Hi, one option (which works in parallel out of the box) is to get the distance by solving the Eikonal equation, see Johan Hake’s answer here. However, if the boundary is complex there are some difficulties in getting the Newton solver to converge on the nonlinear problem that is the Eikonal equation. You can try to make it more robust following the discussion in the linked post. An alternative presented below is to decompose the boundary, solve simpler problems and then get the final distance as the minimum of the distance fields from the boundary pieces.

```
from dolfin import *
def distance_from_bdry_piece(facet_f, tag):
'''Solve Eikonal equation to get distance from tagged facets'''
mesh = facet_f.mesh()
V = FunctionSpace(mesh, 'CG', 1)
bc = DirichletBC(V, Constant(0.0), facet_f, tag)
u, v = TrialFunction(V), TestFunction(V)
phi = Function(V)
f = Constant(1.0)
# Smooth initial guess
a = inner(grad(u), grad(v))*dx
L = inner(f, v)*dx
solve(a == L, phi, bc)
# Eikonal equation with stabilization
eps = Constant(mesh.hmax()/100)
F = sqrt(inner(grad(phi), grad(phi)))*v*dx - inner(f, v)*dx + eps*inner(grad(phi), grad(phi))*v*dx
solve(F == 0, phi, bc)
return phi
def function_min(foos):
'''Function that is min of foos'''
f = foos.pop()
V = f.function_space()
vec = lambda u: as_backend_type(u.vector()).vec()
# The work vector
min_f = f.copy()
f_vec = vec(min_f)
# Reduce
while foos:
g = foos.pop()
f_vec.pointwiseMin(f_vec, vec(g))
# Sync
as_backend_type(min_f.vector()).update_ghost_values()
return min_f
mesh = UnitSquareMesh(32, 32)
facet_f = MeshFunction('size_t', mesh, mesh.topology().dim()-1, 0)
CompiledSubDomain('near(x[0], 0)').mark(facet_f, 1)
CompiledSubDomain('near(x[0], 1)').mark(facet_f, 2)
CompiledSubDomain('near(x[1], 0)').mark(facet_f, 3)
CompiledSubDomain('near(x[1], 1)').mark(facet_f, 4)
phis = [distance_from_bdry_piece(facet_f, tag) for tag in (1, 2, 3, 4)]
for i, phi in enumerate(phis):
File('d%d.pvd' % i) << phi
# Final distance is the min of these
phi = function_min(phis)
File('distance.pvd') << phi
```