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 23 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  13:4515:15 
"Multiscale Finite Elements" (Buck/Kollmann, Willems) Presentation slides (PDFFile, 927KB) 
Mon, Nov 7  13:4515:15 
"Multiscale Finite Elements" (Buck/Kollmann, Willems) Notes (PDFFile, 645KB), Paper (PDFFile, 807KB) 
Tue, Nov 8  13:4515:15 
"Variational Multiscale Method" (Amos/Gangl, Zulehner) Presentation slides (PDFFile, 772KB) 
Mon, Nov 14  13:4515:15 
"Variational Multiscale Method" (Amos/Gangl, Zulehner) Presentation slides (PDFFile, 181KB) 
Mon, Nov 14  15:3017:00 
"Mixed Multiscale Methods" (Hrtus/Kolmbauer, Langer) Presentation slides (PDFFile, 950KB) 
Tue, Nov 15  13:4515:15 
"Mixed Multiscale Methods" (Hrtus/Kolmbauer, Langer) Presentation slides (PDFFile, 188KB) 
Mon, Dec 5  10:1511:45 
"Heterogeneous Multiscale Method" (Gahalaut/Wolfmayr, Nordbotten, Gfrerer) Presentation slides (PDFFile, 288KB) 
Mon, Dec 5  13:4515:15 
"Heterogeneous Multiscale Method" (Gahalaut/Wolfmayr, Nordbotten, Gfrerer) Presentation slides (PDFFile, 195KB) 
Mon, Dec 5  15:3016:30 
"Empirical Upscaling & Stochastic Homogenisation" (Alyaev/Nayak/Teckentrup, Scheichl) Presentation slides (PPTXFile, 9.9MB), Notes (PDFFile, 1MB) Paper (PDFFile, 5.2MB), Paper (PDFFile, 116KB) 
Tue, Dec 6  13:4515:45 
"Empirical Upscaling & Stochastic Homogenisation" (Alyaev/Nayak/Teckentrup, Scheichl) Presentation slides (PDFFile, 250KB) 
Literature:

Multiscale Finite Elements.

Basic methodology and theory for periodic coefficients for secondorder 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]

Basic methodology and theory for periodic coefficients for secondorder elliptic equations.

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 advectiondiffusion problems.
[T.J.R. Hughes and G. Sangalli, "Variational multiscale analysis: the finescale Green’s function, projection, optimization, localization, stabilized methods", SIAM J Num Anal 45:539557, 2007]

Abstract methodology, basic concepts, Green’s functions, residual–free bubbles.

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. 72:541576, 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. 44:11501171, 2006]

Mixed multiscale finite elements for elliptic problems with oscillating coefficients.

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. 4:447459, 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]

HMM for elliptic problems with periodic coefficients.

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 27:699708, 1991] (also [WRR 28:17911800, 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 40:153165, 2004]

Review of Empirical Upscaling Methods.