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# UFL input for the Matrix-free Poisson Demo
from basix.ufl import element
from ufl import (
Coefficient,
Constant,
FunctionSpace,
Mesh,
TestFunction,
TrialFunction,
action,
dx,
grad,
inner,
)
coord_element = element("Lagrange", "triangle", 1, shape=(2,))
mesh = Mesh(coord_element)
# Function Space
e = element("Lagrange", "triangle", 2)
V = FunctionSpace(mesh, e)
# Trial and test functions
u = TrialFunction(V)
v = TestFunction(V)
# Constant RHS
f = Constant(V)
# Bilinear and linear forms according to the variational
# formulation of the equations:
a = inner(grad(u), grad(v)) * dx
L = inner(f, v) * dx
# Linear form representing the action of the form `a`` on the
# coefficient `ui`:`
ui = Coefficient(V)
M = action(a, ui)
# Form to compute the L2 norm of the error
usol = Coefficient(V)
uexact = Coefficient(V)
E = inner(usol - uexact, usol - uexact) * dx
forms = [M, L, E]
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