Divergence conforming discontinuous Galerkin method for the Navier–Stokes equations

This demo (demo_navier-stokes.py) illustrates how to implement a divergence conforming discontinuous Galerkin method for the Navier-Stokes equations in FEniCSx. The method conserves mass exactly and uses upwinding. The formulation is based on a combination of “A fully divergence-free finite element method for magnetohydrodynamic equations” by Hiptmair et al., “A Note on Discontinuous Galerkin Divergence-free Solutions of the Navier-Stokes Equations” by Cockburn et al, and “On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows” by John et al.

Governing equations

We consider the incompressible Navier-Stokes equations in a domain \(\Omega \subset \mathbb{R}^d\), \(d \in \{2, 3\}\), and time interval \((0, \infty)\), given by

\[\begin{split} \partial_t u - \nu \Delta u + (u \cdot \nabla)u + \nabla p &= f \textnormal{ in } \Omega_t, \\ \nabla \cdot u &= 0 \textnormal{ in } \Omega_t, \end{split}\]

where \(u: \Omega_t \to \mathbb{R}^d\) is the velocity field, \(p: \Omega_t \to \mathbb{R}\) is the pressure field, \(f: \Omega_t \to \mathbb{R}^d\) is a prescribed force, \(\nu \in \mathbb{R}^+\) is the kinematic viscosity, and \(\Omega_t := \Omega \times (0, \infty)\).

The problem is supplemented with the initial condition

\[ u(x, 0) = u_0(x) \textnormal{ in } \Omega \]

and boundary condition

\[ u = u_D \textnormal{ on } \partial \Omega \times (0, \infty), \]

where \(u_0: \Omega \to \mathbb{R}^d\) is a prescribed initial velocity field which satisfies the divergence free condition. The pressure field is only determined up to a constant, so we seek the unique pressure field satisfying

\[ \int_\Omega p = 0. \]

Discrete problem

We begin by introducing the function spaces

\[\begin{split} V_h^g &:= \left\{v \in H(\textnormal{div}; \Omega); v|_K \in V_h(K) \; \forall K \in \mathcal{T}, v \cdot n = g \cdot n \textnormal{ on } \partial \Omega \right\} \\ Q_h &:= \left\{q \in L^2_0(\Omega); q|_K \in Q_h(K) \; \forall K \in \mathcal{T} \right\}. \end{split}\]

The local spaces \(V_h(K)\) and \(Q_h(K)\) should satisfy

\[ \nabla \cdot V_h(K) \subseteq Q_h(K), \]

in order for mass to be conserved exactly. Suitable choices on affine simplex cells include

\[ V_h(K) := \mathbb{RT}_k(K) \textnormal{ and } Q_h(K) := \mathbb{P}_k(K), \]

or

\[ V_h(K) := \mathbb{BDM}_k(K) \textnormal{ and } Q_h(K) := \mathbb{P}_{k-1}(K). \]

Let two cells \(K^+\) and \(K^-\) share a facet \(F\). The trace of a piecewise smooth vector valued function \(\phi\) on F taken approaching from inside \(K^+\) (resp. \(K^-\)) is denoted \(\phi^{+}\) (resp. \(\phi^-\)). We now introduce the average \(\renewcommand{\avg}[1]{\left\{\!\!\left\{#1\right\}\!\!\right\}}\)

\[ \avg{\phi} = \frac{1}{2} \left(\phi^+ + \phi^-\right)\]

and jump \(\renewcommand{\jump}[1]{[\![ #1 ]\!]}\)

\[ \jump{\phi} = \phi^+ \otimes n^+ + \phi^- \otimes n^-, \]

operators, where \(n\) denotes the outward unit normal to \(\partial K\). Finally, let the upwind flux of \(\phi\) with respect to a vector field \(\psi\) be defined as

\[\begin{split} \hat{\phi}^\psi := \begin{cases} \lim_{\epsilon \downarrow 0} \phi(x - \epsilon \psi(x)), \; x \in \partial K \setminus \Gamma^\psi, \\ 0, \qquad \qquad \qquad \qquad x \in \partial K \cap \Gamma^\psi, \end{cases} \end{split}\]

where \(\Gamma^\psi = \left\{x \in \Gamma; \; \psi(x) \cdot n(x) < 0\right\}\).

