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# Copyright (C) 2016 Lizao Li
"""Biharmonic Regge demo.
The bilinear form a(u, v) and linear form L(v) for Biharmonic equation in Regge formulation.
"""
import basix.ufl
from ufl import (
Coefficient,
FacetNormal,
FunctionSpace,
Identity,
Mesh,
TestFunctions,
TrialFunctions,
dot,
dS,
ds,
dx,
grad,
inner,
jump,
tr,
)
REG = basix.ufl.element("Regge", "tetrahedron", 1)
P = basix.ufl.element("Lagrange", "tetrahedron", 2)
mixed_element = basix.ufl.mixed_element([REG, P])
domain = Mesh(basix.ufl.element("P", "tetrahedron", 1, shape=(3,)))
mixed_space = FunctionSpace(domain, mixed_element)
p_space = FunctionSpace(domain, P)
(sigma, u) = TrialFunctions(mixed_space)
(tau, v) = TestFunctions(mixed_space)
f = Coefficient(p_space)
def S(mu):
"""The form S."""
return mu - Identity(3) * tr(mu)
def b(mu, v):
"""The form b."""
n = FacetNormal(domain)
return (
inner(S(mu), grad(grad(v))) * dx
- dot(dot(S(mu("+")), n("+")), n("+")) * jump(grad(v), n) * dS
- dot(dot(S(mu), n), n) * dot(grad(v), n) * ds
)
a = inner(S(sigma), S(tau)) * dx - b(tau, u) + b(sigma, v)
L = f * v * dx
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