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Discontinuous Galerkin methods on arbitrarily shaped elements and their application to interface problems.

Emmanuil Georgoulis

Dienstag, 3. November 2020 14:15Uhr

Motivated by the problem of numerical treatment of curved boundaries and interfaces in numerical PDEs, will review some recent work on the development of discontinuous Galerkin (dG) methods which are able to be applied on meshes comprising of essentially arbitrarily-shaped elements [1,2]. The use of such element shapes makes possible the fitted representation of curved geometries, by moving the variational crime challenge from the domain representation (as is the case for classical FEM/dG) to the quadrature evaluations. The second part of my talk will focus on the application of these ideas to the specific problem of proof of a posteriori error bounds for elliptic and parabolic interface problems on curved interfaces [3,4,5].

[1] A. Cangiani, Z. Dong, and E. H. