement, approximation, dt, bcs={}, quad_degree=6, solver_parameters=None, J=None, constant_jacobian=False, **kwargs)

Bases: StokesSolver

Solves the Stokes system assuming a Maxwell viscoelastic rheology.

Parameters:

Name	Type	Description	Default
z	Function	Firedrake function representing mixed Stokes system	required
stress_old	Function	Firedrake function representing deviatoric stress from previous timestep	required
displacement	Function	Firedrake function representing displacement field	required
approximation	BaseApproximation	Approximation describing system of equations	required
dt	Number | Constant	Timestep for viscoelastic rheology	required
bcs	dict[int, dict[str, Number]]	Dictionary of identifier-value pairs specifying boundary conditions	{}
quad_degree	int	Quadrature degree. Default value is 2p + 1, where p is the polynomial degree of the trial space	6
solver_parameters	Optional[dict[str, str | Number] | str]	Either a dictionary of PETSc solver parameters or a string specifying a default set of parameters defined in G-ADOPT	None
J	Optional[Function]	Firedrake function representing the Jacobian of the system	None
constant_jacobian	bool	Whether the Jacobian of the system is constant	False

Source code in g-adopt/gadopt/stokes_integrators.py

def __init__(
    self,
    z: fd.Function,
    stress_old: fd.Function,
    displacement: fd.Function,
    approximation: BaseApproximation,
    dt: Number | fd.Constant,
    bcs: dict[int, dict[str, Number]] = {},
    quad_degree: int = 6,
    solver_parameters: Optional[dict[str, str | Number] | str] = None,
    J: Optional[fd.Function] = None,
    constant_jacobian: bool = False,
    **kwargs,
):
    self.stress_old = stress_old  # Function to store deviatoric stress from previous time step
    self.displacement = displacement
    self.dt = dt

    approximation.mu = approximation.effective_viscosity(self.dt)

    # This isn't used in the viscoelastic code but StokesSolver currently assumes temperature is required as an argument
    T_placeholder = fd.Constant(0)

    super().__init__(
        z,
        T_placeholder,
        approximation,
        bcs=bcs,
        quad_degree=quad_degree,
        solver_parameters=solver_parameters,
        J=J,
        constant_jacobian=constant_jacobian,
        coupled_tstep=self.dt,
        **kwargs,
    )

    scale_mu = fd.Constant(1e10)  # this is a scaling factor roughly size of mantle maxwell time to make sure that solve converges with strong bcs in parallel...
    self.F = (1 / scale_mu)*self.F

set_equations()

Sets up UFL forms for the viscoelastic Stokes equations residual.

Source code in g-adopt/gadopt/stokes_integrators.py

def set_equations(self) -> None:
    """Sets up UFL forms for the viscoelastic Stokes equations residual."""
    u, p = self.solution_split
    stress = self.approximation.stress(u, self.stress_old, self.dt)
    source = self.approximation.buoyancy(self.displacement) * self.k
    eqs_attrs = [
        {"p": p, "stress": stress, "source": source},
        {"u": u, "rho_mass": self.rho_mass},
    ]

    for i in range(len(residual_terms_stokes)):
        self.equations.append(
            Equation(
                self.tests[i],
                self.solution_space[i],
                residual_terms_stokes[i],
                eq_attrs=eqs_attrs[i],
                approximation=self.approximation,
                bcs=self.weak_bcs,
                quad_degree=self.quad_degree,
            )
        )

BoundaryNormalStressSolver(stokes_solver, subdomain_id, **kwargs)

A class for calculating surface forces acting on a boundary.

This solver computes topography on boundaries using the equation:

\[ h = sigma_(rr) / (g delta rho) \]

where \(sigma_(rr)\) is defined as:

\[ sigma_(rr) = [-p I + 2 mu (nabla u + nabla u^T)] . hat n . hat n \]

Instead of assuming a unit normal vector \(hat n\), this solver uses FacetNormal from Firedrake to accurately determine the normal vectors, which is particularly useful for complex meshes like icosahedron meshes in spherical simulations.

