Graduate Seminar on "Multiscale Discretisation Techniques"

Up to 11 times in the weeks between workshops and in January. The idea is to form 5 teams of 2-3 students which will each cover one of the following topics in two 90 minute sessions, with an introductory talk in the first session and a talk on a more advanced topic in the second session.

Location: Science Park 2, 4th Floor, Room 416

Schedule/Topics
Tue, Oct 18 13:45-15:15 "Multiscale Finite Elements" (Buck/Kollmann, Willems)
Presentation slides (PDF-File, 927KB)
Mon, Nov 7 13:45-15:15 "Multiscale Finite Elements" (Buck/Kollmann, Willems)
Notes (PDF-File, 645KB), Paper (PDF-File, 807KB)
Tue, Nov 8 13:45-15:15 "Variational Multiscale Method" (Amos/Gangl, Zulehner)
Presentation slides (PDF-File, 772KB)
Mon, Nov 14 13:45-15:15 "Variational Multiscale Method" (Amos/Gangl, Zulehner)
Presentation slides (PDF-File, 181KB)
Mon, Nov 14 15:30-17:00 "Mixed Multiscale Methods" (Hrtus/Kolmbauer, Langer)
Presentation slides (PDF-File, 950KB)
Tue, Nov 15 13:45-15:15 "Mixed Multiscale Methods" (Hrtus/Kolmbauer, Langer)
Presentation slides (PDF-File, 188KB)
Mon, Dec 5 10:15-11:45 "Heterogeneous Multiscale Method" (Gahalaut/Wolfmayr, Nordbotten, Gfrerer)
Presentation slides (PDF-File, 288KB)
Mon, Dec 5 13:45-15:15 "Heterogeneous Multiscale Method" (Gahalaut/Wolfmayr, Nordbotten, Gfrerer)
Presentation slides (PDF-File, 195KB)
Mon, Dec 5 15:30-16:30 "Empirical Upscaling & Stochastic Homogenisation" (Alyaev/Nayak/Teckentrup, Scheichl)
Presentation slides (PPTX-File, 9.9MB), Notes (PDF-File, 1MB)
Paper (PDF-File, 5.2MB), Paper (PDF-File, 116KB)
Tue, Dec 6 13:45-15:45 "Empirical Upscaling & Stochastic Homogenisation" (Alyaev/Nayak/Teckentrup, Scheichl)
Presentation slides (PDF-File, 250KB)

Literature:

  1. Multiscale Finite Elements.
    • Basic methodology and theory for periodic coefficients for second-order elliptic equations.
      [Y. Efendiev and T.Y. Hou, Multiscale Finite Element Methods: Theory and Applications, Springer, New York, 2009]
    • Extension to higher order elements.
      [G. Allaire and R. Brizzi, "A multiscale finite element method for numerical homogenization", Multiscale Model. Simul. 4:790–812, 2005]
  2. Variational Multiscale Method.
    • Abstract methodology, basic concepts, Green’s functions, residual–free bubbles.
      [T.J.R. Hughes, G.R. Feijóo, L. Mazzei and J.-B. Quincy, "The variational multiscale method – a paradigm for computational mechanics", Comput. Meth. Appl. Mech. Engrg. 166:3–24, 1998]
      [F. Brezzi, "Interacting with the subgrid world", Numerical analysis 1999, CRC Press, 2000]
    • Application to advection-diffusion problems.
      [T.J.R. Hughes and G. Sangalli, "Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization, stabilized methods", SIAM J Num Anal 45:539-557, 2007]
  3. Mixed Multiscale Methods.
    • Mixed multiscale finite elements for elliptic problems with oscillating coefficients.
      [Z. Chen and T.Y. Hou, "A mixed multiscale finite element method for elliptic problems with oscillating coefficients" Math. Comput. 72:541-576, 2002]
    • Mixed variational multiscale methods for elliptic problems with oscillating coefficients.
      [T. Arbogast and K.J. Boyd, "Subgrid upscaling and mixed multiscale finite elements", SIAM J. Numer. Anal. 44:1150-1171, 2006]
  4. Heterogeneous Multiscale Methods.
    • HMM for elliptic problems with periodic coefficients.
      [A. Abdulle, "On a priori error analysis of fully discrete heterogeneous multiscale FEM", Multiscale Model. Simul. 4:447-459, 2005]
    • HMM for the wave equation with periodic coefficients.
      [A. Abdulle and M. Grote, "Finite element heterogeneous multiscale method for the wave equation", Multiscale Model. Simul. 9:766792, 2011]
  5. Empirical Upscaling Methods & Stochastic Homogenisation.
    • Review of Empirical Upscaling Methods.
      [L. Durlofsky, "Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media", Water Res. Research 27:699-708, 1991] (also [WRR 28:1791-1800, 1992])
      [C. Farmer, "Upscaling: A review", Int. J. Numer. Meth. Fluids 40:6378, 2002]
    • Stochastic Homogenisation and Numerical Upscaling.
      [A. Bourgeat and A. Piatnitski, "Approximations of effective coefficients in stochastic homogenization", Ann. I. H. Poincaré PR 40:153-165, 2004]