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
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[1] Lecture notes Wick; Numerical methods for PDEs, Chapter 7, LUH, 2018
[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
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14 lectures à 1.5 hours
7 exerices à 1.5 hours
Topics
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1) Motivation for mesh adaptivity,
examples for goal functionals
Basics of AFEM: solve, estimate, mark, refine
Basics of error estimation [1]
Basics to adjoints [3]
2) Directional derivatives [5]
Lagrangian for optimization problem:
min(J(u) - J(u_h)) s.t. a(u,v) = l(v)
3) DWR for Poisson [1,3,6]
Some of the Original Proofs
Classical localization
PU (partition of unity) localization
Approximation of the adjoint solution
4) Effectivity of some methods [6]
Definition of effectivity and
indicator indices
5) DWR for nonlinear problems [3,1]
Basics
Algorithm
Primal and dual error parts
6) Multiple goal functionals [2]
motivated by towards multiphysics
7) Balancing discretization and iteration errors [7,2]
8) Two-sided error estimates [2]
9) Practical aspects
- refinining elements (triangles, prisms, quads, hexs)
- mesh regularity
- hanging nodes (Oden/Carey)
10) DWR for time-dependent problems
illustrated for ODEs [3,4]
11) Some error estimates for DWR for ODEs
12) DWR for space-time [4,3,8]
for NSE (Besier/Rannacher) [4]
for wave equation [8]
13) DWR for space-time [4,3,8]
14) DWR for space-time [4,3,8]