Prerequisites
- Sobolev and Bochner spaces
- Finite elements for elliptic problems
Topics
- Parabolic and wave problems
- Finite elements and a priori error analysis of time-stepping schemes for parabolic and wave (first and second order formulation) problems
- Error estimators for elliptic, parabolic, and wave problems
Schedule
Time 14:00 (sharp) -- 17:00 (with one exception, see below)Location Sala ``Riunioni/Seminari'' C12/C13 (with one exception, see below), floor C, Department of Mathematics Pavia; online (see below for the Webex links)
- 09.01.2026 (Friday) Lecture 1 – Lorenzo Mascotto (LM).
Introduction of the course. Regularity of solutions to 2D Poisson problems. Singular expansion at the corners. Hints on hp-FEM and XFEM. Structure of adaptive algorithms. The ``obvious'' error estimator (residual in the H^-1 norm). The residual error estimator: meaning of efficiency and reliability. Definition and properties of the Clement quasi-interpolator. Proof of the reliability.
- 12.01.2026 (Monday) Lecture 2 – LM
Inverse estimates based on bubbles. Efficiency of the residual error estimator. Expansion of solutions to parabolic and wave problems into eigenfunctions of the spatial operator. Bochner regularity of solutions to parabolic equations.
- 13.01.2026 (Tuesday) Lecture 3 – Sergio Gomez (SG)
Petrov–Galerkin weak formulation for the heat equation; well-posedness; discontinuous Galerkin time discretization for the heat equation; consistency; time reconstruction operator; discrete inf–sup stability; quasi-optimality in discrete and continuous norms.
- 15.01.2026 (Thursday) Lecture 4 – SG CAREFUL: this lecture takes place at 10:00 (sharp) -- 13:00 in room C29, floor C, Department of Mathematics Pavia
Properties of the Lagrange interpolant associated with the left-sided Gauss-Radau nodes; auxiliary weight functions; robust continuous dependence on the data for the DG time discretization of the heat equation; stability and approximation properties of the Thomée projection operator; the Ritz projection operator; a priori error estimates for the DG time discretization of the heat equation.
- 16.01.2026 (Friday) Lecture 5 – SG
- 19.01.2026 (Monday) Lecture 6 – SG
- 20.01.2026 (Tuesday) Lecture 7 – LM
- 26.01.2026 (Monday) Lecture 8 – LM
Stability and convergence analysis for the CG time discretization of the heat equation (Aziz–Monk); discussion on the lack of unconditional stability of the DG time discretization for the second-order formulation of the wave equation; attempts in the literature to recover unconditional stability.
Stability and convergence analysis for: (i) the DG time discretization for the Hamiltonian formulation of the wave equation (Johnson); (ii) the CG time discretization for the Hamiltonian formulation of the wave equation (French–Peterson); (iii) the DG–CG time discretization for the second-order formulation of the wave equation (Walkington).
Residual error estimators for parabolic problems: the standard approach fails. Makridakis-Nochetto's reconstruction operator. Lifting operator of the temporal jumps. Error estimator involving the reconstruction operator. Error estimator given by the jump terms for the semi-discrete in time formulation of the heat equation. Reliability and efficiency of the two error estimators.
Walkington's formulation. Conforming reconstruction operator for the wave equation (smoother version of Makridakis-Nochetto). Error estimator involving the reconstruction operator. Error estimator based on the jump terms of the first derivative in time for the semi-discrete in time formulation of the wave equation. Reliability of the two error estimators.
Webex Links
Material (for personal use only)
The files are subject to changes (if you find typos, please drop us an email)- Shared folder with notes on the a priori error analysis
- Shared folder with notes on the a posteriori error analysis and some videos