Mathematical Model and Weak Form

State your chosen strong-form PDE or PDE system on a domain \(\Omega \subset \mathbb R^d\), \(d=1,2,\) or \(3\), and define all unknown fields, material parameters, source terms, and units.

The following scalar second-order equation is only a small illustrative example of the kind of notation one might write down:

\[ -\nabla\cdot(A\nabla u) + b\cdot\nabla u + c u = f \quad \text{in } \Omega. \]

Do not treat this as the default project model. Your own project may look quite different: it may be nonlinear, vector-valued, mixed, time-dependent, an eigenvalue problem, or a coupled system.

Describe the boundary decomposition relevant to your model, for example

\[ \partial \Omega = \Gamma_D \cup \Gamma_N \cup \Gamma_R \cup \Gamma_{\mathrm{other}}, \]

where the subsets are disjoint and may be empty. Explain the physical meaning of every boundary part and every boundary condition.

Typical boundary-condition types include:

\[ u = u_D \quad \text{on } \Gamma_D, \]

\[ (A\nabla u)\cdot n = g_N \quad \text{on } \Gamma_N, \]

\[ (A\nabla u)\cdot n + h(u-u_\infty)=0 \quad \text{on } \Gamma_R. \]

Adapt signs and notation to your own PDE. For systems such as Stokes or elasticity, formulate the corresponding vector or mixed boundary conditions instead.

In your report, derive the weak form by multiplying with test functions, integrating by parts where appropriate, and explaining:

1. which function spaces contain the trial and test functions,
2. how essential boundary conditions enter the space or are imposed algebraically,
3. which natural boundary conditions appear in the linear form,
4. which Robin, reaction, coupling, or stabilization terms appear in the bilinear or nonlinear residual form,
5. what finite element spaces you use, for example continuous Lagrange elements, Taylor-Hood elements, or another justified choice.

Required Software Stack

Use the same implementation stack as Lecture 03:

• Python with the fenicsx environment/kernel,
• dolfinx, ufl, basix, mpi4py, petsc4py,
• dolfinx.fem.petsc.LinearProblem, NonlinearProblem, explicit PETSc assembly, SLEPc, or another appropriate PETSc-backed workflow,
• gmsh and dolfinx.io.gmshio.model_to_mesh or dolfinx.io.gmsh.read_from_msh for generated geometries where applicable,
• numpy and matplotlib for convergence data and visualization.

ParaView may also be used for visualization if applicable. If you use ParaView, write XDMF/VTK output from your scripts and document the exact plotting workflow well enough that the figures are reproducible.

For reproducibility, your repository must document the environment and include commands that can be run from a clean checkout. A good minimal check is:

python -c "import dolfinx, gmsh, petsc4py, mpi4py; print('FEniCSx stack ok')"

Document the installation and execution instructions in the README.md.

Exercise 1: Problem Choice, Model, and Weak Form

1. Choose a physical problem or PDE system and explain why it is interesting.
2. State the PDE, unknowns, material parameters, source terms, and physical assumptions.
3. Define the boundary decomposition and explain the role of each boundary subset.
4. Derive the weak form from the strong form, including boundary terms.
5. State the finite element spaces and polynomial orders you use.
6. Explain any simplifications, nondimensionalization, or assumptions needed to keep the project feasible.

Your write-up should be specific enough that another student could identify exactly which mathematical problem you solved.

Exercise 2: FEniCSx Implementation and Verification

Implement your chosen PDE first on a simple domain where verification is possible, such as an interval, unit square, rectangle, unit cube, or another geometry with a known solution.

1. Implement a reusable FEniCSx solver for your chosen PDE.
2. Use the Lecture 03 pattern: build function spaces, define UFL forms/residuals, impose boundary conditions, and solve with PETSc-backed linear algebra.
3. Accept geometry, material data, source terms, and boundary data in a way that lets you change test cases without rewriting the full solver.
4. Represent boundary subsets by facet tags whenever tags are available.
5. Verify the implementation by at least one of the following:
   • manufactured solution: choose an exact solution and derive matching source terms and boundary data,
   • comparison with another method: for example finite differences on a structured-grid problem if applicable,
   • comparison with a trusted reference: for example an analytical solution, benchmark value, or carefully documented literature result.
   • [mesh convergence study: show that the solver converges under mesh refinement]
6. Explain why your chosen verification method is appropriate for your PDE.
7. Include plots or tables that compare numerical and reference results.

Exercise 3: Mesh Convergence Study

Perform a mesh convergence study for the verification problem.

1. Use at least four mesh resolutions.
2. Define the mesh-size measure \(h\) you use.
3. Compute at least one error norm. Use more than one where meaningful, for example \(L^2\), \(H^1\) seminorm, energy norm, maximum norm, divergence error, eigenvalue error, or an application-specific quantity.
4. Plot error versus \(h\) on a log-log scale.
5. Estimate experimental convergence rates.
6. Compare the observed rates with theory or with a justified expectation for your element choice and PDE.
7. If exact error norms are not available, compare against a sufficiently refined reference solution and state the limitations.
8. If no theoretical rates are known for your problem, study the changes between consecutive mesh refinements. For example, compare quantities of interest or solution differences from mesh to mesh and explain whether the sequence appears to stabilize.

For many smooth scalar elliptic problems with P1 elements, one often expects approximately

\[ \|u-u_h\|_{L^2} = \mathcal O(h^2), \qquad \|u-u_h\|_{H^1} = \mathcal O(h). \]

These rates are not universal. State the expected behavior for your own PDE, element, regularity, and boundary conditions.

Backup Option: Stationary Heat Conduction

The intended project is an own-initiative FEM study. If you really do not know what to do, you may use a stationary heat-conduction project only as a backup option after talking to us.

A possible backup setup is:

• solve a stationary heat or diffusion equation,
• derive the weak form with Dirichlet, Neumann, and Robin boundary conditions,
• verify by manufactured solution or finite-difference comparison,
• perform a mesh convergence study with \(L^2\) and \(H^1\)-type errors,
• create a realistic thermal scenario such as a thermal bridge, heat sink, pipe insulation defect, or bracket-like geometry,
• compare at least two geometries or boundary-condition variants using a quantity such as total heat flux, maximum temperature, or effective thermal resistance.

This fallback is deliberately less open-ended. Use it only if you have discussed the topic choice with us and cannot identify a suitable own project.

Optional Extensions

If the mandatory scope is complete, you may extend the project with one of the following:

• time-dependent simulation,
• nonlinear material laws or nonlinear PDEs,
• spatially varying coefficients,
• multi-material subdomains,
• mixed finite element methods,
• eigenvalue problems,
• coupling between two physical fields,
• solver-performance comparison for larger meshes,
• a 3D geometry instead of a 2D cross-section,
• comparison of different numerical solvers and/or preconditioners for the linear system.

Suggested Repository Layout

This is only a suggestion. The final structure will look different depending on your PDE, geometry, solver workflow, and plotting pipeline. Choose a layout that makes your own project reproducible and understandable.

project01/
  pyproject.toml or environment.yml
  README.md
  src/
    solver.py
    geometry.py
    ... more Python package files
  tests/
    test_verification.py
  scripts/
    run_convergence.py
    run_scenario_study.py
  results/
    figures/
  report/
    report.qmd

Keep generated results separate from solver source code. Commit the scripts and small configuration files needed to regenerate figures; avoid committing large generated mesh or output files unless you have a clear reason.

TODO (for next year)

• Mention that for nonlinear Problems, residual checking and convergence monitoring is important and shall be included.