The semi-discrete version problem (in dimensionless form) is: find \((u_h, p_h) \in V_h^{u_D} \times Q_h\) such that

\[\begin{split} \int_\Omega \partial_t u_h \cdot v + a_h(u_h, v_h) + c_h(u_h; u_h, v_h) + b_h(v_h, p_h) &= \int_\Omega f \cdot v_h + L_{a_h}(v_h) + L_{c_h}(v_h) \quad \forall v_h \in V_h^0, \\ b_h(u_h, q_h) &= 0 \quad \forall q_h \in Q_h, \end{split}\]

where \(\renewcommand{\sumK}[0]{\sum_{K \in \mathcal{T}_h}}\) \(\renewcommand{\sumF}[0]{\sum_{F \in \mathcal{F}_h}}\)

\[\begin{split} a_h(u, v) &= Re^{-1} \left(\sumK \int_K \nabla u : \nabla v - \sumF \int_F \avg{\nabla u} : \jump{v} - \sumF \int_F \avg{\nabla v} : \jump{u} \\ + \sumF \int_F \frac{\alpha}{h_K} \jump{u} : \jump{v}\right), \\ c_h(w; u, v) &= - \sumK \int_K u \cdot \nabla \cdot (v \otimes w) + \sumK \int_{\partial_K} w \cdot n \hat{u}^{w} \cdot v, \\ L_{a_h}(v_h) &= Re^{-1} \left(- \int_{\partial \Omega} u_D \otimes n : \nabla_h v_h + \frac{\alpha}{h} u_D \otimes n : v_h \otimes n \right), \\ L_{c_h}(v_h) &= - \int_{\partial \Omega} u_D \cdot n \hat{u}_D \cdot v_h, \\ b_h(v, q) &= - \int_K \nabla \cdot v q. \end{split}\]

Implementation

We begin by importing the required modules and functions

from mpi4py import MPI
from petsc4py import PETSc

import numpy as np

from dolfinx import default_real_type, fem, io, mesh
from dolfinx.fem.petsc import assemble_matrix_block, assemble_vector_block
from ufl import (CellDiameter, FacetNormal, TestFunction, TrialFunction, avg,
                 conditional, div, dot, dS, ds, dx, grad, gt, inner, outer)

if np.issubdtype(PETSc.ScalarType, np.complexfloating):  # type: ignore
    print("Demo should only be executed with DOLFINx real mode")
    exit(0)

We also define some helper functions that will be used later

def norm_L2(comm, v):
    """Compute the L2(Ω)-norm of v"""
    return np.sqrt(comm.allreduce(fem.assemble_scalar(fem.form(inner(v, v) * dx)), op=MPI.SUM))


def domain_average(msh, v):
    """Compute the average of a function over the domain"""
    vol = msh.comm.allreduce(fem.assemble_scalar(fem.form(fem.Constant(msh, default_real_type(1.0)) * dx)), op=MPI.SUM)
    return (1 / vol) * msh.comm.allreduce(fem.assemble_scalar(fem.form(v * dx)), op=MPI.SUM)


def u_e_expr(x):
    """Expression for the exact velocity solution to Kovasznay flow"""
    return np.vstack((1 - np.exp(
        (Re / 2 - np.sqrt(Re**2 / 4 + 4 * np.pi**2)) * x[0]) * np.cos(2 * np.pi * x[1]),
        (Re / 2 - np.sqrt(Re**2 / 4 + 4 * np.pi**2)) / (2 * np.pi) * np.exp(
            (Re / 2 - np.sqrt(Re**2 / 4 + 4 * np.pi**2)) * x[0])
        * np.sin(2 * np.pi * x[1])))


def p_e_expr(x):
    """Expression for the exact pressure solution to Kovasznay flow"""
    return (1 / 2) * (1 - np.exp(2 * (Re / 2 - np.sqrt(Re**2 / 4 + 4 * np.pi**2)) * x[0]))


def f_expr(x):
    """Expression for the applied force"""
    return np.vstack((np.zeros_like(x[0]), np.zeros_like(x[0])))


def boundary_marker(x):
    return np.isclose(x[0], 0.0) | np.isclose(x[0], 1.0) | np.isclose(x[1], 0.0) | np.isclose(x[1], 1.0)

