Thomas Wick
Institute for Applied Mathematics
Leibniz University Hannover
WS 2019/2020

Goal-oriented a posteriori error estimation and adaptive finite elements
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Info: Goal-oriented is realized via the
      dual-weighted residual (DWR) method


Literature
----------
[1] Lecture notes Wick; Numerical methods for PDEs, Chapter 8, LUH, Oct 2019
    www.thomaswick.org/links/lecture_notes_Numerics_PDEs_Oct_12_2019.pdf
[2] Multiple goal functionals with Bernhard Endtmayer, 2017-2019
[3] Bangerth/Rannacher; LN, Birkhaeuser, 2003
[4] Besier/Rannacher; 2012
[5] P.G. Ciarlet; Linear and nonlinear functional analysis; 2013
[6] Richter/Wick; JCAM, 2015
[7] Rannacher/Vihharev; JNUM, 2013
[8] Bangerth/Geiger/Rannacher; CMAM, 2010


Formal schedule
---------------
14 lectures a 1.5 hours
7  exercises a 1.5 hours


Topics
------

1) Motivation for mesh adaptivity,
   and adaptivity in general;
   examples for goal functionals
   Basics of AFEM: solve, estimate, mark, refine

2) Basics to error estimation and duality [3], Sec. 1.4
   Directional derivatives [5]
   Lagrangian for optimization problem:
   min(J(u) - J(u_h)) s.t. a(u,v) = l(v)

3) DWR for ODE problems [3], Chapter 2


4) DWR for Poisson [1,3,6]
   Some of the Original Proofs
   Classical localization
   PU (partition of unity) localization
   Approximation of the adjoint solution

5) Effectivity of some methods [6]
   Definition of effectivity and 
   indicator indices

6) DWR for nonlinear problems [3,1]
   Basics 
   Algorithm
   Primal and dual error parts

7) Multiple goal functionals [2]
   motivated by towards multiphysics

8) Balancing discretization and iteration errors [7,2]

9) Two-sided error estimates [2]

10) Practical aspects
   - refinining elements (triangles, prisms, quads, hexs)
   - mesh regularity
   - hanging nodes (Oden/Carey)


11) DWR for space-time [4,3,8]
    for NSE (Besier/Rannacher) [4]
    for wave equation [8]

12) DWR for space-time [4,3,8]

13) DWR for space-time [4,3,8]

14) Revision of the most important topics