Parameters:

Name	Type	Description	Default
stokes_solver	StokesSolver	The Stokes solver providing the necessary fields for calculating stress.	required
subdomain_id	str | int	The subdomain ID of a physical boundary.	required
**kwargs		Optional keyword arguments. You can provide: - solver_parameters (dict[str, str | Number]): Parameters to control the variational solver. If not provided, defaults are chosen based on whether the Stokes solver is direct or iterative.	{}

Source code in g-adopt/gadopt/stokes_integrators.py

def __init__(self,
             stokes_solver: StokesSolver,
             subdomain_id: int | str,
             **kwargs
             ):
    # pressure and velocity together with viscosity are needed
    self.u, self.p, *self.eta = stokes_solver.solution.subfunctions

    # geometry
    self.mesh = stokes_solver.mesh
    self.dim = self.mesh.geometric_dimension()

    # approximation tells us if we need to consider compressible formulation or not
    self.approximation = stokes_solver.approximation

    # Domain id that we want to use for boundary force
    self.subdomain_id = subdomain_id

    self._kwargs = kwargs

    self.solver_parameters = self._kwargs.get(
        "solver_parameters",
        BoundaryNormalStressSolver.direct_solve_parameters
        if stokes_solver.solver_parameters == direct_stokes_solver_parameters
        else BoundaryNormalStressSolver.iterative_solver_parameters
    )

    self._solver_is_set_up = False

solve()

Solves a linear system for the force and applies necessary boundary conditions.

Returns:

Type	Description
	The modified force after solving the linear system and applying boundary conditions.

Source code in g-adopt/gadopt/stokes_integrators.py

def solve(self):
    """
    Solves a linear system for the force and applies necessary boundary conditions.

    Returns:
        The modified force after solving the linear system and applying boundary conditions.
    """
    # Solve a linear system
    if not self._solver_is_set_up:
        self.setup_solver()
    # Solve for the force
    self.solver.solve()

    # Take the average out
    vave = fd.assemble(self.force * self.ds) / fd.assemble(1 * self.ds(self.mesh))
    self.force.assign(self.force - vave)

    # Re-apply the zero condition everywhere except for the boundary
    self.interior_null_bc.apply(self.force)

    return self.force

create_stokes_nullspace(Z, closed=True, rotational=False, translations=None, ala_approximation=None, top_subdomain_id=None)

Create a null space for the mixed Stokes system.

Parameters:

Name	Type	Description	Default
Z	WithGeometry	Firedrake mixed function space associated with the Stokes system	required
closed	bool	Whether to include a constant pressure null space	True
rotational	bool	Whether to include all rotational modes	False
translations	Optional[list[int]]	List of translations to include	None
ala_approximation	Optional[AnelasticLiquidApproximation]	AnelasticLiquidApproximation for calculating (non-constant) right nullspace	None
top_subdomain_id	Optional[str | int]	Boundary id of top surface. Required when providing ala_approximation.	None

Returns:

Type	Description
MixedVectorSpaceBasis	A Firedrake mixed vector space basis incorporating the null space components

Source code in g-adopt/gadopt/stokes_integrators.py

def create_stokes_nullspace(
    Z: fd.functionspaceimpl.WithGeometry,
    closed: bool = True,
    rotational: bool = False,
    translations: Optional[list[int]] = None,
    ala_approximation: Optional[AnelasticLiquidApproximation] = None,
    top_subdomain_id: Optional[str | int] = None,
) -> fd.nullspace.MixedVectorSpaceBasis:
    """Create a null space for the mixed Stokes system.

    Arguments:
      Z: Firedrake mixed function space associated with the Stokes system
      closed: Whether to include a constant pressure null space
      rotational: Whether to include all rotational modes
      translations: List of translations to include
      ala_approximation: AnelasticLiquidApproximation for calculating (non-constant) right nullspace
      top_subdomain_id: Boundary id of top surface. Required when providing
                        ala_approximation.

