# 19. Stokes equations¶

This demo is implemented in a single Python file, demo_stokes-iterative.py, which contains both the variational forms and the solver.

This demo illustrates how to:

• Maintain symmetry when assembling a system of symmetric equations with essential (Dirichlet) boundary conditions
• Use a iterative solver explicitly for solving a linear system of equations
• Define a preconditioner explicitly using a form

## 19.1. Strong form of the Stokes equations¶

The incompressible Stokes equations in strong form read: for a domain $$\Omega \subset \mathbb{R}^n$$, find the velocity $$u$$ and the pressure $$p$$ satisfying

$\begin{split}- \nabla \cdot (\nabla u + p I) &= f \quad {\rm in} \ \Omega, \\ \nabla \cdot u &= 0 \quad {\rm in} \ \Omega. \\\end{split}$

Note

The sign of the pressure has been flipped from the classical definition. This is done in order to have a symmetric (but not positive-definite) system of equations rather than a non-symmetric (but positive-definite) system of equations.

A typical set of boundary conditions on the boundary $$\partial \Omega = \Gamma_{D} \cup \Gamma_{N}$$ can be:

$\begin{split}u &= u_0 \quad {\rm on} \ \Gamma_{D}, \\ \nabla u \cdot n + p n &= g \quad {\rm on} \ \Gamma_{N}. \\\end{split}$

## 19.2. Weak form of the Stokes equations¶

The Stokes equations can easily formulated in a mixed variational form; that is, a form where the two variables, the velocity and the pressure, are approximated simultaneously. Using the abstract framework, we have the problem: find $$(u, p) \in W$$ such that

$a((u, p), (v, q)) = L((v, q))$

for all $$(v, q) \in W$$ where

$\begin{split}a((u, p), (v, q)) &= \int_{\Omega} \nabla u \cdot \nabla v + \nabla \cdot v \ p + \nabla \cdot u \ q \, {\rm d} x, \\ L((v, q)) &= \int_{\Omega} f \cdot v \, {\rm d} x + \int_{\partial \Omega_N} g \cdot v \, {\rm d} s. \\\end{split}$

The space W should be a mixed (product) function space: $$W = V \times Q$$ such that $$u \in V$$ and $$q \in Q$$.

## 19.3. Preconditioning of the linear system of equations¶

For the resulting linear system of equations, the following form defines a suitable preconditioner:

$b((u, p), (v, q)) = \int_{\Omega} \nabla u \cdot \nabla v + p \, q \, {\rm d} x$

## 19.4. Domain and boundary conditions¶

In this demo, we shall consider the following definitions of the input functions, the domain, and the boundaries:

• $$\Omega = [0,1]^3$$ (a unit cube)
• $$\Omega_D = \{(x_0, x_1, x_2) \, | \, x_0 = 0 \, \text{or} \, x_0 = 1 \, \text{or} \, x_1 = 0 \, \text{or} \, x_1 = 1 \}$$
• $$u_0 = (- \sin(\pi x_1), 0.0, 0.0)$$ for $$x_0 = 1$$ and $$u_0 = (0.0, 0.0, 0.0)$$ otherwise
• $$f = (0.0, 0.0, 0.0)$$
• $$g = (0.0, 0.0, 0.0)$$

## 19.5. Implementation¶

This description goes through the implementation (in demo_stokes-iterative.py) of a solver for the above described Stokes equations. Some of the standard steps will be described in less detail, so before reading this, we suggest that you are familiarize with the Poisson demo (for the very basics) and the Mixed Poisson demo (for how to deal with mixed function spaces). Also, the Navier–Stokes demo illustrates how to use iterative solvers in a more implicit manner (typically only suitable for positive-definite systems of equations).

The Stokes equations as formulated above result in a system of linear equations that is not positive-definite. Standard iterative linear solvers typically fail to converge for such systems. Some care must therefore be taken in preconditioning the systems of equations. Moreover, not all of the linear algebra backends support this. We therefore start by checking that either “PETSc” or “Epetra” (from Trilinos) is available. We also try to pick MINRES Krylov subspace method which is suitable for symmetric indefinite problems. If not available, costly QMR method is choosen.

from dolfin import *

# Test for PETSc or Epetra
if not has_linear_algebra_backend("PETSc") and not has_linear_algebra_backend("Epetra"):
info("DOLFIN has not been configured with Trilinos or PETSc. Exiting.")
exit()

if not has_krylov_solver_preconditioner("amg"):
info("Sorry, this demo is only available when DOLFIN is compiled with AMG "
"preconditioner, Hypre or ML.")
exit()

if has_krylov_solver_method("minres"):
krylov_method = "minres"
elif has_krylov_solver_method("tfqmr"):
krylov_method = "tfqmr"
else:
info("Default linear algebra backend was not compiled with MINRES or TFQMR "
"Krylov subspace method. Terminating.")
exit()


Next, we define the mesh (a UnitCubeMesh) and a MixedFunctionSpace composed of a VectorFunctionSpace of continuous piecewise quadratics and a FunctionSpace of continuous piecewise linears. (This mixed finite element space is known as the Taylor–Hood elements and is a stable, standard element pair for the Stokes equations.)

# Load mesh
mesh = UnitCubeMesh(16, 16, 16)

# Define function spaces
V = VectorFunctionSpace(mesh, "CG", 2)
Q = FunctionSpace(mesh, "CG", 1)
W = V * Q


Next, we define the boundary conditions.

