A bug on boundary condition for i-tutorial unit 3.3.1, and related questions


I am looking into traceoperator/geom_free implementation of DG.

My starting point is this i-tutorial https://ngsolve.org/docu/latest/i-tutorials/unit-3.3.1-wavedg/wavedg.html?highlight=traceoperator

In version 1, you are using the wall boundary condition u.n=0. The numerical flux on the boundary is

phat = p uhat.n = 0
(Here maybe make a note that

returns p, not zero on the boundary edge, as it is sometimes confusing what the magic Other() operator is doing on domain boundary. See this discussionhttps://ngsolve.org/forum/ngspy-forum/28-inconsistency-between-apply-and-assemble-when-using-other-in-bnd-integral#146)

By increasing final time tend = 0.3, we find the boundary condition is indeed correctly implemented.

Then in Version 2, phat is the average on interior edges, but phat=p on boundary edges. Hence, to have a correct implementation of the boundary term Btr in block 11, I added the following hack to have the correct boundary condition:

gf = GridFunction(FacetFESpace(mesh, dirichlet=".*"))
gf.Set(1, BND)

Btr = BilinearForm(trialspace=fes_tr, testspace=fes_u)
Btr += 0.5 * (1+gf)*phat * (v*n) * dx(element_boundary=True) 

With this correction, the version 2 results are identical to version 1 up to time tend=0.3. (Otherwise, we observe something wrong when wave hit the boundary)

Unfortunately, the same trick does not work for the geom_free version.
Do you have any suggestions to fix the boundary condition for Version 3?

Another question: does traceoperator and/or geom_free work on a periodic mesh?
I have some trouble getting consistent results on periodic mesh for the traceoperator.


I like the HDG-like implementation of an explicit DG scheme.
The traceop can be explicitly realized via a matrix-vector form (although the performance is a bit slower than traceop, but it handles the boundary condition correctly)

    # test/trial functions
    u,v = V.TnT()
    uhat, vhat = VF.TnT()
    ### mass matrix for uhat (diagonal)
    Mtr = BilinearForm(VF)
    Mtr += uhat*vhat*dx(element_boundary=True)
    invuhat = Mtr.mat.CreateSmoother(VF.FreeDofs())

    ### u-->uhat (uhat=average of u)
    Bhat = BilinearForm(trialspace=V, testspace=VF)
    Bhat += u*vhat*dx(element_boundary=True)
    traceop = invuhat @ Bhat.mat

But the geom_free flag still does not work for a periodic facet fespace: the following operator shall allow for geom_free implementation, but it fails for a periodic facet fespace VF.

Btr = BilinearForm(trialspace=VF, testspace=W, geom_free=True)
Btr += uhat * (r*n) * dx(element_boundary=True)