We define some simulation parameters

n = 16
num_time_steps = 25
t_end = 10
Re = 25  # Reynolds Number
k = 1  # Polynomial degree

Next, we create a mesh and the required functions spaces over it. Since the velocity uses an \(H(\textnormal{div})\)-conforming function space, we also create a vector valued discontinuous Lagrange space to interpolate into for artifact free visualisation.

msh = mesh.create_unit_square(MPI.COMM_WORLD, n, n)

# Function spaces for the velocity and for the pressure
V = fem.functionspace(msh, ("Raviart-Thomas", k + 1))
Q = fem.functionspace(msh, ("Discontinuous Lagrange", k))

# Funcion space for visualising the velocity field
gdim = msh.geometry.dim
W = fem.functionspace(msh, ("Discontinuous Lagrange", k + 1, (gdim,)))

# Define trial and test functions

u, v = TrialFunction(V), TestFunction(V)
p, q = TrialFunction(Q), TestFunction(Q)

delta_t = fem.Constant(msh, default_real_type(t_end / num_time_steps))
alpha = fem.Constant(msh, default_real_type(6.0 * k**2))

h = CellDiameter(msh)
n = FacetNormal(msh)


def jump(phi, n):
    return outer(phi("+"), n("+")) + outer(phi("-"), n("-"))

We solve the Stokes problem for the initial condition, omitting the convective term:

a_00 = (1.0 / Re) * (inner(grad(u), grad(v)) * dx
                     - inner(avg(grad(u)), jump(v, n)) * dS
                     - inner(jump(u, n), avg(grad(v))) * dS
                     + (alpha / avg(h)) * inner(jump(u, n), jump(v, n)) * dS
                     - inner(grad(u), outer(v, n)) * ds
                     - inner(outer(u, n), grad(v)) * ds
                     + (alpha / h) * inner(outer(u, n), outer(v, n)) * ds)
a_01 = - inner(p, div(v)) * dx
a_10 = - inner(div(u), q) * dx

a = fem.form([[a_00, a_01],
              [a_10, None]])

f = fem.Function(W)
u_D = fem.Function(V)
u_D.interpolate(u_e_expr)
L_0 = inner(f, v) * dx + (1 / Re) * (- inner(outer(u_D, n), grad(v)) * ds
                                     + (alpha / h) * inner(outer(u_D, n), outer(v, n)) * ds)
L_1 = inner(fem.Constant(msh, default_real_type(0.0)), q) * dx
L = fem.form([L_0, L_1])

# Boundary conditions
boundary_facets = mesh.locate_entities_boundary(msh, msh.topology.dim - 1, boundary_marker)
boundary_vel_dofs = fem.locate_dofs_topological(V, msh.topology.dim - 1, boundary_facets)
bc_u = fem.dirichletbc(u_D, boundary_vel_dofs)
bcs = [bc_u]

# Assemble Stokes problem
A = assemble_matrix_block(a, bcs=bcs)
A.assemble()
b = assemble_vector_block(L, a, bcs=bcs)

# Create and configure solver
ksp = PETSc.KSP().create(msh.comm)  # type: ignore
ksp.setOperators(A)
ksp.setType("preonly")
ksp.getPC().setType("lu")
ksp.getPC().setFactorSolverType("mumps")
opts = PETSc.Options()  # type: ignore
opts["mat_mumps_icntl_14"] = 80  # Increase MUMPS working memory
opts["mat_mumps_icntl_24"] = 1  # Option to support solving a singular matrix (pressure nullspace)
opts["mat_mumps_icntl_25"] = 0  # Option to support solving a singular matrix (pressure nullspace)
opts["ksp_error_if_not_converged"] = 1
ksp.setFromOptions()

