Stokes problem with only Dirichlet boundary condition

variational formulation
2

Here is a minimal example highlighting the two issues of your code

  1. The boundary condition is wrong
  2. The nullspace creation is overly complicated
import dolfinx
import numpy as np
import ufl
from mpi4py import MPI
from petsc4py import PETSc

N = 10
mesh = dolfinx.mesh.create_unit_cube(MPI.COMM_WORLD, N, N, N,
                                     cell_type=dolfinx.mesh.CellType.hexahedron)

P2 = ufl.VectorElement("Lagrange", mesh.ufl_cell(), 2)
P1 = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
TH = ufl.MixedElement([P2, P1])
W = dolfinx.fem.FunctionSpace(mesh, TH)

def u_ex(x): 
    return np.vstack((4*(x[1]-1)*x[1]*(x[0]-1)*(x[0]-1)*x[0]*x[0]*(x[1]-0.5), -4*(x[0]-0.5)*x[1]*x[1]*x[0]*(x[1]-1)*(x[1]-1)*(x[0]-1), np.zeros_like(x[0])))
def p_ex(x):
    return x[0]+x[1]+x[2]-3./2

V, V_to_W = W.sub(0).collapse()
Q, Q_to_W = W.sub(1).collapse()

### Define the boundary condition -> which in this case are only of Dirichlet type 
uD = dolfinx.fem.Function(V)
uD.interpolate(u_ex)
# Create facet to cell connectivity required to determine boundary facets
tdim = mesh.topology.dim
fdim = tdim - 1
mesh.topology.create_connectivity(fdim, tdim)
boundary_facets = dolfinx.mesh.exterior_facet_indices(mesh.topology)
boundary_dofs = dolfinx.fem.locate_dofs_topological((W.sub(0),V), fdim, boundary_facets)
bcs = [dolfinx.fem.dirichletbc(uD, boundary_dofs, W.sub(0))]

(u, p) = ufl.TrialFunctions(W)
(v, q) = ufl.TestFunctions(W)

dx = ufl.Measure("dx", domain=mesh)
a = ufl.inner(ufl.grad(u), ufl.grad(v)) * dx + \
    ufl.div(v)*p*dx + q*ufl.div(u)*dx

a_f = dolfinx.fem.form(a)
A = dolfinx.fem.petsc.assemble_matrix(a_f, bcs=bcs)
A.assemble()
ns_vec = dolfinx.fem.Function(W)
ns_vec.x.array[Q_to_W] = 1
dolfinx.la.orthonormalize([ns_vec.vector])

nullspace = PETSc.NullSpace().create(vectors=[ns_vec.vector])

assert nullspace.test(A)

A.setNearNullSpace(nullspace)