Winter semester 2026/2027 (3 VO + 1 UE)

Organization (as agreed with the participants)

• the lecture as well as the exercise class will be taught in English
• the preliminary meeting for the lecture and the exercise class takes place on Monday, October 5th, 2026 at 12:00 p.m. at the seminar room DA04G10 (Freihaus, green area, 4th floor)
• lecture: tba
• exercises: tba
• office hour: upon request (for potential questions concerning lecture or exercises)
• prerequisites: PDE theory, numerics of PDEs: stationary problems (in case of questions/doubts, please write an email)

Topics

• Finite Element Methods (FEMs) for elliptic PDEs (brief recap)
• residual a-posteriori error estimator (brief recap) and properties
• the idea of adaptive mesh-refinement
• plain convergence of adaptive FEM
• axioms of adaptivity
• linear convergence of adaptive FEM
• optimal convergence rates with respect to degrees of freedom / overall computational cost
• optimal interplay of adaptive mesh-refinement and iterative solvers
• newest-vertex bisection as a refinement strategy
• optimal adaptivity for nonlinear PDEs
• optimal goal-oriented adaptivity

Learning objectives

After successful completion of the course, students will be able to

• explain how a posteriori error estimators for elliptic differential equations are derived and mathematically justified
• explain the idea of adaptive mesh refinement
• formulate convergence and optimality of adaptive algorithms
• write a bachelor’s or master’s thesis in this area

Building on the lecture, topics for bachelor’s theses, master’s theses, and doctoral dissertations can be assigned under the supervision of Prof. Dirk Praetorius.