DG advection in periodic domain

Question

Hi
I am trying to use DG method with FEnics. it doesnt work at all. I dont know if it is problem of periodic BC?

from dolfin import *
mesh = UnitInterval(16)
c0=Expression('exp(-pow((x[0]-0.5),2)/0.025)')

# Sub domain for Periodic boundary condition
class PeriodicBoundary(SubDomain):

    def inside(self, x, on_boundary):
        return bool(x[0] < DOLFIN_EPS and x[0] > -DOLFIN_EPS and on_boundary)

    def map(self, x, y):
        y[0] = x[0] - 1.0

# Create function space constrained by periodic BC
pbc = PeriodicBoundary()
V = FunctionSpace(mesh, 'DG', 1, constrained_domain=pbc)

c=TrialFunction(V)
v=TestFunction(V)

a_M = c*v*dx
M=assemble(a_M)

# Define normal component, mesh size and right-hand side
n = FacetNormal(mesh)

c=Function(V)

c_1=interpolate(c0,V)
c.assign(c_1)

k=[]
for i in range(4):
    k.append(Function(V))

T = 0.002
t = 0
dt =0.001
alpha=0
co=(1.-alpha)/2.
while t<=T:
    flux = avg(c)+co*jump(c)
    a=v*c.dx(0)*dx - dot(n,flux*v)*dS 
    b=assemble(a)
    solve(M,k[0].vector(),b)

    c.vector()[:]=k[0].vector()
    t += dt
    plot(c,rescale=False)

Comments

I wondered about this too. Apparently it does not work. See here. I would be interested in getting it to work though.

Answer 1

Discontinuous Lagrange elements + periodic boundary conditions are not presently supported.