Lecture WS 16/17 Selected Topics in Numerical Methods in Science and Technology

Adaptive Methods for Elliptic Operator Equations

Lecturer
Prof. Markus Bachmayr
Adaptive solvers are especially well-known in the context of finite element methods for partial differential equations, where they are used to guide the refinement of a mesh such that the solution meets a certain error criterion. In this lecture, we focus on basic concepts and on optimality properties: here the central question is whether a method can indeed be guaranteed to always make the best of a certain computational budget, even without any explicit knowledge on the sought solution. We consider basics from approximation theory and results for adaptive finite element methods, for wavelet methods (which can also be applied to problems with nonlocal operators), and for high-dimensional problems.

When & where:

Fr 10 (c.t.) - 12, Wegelerstr 6, SemR We 5.002

Additional lecture (substitute for Dec 23): Jan 30, 14:00, We 5.002

Lecture notes

  • Lecture notes (complete, version of Feb 10).

Topics and Literature

  • Elliptic boundary value problems, discretization and error estimates
  • Basics on the theory of nonlinear and best N-term approximation [DeV98], [C03]
  • A posteriori error estimators and mesh refinement for finite elements [B07], [NSV09]
  • Convergence and complexity of adaptive finite element methods [BDD04], [CKNS08], [NSV09], [N09]
  • Optimality theory of adaptive wavelet methods [CDD01], [CDD02], [GHS07], [S09]
  • Adaptivity for high-dimensional problems

Prerequisites

Basic knowledge on elliptic partial differential equations and Sobolev spaces.

Final Exam

Oral exam, by individual appointment.

P4E1   Practical Lab Numerical Simulation

The lecture is complemented by a practical lab devoted to the numerical implementation of adaptive finite elements and adaptive wavelet methods. The lab also offers the opportunity to try the Julia programming language, which is designed specifically for numerical purposes.

When & where:

Fr 14 (c.t.) - 16, Wegelerstr 6, SemR We 5.002

Course materials

  • Overview of Julia [View] [Download]
  • Julia example code: Mesh.jl, MeshPyPlot.jl, Quadrature.jl, FEMP1.jl, AFEM.jl
  • Wavelets [View] [Download]
  • Adaptive finite elements [View] [Download]