1D discontinuous galerkin boudary conditions

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1

Hi all! I am recently trying to solve the 1D advection equations with discontinuous galerkin method. The 1D boudary conditions is not working properly. The attached is the code that explains the problem:

m_size = 32
mesh = UnitIntervalMesh(m_size)

beta = Constant((0.5,))

parameters[“ghost_mode”] = “shared_facet”

V_dg = FunctionSpace(mesh, ‘DG’, 2)
phi, v = TrialFunction(V_dg), TestFunction(V_dg)
u_new, u_old = Function(V_dg), Function(V_dg)

u0 = Expression(“sin(2pix[0])”, degree=1, t=0, name=‘u0’) #
u_old = project(u0, V_dg)

u_exact = Expression("sin(2pi(x[0] - c*t)) ", degree=2, t=0, c=float(beta[0])) #

bcs = DirichletBC(V_dg, u_exact, “on_boundary”)

u_exact.t = 0.1
bcs.apply(u_old.vector())
print(u_old(0))

Is there a problem with the boundary conditions definition or application? Thanks a lot in advance!

2

Hi, it would make your question more readable if you could encapsulate your code between ``` .

This being said, boundary conditions must be imposed ‘‘weakly’’ in Discontinuous Galerkin discretizations. There is a very good package dolfin-dg that makes these things easier, the paper where the authors introduce it is also a good reference, you can check the references in there.

3

Thanks a lot for your reponse! I will encapsulate my code in the later questions!

Is there a reference that you may recommend where the reasonings behind why BC can only be imposed weakly for DG?

Thanks!

4

The paper by Nitsche and the classic review paper by Arnold et al.