Is this the correct way to use grad() in an H(curl) space?

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1

I have a system of coupled PDEs, and I use the following to generate test and trial functions:

fesm = HCurl(mesh, order=ord, complex=True, dirichlet="outer")
fes = FESpace([fesm,fesm])
# Define test and trial functions:
E, F = fes.TrialFunction()
Et, Ft = fes.TestFunction()

Now, I want to calculate the following term of a weak form: [tex]\int E \nabla (Et) [/tex].

When I naively try the following for my bilinear form, I get a dimensional mismatch error:

a = BilinearForm(fes, symmetric=True, eliminate_internal=True)
a += SymbolicBFI(InnerProduct(E,grad(Et)))

but, the following seems to work:

a += SymbolicBFI(InnerProduct(E,(grad(Et)[0],grad(Et)[1],grad(Et)[2])))

Is this the correct thing to do? I am still learning NGSolve. Thank you all for your hard work.

I saw a different post [1] saying that VectorH1 spaces will allow all div and curl and grad to be used simultaneously, but I wanted to make sure my attempt makes sense too.

[1] /forum/ngspy-forum/781-fe-space-with-curl-and-div

2

Hi Thauwa,

the code

a += SymbolicBFI(InnerProduct(E,grad(Et)))

leads to an Exception as E is a vector (with dimension three as I guess that you have a 3D mesh) whereas grad(Et) is a 3x3 matrix and InnerProduct expects that both inputs have the same dimension. Thus, your second code works, but then you have a different integral.

All integrators need at the end a scalar valued function as input, but your integral [tex]\int E\nabla Et[/tex] seems to be vector valued?

Note that if you want the Jacobian you need the “Grad” operator. With “grad” you obtain the transposed Jacobian: grad() = Grad().trans

Best
Michael

3

Hi Michael,

Thank you kindly for your response.
So, my weak formulation appears to be incorrect. In my system of coupled PDEs that focuses on curl of associated vectors (derived from a standard Maxwell system), I have the following term:
[tex]
\nabla\cdot E
[/tex]

According to 2, I must use the following identity to get the variational formulation:
[tex]
\int_{\Omega} \nabla\cdot E E_t dx =
-\int_{\Omega} E\cdot \nabla E_t dx +
\int_{\partial\Omega} n \cdot E E_t dS,
[/tex]

where n is the unit outwards normal vector. So, it seems to me that I cannot avoid a dimensional mismatch in the inner product because of the gradient operation. I thought I could just have a derivative taken on each row of my vector and summed up, as usual. That is, for elements v_i of E_t:

[tex](v_1, v_2, v_3) = (\partial_x v_1 + \partial_y v_1 + \partial_z v_1, \partial_x v_2 + \partial_y v_2 + \partial_z v_2, \partial_x v_3 + \partial_y v_3 + \partial_z v_3).[/tex]

But I am not sure whether this is valid anymore.
Do you have any advice on
how I can implement this [tex]E \nabla E_t[/tex] properly in NGSolve?
In the NGSolve forum post 3, they suggest using VectorH1 spaces for situations where grad(), curl() and div() might have to be used simultaneously, but I am concerned it will conflict with HCurl()'s reputation of being suitable for Maxwell-type problems involving curl terms.

4

Hi Thauwa,

for this integration by parts formula to hold you have (normally) that E is vector valued and Et is scalar valued. Then the term [tex]\int E\cdot\nabla Et[/tex] in NGSolve

a += SymbolicBFI( InnerProduct(E,grad(Et)))

works.

Without knowing the full (coupled) PDE in strong form (+ dimension of the unknowns) I can only guess that one of the unknowns is vector valued (in HCurl) and the other scalar valued (in H1), something like

fes1 = HCurl(mesh, order=1)
fes2 = H1(mesh,order=1)
fes = FESpace( [fes1,fes2] )

Best
Michael

5

Hi Michael,

Thank you very much for your clarifications!

Best,
Thauwa

6

mneunteufel, thanks for explanation.