    Returns:
      A Firedrake mixed vector space basis incorporating the null space components

    """
    # ala_approximation and top_subdomain_id are both needed when calculating right nullspace for ala
    if (ala_approximation is None) != (top_subdomain_id is None):
        raise ValueError("Both ala_approximation and top_subdomain_id must be provided, or both must be None.")

    X = fd.SpatialCoordinate(Z.mesh())
    dim = len(X)
    stokes_subspaces = Z.subspaces

    if rotational:
        if dim == 2:
            rotV = fd.Function(stokes_subspaces[0]).interpolate(fd.as_vector((-X[1], X[0])))
            basis = [rotV]
        elif dim == 3:
            x_rotV = fd.Function(stokes_subspaces[0]).interpolate(fd.as_vector((0, -X[2], X[1])))
            y_rotV = fd.Function(stokes_subspaces[0]).interpolate(fd.as_vector((X[2], 0, -X[0])))
            z_rotV = fd.Function(stokes_subspaces[0]).interpolate(fd.as_vector((-X[1], X[0], 0)))
            basis = [x_rotV, y_rotV, z_rotV]
        else:
            raise ValueError("Unknown dimension")
    else:
        basis = []

    if translations:
        for tdim in translations:
            vec = [0] * dim
            vec[tdim] = 1
            basis.append(fd.Function(stokes_subspaces[0]).interpolate(fd.as_vector(vec)))

    if basis:
        V_nullspace = fd.VectorSpaceBasis(basis, comm=Z.mesh().comm)
        V_nullspace.orthonormalize()
    else:
        V_nullspace = stokes_subspaces[0]

    if closed:
        if ala_approximation:
            p = ala_right_nullspace(W=stokes_subspaces[1], approximation=ala_approximation, top_subdomain_id=top_subdomain_id)
            p_nullspace = fd.VectorSpaceBasis([p], comm=Z.mesh().comm)
            p_nullspace.orthonormalize()
        else:
            p_nullspace = fd.VectorSpaceBasis(constant=True, comm=Z.mesh().comm)
    else:
        p_nullspace = stokes_subspaces[1]

    null_space = [V_nullspace, p_nullspace]

    # If free surface unknowns, add dummy free surface nullspace
    null_space += stokes_subspaces[2:]

    return fd.MixedVectorSpaceBasis(Z, null_space)

ala_right_nullspace(W, approximation, top_subdomain_id)

Compute pressure nullspace for Anelastic Liquid Approximation.

Parameters:

Name	Type	Description	Default
W	WithGeometry	pressure function space	required
approximation	AnelasticLiquidApproximation	AnelasticLiquidApproximation with equation parameters	required
top_subdomain_id	str | int	boundary id of top surface	required

Returns:

Type	Description
	pressure nullspace solution

To obtain the pressure nullspace solution for the Stokes equation in Anelastic Liquid Approximation, which includes a pressure-dependent buoyancy term, we try to solve the equation:

\[ -nabla p + g "Di" rho chi c_p/(c_v gamma) hatk p = 0 \]

Taking the divergence:

\[ -nabla * nabla p + nabla * (g "Di" rho chi c_p/(c_v gamma) hatk p) = 0, \]

then testing it with q:

\[ int_Omega -q nabla * nabla p dx + int_Omega q nabla * (g "Di" rho chi c_p/(c_v gamma) hatk p) dx = 0 \]

followed by integration by parts:

\[ int_Gamma -bb n * q nabla p ds + int_Omega nabla q cdot nabla p dx + int_Gamma bb n * hatk q g "Di" rho chi c_p/(c_v gamma) p dx - int_Omega nabla q * hatk g "Di" rho chi c_p/(c_v gamma) p dx = 0 \]

This elliptic equation can be solved with natural boundary conditions by imposing our original equation above, which eliminates all boundary terms:

\[ int_Omega nabla q * nabla p dx - int_Omega nabla q * hatk g "Di" rho chi c_p/(c_v gamma) p dx = 0. \]

However, if we do so on all boundaries we end up with a system that has the same nullspace, as the one we are after (note that we ended up merely testing the original equation with \(nabla q\)). Instead we use the fact that the gradient of the null mode is always vertical, and thus the null mode is constant at any horizontal level (geoid), specifically the top surface. Choosing any nonzero constant for this surface fixes the arbitrary scalar multiplier of the null mode. We choose the value of one and apply it as a Dirichlet boundary condition.

