Add example for mixed Dirichlet–Neumann boundary conditions (2D Poisson equation)

[akarve1507]

Describe new/missing feature

I would like to contribute a tutorial demonstrating mixed boundary conditions (Dirichlet + Neumann) for the 2D Poisson equation in DOLFINx using Python.

The example will show how to:

• Apply Dirichlet and Neumann boundary conditions on different boundaries of the same domain
• Formulate the weak form including both terms
• Visualize the computed solution using PyVista

Mathematical Problem Setup

−∇²u = f in Ω = [0,1] × [0,1]

u = 0 on Γ_D : y = 0 (Dirichlet BC)
∂u/∂n = 1 on Γ_N : y = 1 (Neumann BC)
∂u/∂n = 0 on Γ_nat : x = 0,1 (Natural no-flux BC)

f = 1

Environment
I am working on macOS Sonoma 14.x using Python 3.11 installed through conda-forge.
DOLFINx is installed via the fenics-dolfinx package, along with dependencies including pyvista, mpi4py, and petsc4py.
The environment is set up using a Conda environment on an Apple M-series (ARM64) machine, and visualization is handled locally through PyVista.

I searched existing issues and did not find any example or discussion related to mixed Dirichlet–Neumann boundary condition tutorials in DOLFINx.

Suggested user interface

[jorgensd]

There is a tutorial for this at:
https://jsdokken.com/dolfinx-tutorial/chapter3/neumann_dirichlet_code.html

[akarve1507]

Thank you for the clarification and for pointing out that the mixed Dirichlet–Neumann boundary condition example already exists in the FEniCSx tutorials.
While reviewing that example, I noticed that it currently uses a static PyVista for visualization and does not include an interactive PyVista workflow or instructions for exporting results to ParaView.
I would like to propose extending or supplementing the existing tutorial with a short section focused on visualization and post-processing.

This could include:

• Demonstrating how to open an interactive PyVista viewer (rotation, color maps, contours, etc.)
• Explaining how to export solutions to .xdmf or .vtu for ParaView visualization

This addition would improve usability and help new users better understand how to explore and present their results.
Please let me know if this would be a useful contribution.

[jorgensd]

Thank you for the clarification and for pointing out that the mixed Dirichlet–Neumann boundary condition example already exists in the FEniCSx tutorials. While reviewing that example, I noticed that it currently uses a static PyVista for visualization and does not include an interactive PyVista workflow or instructions for exporting results to ParaView. I would like to propose extending or supplementing the existing tutorial with a short section focused on visualization and post-processing.

This could include:

• Demonstrating how to open an interactive PyVista viewer (rotation, color maps, contours, etc.)
• Explaining how to export solutions to .xdmf or .vtu for ParaView visualization

This addition would improve usability and help new users better understand how to explore and present their results. Please let me know if this would be a useful contribution.

In: http://jsdokken.com/FEniCS-workshop/src/outputting.html I explain outputting formats and how to open pyvista.
A brief discussion on Paraview exists at: https://jsdokken.com/dolfinx-tutorial/chapter1/membrane_paraview.html
There is also a demo on visualization with pyvista at https://docs.fenicsproject.org/dolfinx/main/python/demos/demo_pyvista.html

Not all demos can cover all visualization aspects. If you find things to improve in the aforementioned demos, you can open a pull request at
https://github.com/jorgensd/dolfinx-tutorial/pulls
or at:
https://github.com/jorgensd/FEniCS-workshop/pulls