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I am using interior penalty discontinuous Galerkin to solve a simple Laplace problem: \begin{align*} \nabla u=0 \end{align*} with prescribed 0 and 1 Dirichlet boundary conditions on opposite edges of a unit square. The other edges have 0 Neumann. enter image description here You'll see above that I have an image of the solution and it's gradient on the right.

The exact solution should be linear and it's gradient should be a constant value which is what I would get with continuous Galerkin. But with DG this isn't the case. I am getting some pollution of my solution and it's gradient at the corner. This won't happen if I defined continuous boundary conditions.

I need to integrate on the edge to queue for forces but the corner elements are causing me problems. I am using an edited version of step 74 of dealii for this and I can't impose BCs strongly in DG.

Any hints or tips on how to circumvent this issue? Other formulations and ways to impose BCs?

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  • $\begingroup$ How did you get the plot on the right? Maybe it's just a post processing problem $\endgroup$
    – FEGirl
    Commented Jan 5, 2023 at 15:20
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    $\begingroup$ A more important question would be the scaling of the axis first. $\endgroup$
    – ConvexHull
    Commented Jan 5, 2023 at 15:51
  • $\begingroup$ Moreover, what is the polynomial degree in your calculation and how did you choose the penalties for the Inner- and the BC-sides? For accurate results, the penalties should be as high as possible, but less enough to prevent a bad condition number. Also be aware that DG does not even directly impose $C^0$ solutions, which makes the result in this particular case extremely sensitive on the penalties, especially with a high polynomial degree. $\endgroup$
    – ConvexHull
    Commented Jan 5, 2023 at 16:08
  • $\begingroup$ #bobinthebox The plot on the right is directly using the Paraview Gradient filter of the solution. I get the same looking solution if I were to use the in-built dealii functions @ConvexHull the gradients aren't too high. The gradient in the domain is at 1, and at the corners goes form 0.2 to 1.5. Polynomial degree=1, the penalties are the same ones used in dealii step 74. $\endgroup$
    – CuteCompute
    Commented Jan 5, 2023 at 16:34
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    $\begingroup$ Importantly, if you use polynomial degree 1, the exact solution is in the discrete solution space and the discrete solution needs to be exact. If it isn't, something is wrong. $\endgroup$
    – Wolfgang Bangerth
    Commented Jan 6, 2023 at 6:04

2 Answers 2

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I'm not familiar with deal.II. However, to show that DG is able to reproduce the constant gradient solution I will post some results with a different tool using IP. The penalty is about $\sigma\approx1\times10^8$, the polynomial degree is $P=1$.

Edit:

The solver is based on a "grad/div" or "strong-weak/weak" formulation. The system is assembled using the flux form and solved in primal form (applying the Schur complement). Using this approach all contributions in the system matrix from element-outside are set to zero using Neumann BC's. The matrix system is solved using direct LU.

Coarse mesh: 2x2 elements

a

Medium mesh: 8x8 elements

b

Fine mesh: 32x32 elements

enter image description here

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  • $\begingroup$ Is there a term in the weak from associated with the zero Neumann BC? $\endgroup$
    – CuteCompute
    Commented Jan 11, 2023 at 8:58
  • $\begingroup$ @CuteCompute The solver is based on a "grad/div" or "strong-weak/weak" formulation. The system is assembled using the flux form and solved in primal form (applying the Schur complement). Using this approach all contributions in the system matrix from element-outside are set to zero using Neumann BC's. The matrix system is solved using direct LU. $\endgroup$
    – ConvexHull
    Commented Jan 11, 2023 at 17:05
  • $\begingroup$ Can you share a paper that best illustrates this formulation? $\endgroup$
    – CuteCompute
    Commented Jan 12, 2023 at 5:52
  • $\begingroup$ I think the methodology is quite standard. $\endgroup$
    – ConvexHull
    Commented Jan 13, 2023 at 21:39
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I fixed it after help from the dealii user group. It was a coding error on my part. Thank you all for your inputs

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