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Multiscale DG Methods

Numerical results at intermediate Peclet number for the MDG solution (light blue), and the coarse-scale solution (blue). The exact solution is in red

Proliferation of degrees-of-freedom has plagued the Discontinuous Galerkin method (DG) since its inception over thirty years ago. We developed a new class of Discontinuous Galerkin methods based on variational multiscale ideas. Our approach begins with an additive decomposition of the discontinuous finite element space into continuous (coarse) and discontinuous (fine) components. Variational multiscale analysis is used to define an interscale transfer operator that associates coarse and fine scale functions [3]. Composition of this operator with a donor DG method yields a new formulation (MDG) that combines the advantages of DG methods with the attractive and more efficient computational structure of a continuous Galerkin method [1,2].

References:

[1]	Guglielmo Scovazzi, Ph.D. thesis, part II, "Multiscale Methods in Science and Engineering", Mechanical Engineering Department, Stanford University, August 2004. PDF file
[2]	Thomas J. R. Hughes, Guglielmo Scovazzi, Pavel B. Bochev, and Annalisa Buffa, "A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method", Sandia report SAND-2005-2124J (Comp. Meth. Appl. Mech. Eng., 195(19-22), April 2006, pp. 2761-2787).
[3]	Pavel B. Bochev, Thomas J. R. Hughes, Guglielmo Scovazzi, "A multiscale discontinous Galerkin method", SAND-2005-2587C (in Lecture Notes in Computer Science, Vol. 3743, pp. 84-93, Springer, 2006).

Guglielmo Scovazzi

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