# Solve Stokes for initial condition
x = A.createVecRight()
try:
    ksp.solve(b, x)
except PETSc.Error as e:  # type: ignore
    if e.ierr == 92:
        print("The required PETSc solver/preconditioner is not available. Exiting.")
        print(e)
        exit(0)
    else:
        raise e


# Split the solution
u_h = fem.Function(V)
p_h = fem.Function(Q)
p_h.name = "p"
offset = V.dofmap.index_map.size_local * V.dofmap.index_map_bs
u_h.x.array[:offset] = x.array_r[:offset]
u_h.x.scatter_forward()
p_h.x.array[:(len(x.array_r) - offset)] = x.array_r[offset:]
p_h.x.scatter_forward()
# Subtract the average of the pressure since it is only determined up to
# a constant
p_h.x.array[:] -= domain_average(msh, p_h)

u_vis = fem.Function(W)
u_vis.name = "u"
u_vis.interpolate(u_h)

# Write initial condition to file
t = 0.0
try:
    u_file = io.VTXWriter(msh.comm, "u.bp", [u_vis._cpp_object])
    p_file = io.VTXWriter(msh.comm, "p.bp", [p_h._cpp_object])
    u_file.write(t)
    p_file.write(t)
except AttributeError:
    print("File output requires ADIOS2.")

# Create function to store solution and previous time step
u_n = fem.Function(V)
u_n.x.array[:] = u_h.x.array

Now we add the time stepping and convective terms

lmbda = conditional(gt(dot(u_n, n), 0), 1, 0)
u_uw = lmbda("+") * u("+") + lmbda("-") * u("-")
a_00 += inner(u / delta_t, v) * dx - \
    inner(u, div(outer(v, u_n))) * dx + \
    inner((dot(u_n, n))("+") * u_uw, v("+")) * dS + \
    inner((dot(u_n, n))("-") * u_uw, v("-")) * dS + \
    inner(dot(u_n, n) * lmbda * u, v) * ds
a = fem.form([[a_00, a_01],
              [a_10, None]])

L_0 += inner(u_n / delta_t, v) * dx - inner(dot(u_n, n) * (1 - lmbda) * u_D, v) * ds
L = fem.form([L_0, L_1])

# Time stepping loop
for n in range(num_time_steps):
    t += delta_t.value

    A.zeroEntries()
    fem.petsc.assemble_matrix_block(A, a, bcs=bcs)  # type: ignore
    A.assemble()

    with b.localForm() as b_loc:
        b_loc.set(0)
    fem.petsc.assemble_vector_block(b, L, a, bcs=bcs)  # type: ignore

    # Compute solution
    ksp.solve(b, x)

    u_h.x.array[:offset] = x.array_r[:offset]
    u_h.x.scatter_forward()
    p_h.x.array[:(len(x.array_r) - offset)] = x.array_r[offset:]
    p_h.x.scatter_forward()
    p_h.x.array[:] -= domain_average(msh, p_h)

    u_vis.interpolate(u_h)

    # Write to file
    try:
        u_file.write(t)
        p_file.write(t)
    except NameError:
        pass

    # Update u_n
    u_n.x.array[:] = u_h.x.array

try:
    u_file.close()
    p_file.close()
except NameError:
    pass

Now we compare the computed solution to the exact solution

# Function spaces for exact velocity and pressure
V_e = fem.functionspace(msh, ("Lagrange", k + 3, (gdim,)))
Q_e = fem.functionspace(msh, ("Lagrange", k + 2))

u_e = fem.Function(V_e)
u_e.interpolate(u_e_expr)

p_e = fem.Function(Q_e)
p_e.interpolate(p_e_expr)

# Compute errors
e_u = norm_L2(msh.comm, u_h - u_e)
e_div_u = norm_L2(msh.comm, div(u_h))

# This scheme conserves mass exactly, so check this
assert np.isclose(e_div_u, 0.0, atol=float(1.0e5 * np.finfo(default_real_type).eps))
p_e_avg = domain_average(msh, p_e)
e_p = norm_L2(msh.comm, p_h - (p_e - p_e_avg))

if msh.comm.rank == 0:
    print(f"e_u = {e_u}")
    print(f"e_div_u = {e_div_u}")
    print(f"e_p = {e_p}")