Analytical Solutions to Advection-Diffusion Problems for Verification of FEM Implementation

1

I am currently working on solving advection-diffusion problems using different DG formulations in FEniCS (Interior Penalty Method and Local DG). I would like to validate my implementation by comparing the simulated result to an exact/analytical solution for a 1D, time-dependent case. However, most examples that I have found in the literature involve periodic boundary conditions, which I understand are not currently possible to implement with DG elements using FEniCS.

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Has anyone found a way to successfully implement periodic boundary conditions with DG elements, or could perhaps suggest a different time-dependent analytical solution to the 1D advection-diffusion equation so I can see if my implementation is correct?

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As you know the exact solution on the boundary, why not implement the periodic condition as a Dirichlet condition?

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3

Great idea! I will try that, thanks!

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4

There are lots of analytical solutions to this problem in chromatography and adsorption applications

See for example https://pubs.acs.org/doi/10.1021/j150500a014

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5

I benchmarked some advection diffusion problems in the book Finite Element Methods for Flow Problems by Jean Donea and Antonio Huerta using FEniCS. The main focus of the book is continuous Galerkin methods. However, I sprinkled some finite volume methods using DG0 here and there.

You can check out my github page to find some benchmark problems:

Cheers!

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