Note that this procedure does not necessarily compute the exact nullspace of the discretised Stokes system. In particular, since not every test function \(v in V\), the velocity test space, can be written as \(v=nabla q\) with \(q in W\), the pressure test space, the two terms do not necessarily exactly cancel when tested with \(v\) instead of \(nabla q\) as in our final equation. However, in practice the discrete error appears to be small enough, and providing this nullspace gives an improved convergence of the iterative Stokes solver.

Source code in g-adopt/gadopt/stokes_integrators.py

def ala_right_nullspace(
        W: fd.functionspaceimpl.WithGeometry,
        approximation: AnelasticLiquidApproximation,
        top_subdomain_id: str | int):
    r"""Compute pressure nullspace for Anelastic Liquid Approximation.

        Arguments:
          W: pressure function space
          approximation: AnelasticLiquidApproximation with equation parameters
          top_subdomain_id: boundary id of top surface

        Returns:
          pressure nullspace solution

        To obtain the pressure nullspace solution for the Stokes equation in Anelastic Liquid Approximation,
        which includes a pressure-dependent buoyancy term, we try to solve the equation:

        $$
          -nabla p + g "Di" rho chi c_p/(c_v gamma) hatk p = 0
        $$

        Taking the divergence:

        $$
          -nabla * nabla p + nabla * (g "Di" rho chi c_p/(c_v gamma) hatk p) = 0,
        $$

        then testing it with q:

        $$
            int_Omega -q nabla * nabla p dx + int_Omega q nabla * (g "Di" rho chi c_p/(c_v gamma) hatk p) dx = 0
        $$

        followed by integration by parts:

        $$
            int_Gamma -bb n * q nabla p ds + int_Omega nabla q cdot nabla p dx +
            int_Gamma bb n * hatk q g "Di" rho chi c_p/(c_v gamma) p dx -
            int_Omega nabla q * hatk g "Di" rho chi c_p/(c_v gamma) p dx = 0
        $$

        This elliptic equation can be solved with natural boundary conditions by imposing our original equation above, which eliminates
        all boundary terms:

        $$
          int_Omega nabla q * nabla p dx - int_Omega nabla q * hatk g "Di" rho chi c_p/(c_v gamma) p dx = 0.
        $$

        However, if we do so on all boundaries we end up with a system that has the same nullspace, as the one we are after (note that
        we ended up merely testing the original equation with $nabla q$). Instead we use the fact that the gradient of the null mode
        is always vertical, and thus the null mode is constant at any horizontal level (geoid), specifically the top surface. Choosing
        any nonzero constant for this surface fixes the arbitrary scalar multiplier of the null mode. We choose the value of one
        and apply it as a Dirichlet boundary condition.

        Note that this procedure does not necessarily compute the exact nullspace of the *discretised* Stokes system. In particular,
        since not every test function $v in V$, the velocity test space, can be written as $v=nabla q$ with $q in W$, the
        pressure test space, the two terms do not necessarily exactly cancel when tested with $v$ instead of $nabla q$ as in our
        final equation. However, in practice the discrete error appears to be small enough, and providing this nullspace gives
        an improved convergence of the iterative Stokes solver.
    """
    W = fd.FunctionSpace(mesh=W.mesh(), family=W.ufl_element())
    q = fd.TestFunction(W)
    p = fd.Function(W, name="pressure_nullspace")

    # Fix the solution at the top boundary
    bc = fd.DirichletBC(W, 1., top_subdomain_id)

    F = fd.inner(fd.grad(q), fd.grad(p)) * fd.dx

    k = upward_normal(W.mesh())

    F += - fd.inner(fd.grad(q), k * approximation.dbuoyancydp(p, fd.Constant(1.0)) * p) * fd.dx

    fd.solve(F == 0, p, bcs=bc)
    return p