I have to assemble and solve a large number of local patch problems. This is (of course) very inefficient as done in the example below and becomes infeasible for larger meshes.
Is there a way to speed things up? Ideally, one would want to build more than one local problem in one call to assemble/solve. Maybe this could be achieved by identifying non-overlapping patches, defining a set of measures respectively etc. However, this seems rather complicated so I would be glad I there was another way to improve efficiency.
from dolfin import *
# setup
mesh = UnitSquareMesh(50, 50)
mesh.init()
V = FunctionSpace(mesh, 'CG', 1)
f = Constant(1.0)
cf = CellFunction('size_t', mesh)
# solve patch problems one after another, argh...
for v in vertices(mesh):
# mark patch
cf.set_all(0)
for cid in v.entities(2).tolist():
cf[int(cid)] = 1
# define restricted space
restriction = Restriction(cf, 1)
Vr = FunctionSpace(restriction, "CG", 1)
# create patch element measure
dx = Measure('dx')[cf]
# define functions
u = TrialFunction(Vr)
v = TestFunction(Vr)
# define forms
a = dot(grad(u),grad(v))*dx(1)
L = f*v*dx(1)
# solve system
# (doesn't make much sense like this, just for demonstration)
u_local = Function(Vr)
solve(a == L, u_local)
