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Final projects

Seminars

Seminars based on articles

Your presentation should include all proofs, and a brief introduction for those topics that we did not do in class.

  1. Babuska ⟺\Longleftrightarrow Brezzi?

  2. On some techniques for approximating boundary conditions in the finite element method Stenberg, 1995

  3. Analysis of the Shifted Boundary Method for the Poisson Problem in General Domains

  4. Static Condensation, Hybridization, and HDG Methods

  5. A unified study of continuous and discontinuous Galerkin methods

Seminars based on advanced theory

Your task is to prepare a one-hour lecture on one of the following topics:

  1. Drift-diffusion-reaction problems (chap. 8 of Quarteroni & Valli (1994))

    • Well posedness

    • Discretization

    • Inconsistent stabilizations (i.e., artificial diffusion)

    • Consistent stabilizations (i.e., GLS, SUPG)

  2. A Posteriori Error Estimates and Adaptive Meshes (chap. 10 of Ern & Guermond (2004))

    • (A review of) Residual-Based Error Estimates (done in class)

    • Hierarchical Error Estimates

    • Duality-Based Error Estimates

Coding projects

  1. Implement the shifted boundary method (starting from step-60)

  2. Implement the SQH scheme to Solve Nonsmooth PDE Optimal Control Problems

  3. Implement a coupled Laplace-Beltrami -- Poisson problem (step-38 + step-6)

  4. Add support for Chapel in deal.II* (step-64 con chapel)

  5. Implement step-64 using Kokkos.

  6. A time dependent problem, using SUNDIALS, and Nitsche boundary conditions.

References
  1. Stenberg, R. (1995). On some techniques for approximating boundary conditions in the finite element method. Journal of Computational and Applied Mathematics, 63(1–3), 139–148. 10.1016/0377-0427(95)00057-7
  2. Cockburn, B., Gopalakrishnan, J., & Lazarov, R. (2009). Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems. SIAM Journal on Numerical Analysis, 47(2), 1319–1365. 10.1137/070706616
  3. Hong, Q., Wang, F., Wu, S., & Xu, J. (2018). A unified study of continuous and discontinuous Galerkin methods. Science China Mathematics, 62(1), 1–32. 10.1007/s11425-017-9341-1
  4. Quarteroni, A., & Valli, A. (1994). Numerical Approximation of Partial Differential Equations. Clarendon Press.
  5. Ern, A., & Guermond, J.-L. (2004). Theory and Practice of Finite Elements (S. S. Antman, J. E. Marsden, & L. Sirovich, Eds.; Vol. 159). Springer New York. 10.1007/978-1-4757-4355-5
  6. Breitenbach, T., & Borzì, A. (2019). On the SQH Scheme to Solve Nonsmooth PDE Optimal Control Problems. Numerical Functional Analysis and Optimization, 40(13), 1489–1531. 10.1080/01630563.2019.1599911
Numerical Methods for Partial Differential Equations
Laboratory 2 — Meshes, connectivity, and deal.II abstractions