# Poisson equation¶

It is what it is, a conforming discretization on a regular mesh using piecewise quadratic elements.

As usual we start by importing firedrake and setting up the problem.:

from firedrake import *

N = 128

mesh = UnitSquareMesh(N, N)

V = FunctionSpace(mesh, "CG", 2)

u = TrialFunction(V)
v = TestFunction(V)

a = inner(grad(u), grad(v)) * dx

x = SpatialCoordinate(mesh)
F = Function(V)
F.interpolate(sin(x*pi)*sin(2*x*pi))
L = F*v*dx

bcs = [DirichletBC(V, Constant(2.0), (1,))]

uu = Function(V)


With the setup out of the way, we now demonstrate various ways of configuring the solver. First, a direct solve with an assembled operator.:

solve(a == L, uu, bcs=bcs, solver_parameters={"ksp_type": "preonly",
"pc_type": "lu"})


Next, we use unpreconditioned conjugate gradients using matrix-free actions. This is not very efficient due to the $$h^{-2}$$ conditioning of the Laplacian, but demonstrates how to request an unassembled operator using the "mat_type" solver parameter.:

uu.assign(0)
solve(a == L, uu, bcs=bcs, solver_parameters={"mat_type": "matfree",
"ksp_type": "cg",
"pc_type": "none",
"ksp_monitor": None})


Finally, we demonstrate the use of a AssembledPC preconditioner. This uses matrix-free actions but preconditions the Krylov iterations with an incomplete LU factorisation of the assembled operator.:

uu.assign(0)
solve(a == L, uu, bcs=bcs, solver_parameters={"mat_type": "matfree",
"ksp_type": "cg",
"ksp_monitor": None,


To use the assembled matrix for the preconditioner we select a "python" type:

"pc_type": "python",


and set its type, by providing the name of the class constructor to PETSc.:

"pc_python_type": "firedrake.AssembledPC",


Finally, we set the preconditioner type for the assembled operator:

"assembled_pc_type": "ilu"})


This demo is available as a runnable python file here.