Thomas Wick Institute for Applied Mathematics Leibniz University Hannover WS 2019/2020 Goal-oriented a posteriori error estimation and adaptive finite elements ======================================================================== 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