Space-time methods
   ==================
   WS 2023/2024
   Lecturer: Wick/Fischer/Roth
   Leibniz University Hannover, Institute of Applied Mathematics

   Format: 2+1 (90 minutes lecture per week; 45 minutes exercise per week)
   Times: Thu 17:15 - 18:45
   Place: Room C311
   First lecture:  Oct 12, 2023
   Last  lecture:  Jan 25, 2024


   Summary
   =======
   This lecture is devoted to space-time finite element methods. First, we concentrate on formulations, 
   numerical modeling and their discretization. Therein, the focus is on space-time 
   finite element discretizations with discontinuous Galerkin (dG) methods in time 
   and continuous Galerkin (cG) schemes in space. The governing concepts of the 
   numerical solution will be introduced as well. Afterward, we dive into concepts 
   of goal-oriented error control and adaptivity. We finish with current developments 
   of space-time goal-oriented error-controlled adaptivity applied to model order reduction.


   Note
   ====
   The main focus of this course is on the computer-based numerical solution  of 
   time-dependent partial differential equations in space-time. Prototypical applications 
   include the heat, wave, elastodynamics or Navier-Stokes equations. 
   Please do not confuse this course with a physics lecture on special or general relativity.
   

   Main literature
   ===============
   [1] T. Wick; Space-time methods, book, to appear, 2023; current version
       https://thomaswick.org/links/Wi23_st_book_preprint_Aug_8_2023.pdf 
   [2] H. Fischer, J. Roth, T. Wick, L. Chamoin, A. Fau;  
       MORe DWR: Space-time goal-oriented error control for incremental POD-based ROM;
       arXiv 2023, https://arxiv.org/abs/2304.01140

      
   Plan
   ====
   
   A) Introduction to space-time
   -----------------------------

   1. Background and motivation of this lecture
      Notation, space-time Sobolev spaces, Bochner spaces
      Heat equation in classical and space-time form
          
   2. ODEs in space-time
      Time-stepping schemes 
    
   3. PDEs in space-time form: discretization in space and time
  
   4. Space-time solution (I): nonlinear solution

   5. Space-time solution (II): linear solution 



   B) Error estimation and adaptivity part for ODEs
   ------------------------------------------------

   6. Goal-oriented error estimation ODEs
      Introduction via Ax=b
   
   7. Goal-oriented error estimation ODEs
      Final adaptive algorithm

   

   C) Error estimation and adaptivity part for PDEs
   ------------------------------------------------
   8. Space-time adaptivity for PDEs
      Optimization problems, Lagrange formalism, goal functionals

   9. Discussions of adjoint 
      (linear, running backwards in time, partial integration in time)
    
   10. Error control and localization
  
   11. Adaptive algorithm (chapter 12), 
       road map to model problems (Chapter 12.3)


   
   D) Advanced techniques and current state-of-the-art
   ---------------------------------------------------
   12. MORe DWR: Model order reduction with DWR (I)
       Lit.: https://arxiv.org/abs/2304.01140

   13. MORe DWR: Model order reduction with DWR (II)
       Lit.: https://arxiv.org/abs/2304.01140

   14. Quiz

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