# Boundaries
def right(x, on_boundary): return x > (1.0 - DOLFIN_EPS)
def left(x, on_boundary): return x < DOLFIN_EPS
def top_bottom(x, on_boundary):
return x > 1.0 - DOLFIN_EPS or x < DOLFIN_EPS

# No-slip boundary condition for velocity
noslip = Constant((0.0, 0.0, 0.0))
bc0 = DirichletBC(W.sub(0), noslip, top_bottom)

# Inflow boundary condition for velocity
inflow = Expression(("-sin(x*pi)", "0.0", "0.0"))
bc1 = DirichletBC(W.sub(0), inflow, right)

# Boundary condition for pressure at outflow
zero = Constant(0)
bc2 = DirichletBC(W.sub(1), zero, left)

# Collect boundary conditions
bcs = [bc0, bc1, bc2]


The bilinear and linear forms corresponding to the weak mixed formulation of the Stokes equations are defined as follows:

# Define variational problem
(u, p) = TrialFunctions(W)
(v, q) = TestFunctions(W)
f = Constant((0.0, 0.0, 0.0))
L = inner(f, v)*dx


We can now use the same TrialFunctions and TestFunctions to define the preconditioner matrix. We first define the form corresponding to the expression for the preconditioner (given in the initial description above):

# Form for use in constructing preconditioner matrix


Next, we want to assemble the matrix corresponding to the bilinear form and the vector corresponding to the linear form of the Stokes equations. Moreover, we want to apply the specified boundary conditions to the linear system. However, assembling the matrix and vector and applying a DirichletBC separately will possibly result in a non-symmetric system of equations. Instead, we can use the assemble_system function to assemble both the matrix A, the vector bb, and apply the boundary conditions bcs in a symmetric fashion:

# Assemble system
A, bb = assemble_system(a, L, bcs)


We do the same for the preconditioner matrix P using the linear form L as a dummy form:

# Assemble preconditioner system
P, btmp = assemble_system(b, L, bcs)


Next, we specify the iterative solver we want to use, in this case a KrylovSolver. We associate the left-hand side matrix A and the preconditioner matrix P with the solver by calling solver.set_operators.

# Create Krylov solver and AMG preconditioner
solver = KrylovSolver(krylov_method, "amg")

# Associate operator (A) and preconditioner matrix (P)
solver.set_operators(A, P)


We are now almost ready to solve the linear system of equations. It remains to specify a Vector for storing the result. For easy manipulation later, we can define a Function and use the vector associated with this Function. The call to solver.solve then looks as follows

# Solve
U = Function(W)
solver.solve(U.vector(), bb)


Finally, we can play with the result in different ways:

# Get sub-functions
u, p = U.split()

# Save solution in VTK format
ufile_pvd = File("velocity.pvd")
ufile_pvd << u
pfile_pvd = File("pressure.pvd")
pfile_pvd << p

# Plot solution
plot(u)
plot(p)
interactive()


## 19.6. Complete code¶


from dolfin import *

# Test for PETSc or Epetra
if not has_linear_algebra_backend("PETSc") and not has_linear_algebra_backend("Epetra"):
info("DOLFIN has not been configured with Trilinos or PETSc. Exiting.")
exit()

if not has_krylov_solver_preconditioner("amg"):
info("Sorry, this demo is only available when DOLFIN is compiled with AMG "
"preconditioner, Hypre or ML.")
exit()

if has_krylov_solver_method("minres"):
krylov_method = "minres"
elif has_krylov_solver_method("tfqmr"):
krylov_method = "tfqmr"
else:
info("Default linear algebra backend was not compiled with MINRES or TFQMR "
"Krylov subspace method. Terminating.")
exit()

mesh = UnitCubeMesh(16, 16, 16)

# Define function spaces
V = VectorFunctionSpace(mesh, "CG", 2)
Q = FunctionSpace(mesh, "CG", 1)
W = V * Q

# Boundaries
def right(x, on_boundary): return x > (1.0 - DOLFIN_EPS)
def left(x, on_boundary): return x < DOLFIN_EPS
def top_bottom(x, on_boundary):
return x > 1.0 - DOLFIN_EPS or x < DOLFIN_EPS

# No-slip boundary condition for velocity
noslip = Constant((0.0, 0.0, 0.0))
bc0 = DirichletBC(W.sub(0), noslip, top_bottom)

# Inflow boundary condition for velocity
inflow = Expression(("-sin(x*pi)", "0.0", "0.0"))
bc1 = DirichletBC(W.sub(0), inflow, right)

# Boundary condition for pressure at outflow
zero = Constant(0)
bc2 = DirichletBC(W.sub(1), zero, left)

# Collect boundary conditions
bcs = [bc0, bc1, bc2]

# Define variational problem
(u, p) = TrialFunctions(W)
(v, q) = TestFunctions(W)
f = Constant((0.0, 0.0, 0.0))
L = inner(f, v)*dx

# Form for use in constructing preconditioner matrix

# Assemble system
A, bb = assemble_system(a, L, bcs)

# Assemble preconditioner system
P, btmp = assemble_system(b, L, bcs)

# Create Krylov solver and AMG preconditioner
solver = KrylovSolver(krylov_method, "amg")

# Associate operator (A) and preconditioner matrix (P)
solver.set_operators(A, P)

# Solve
U = Function(W)
solver.solve(U.vector(), bb)

# Get sub-functions
u, p = U.split()

# Save solution in VTK format
ufile_pvd = File("velocity.pvd")
ufile_pvd << u
pfile_pvd = File("pressure.pvd")
pfile_pvd << p

# Plot solution
plot(u)
plot(p)
interactive()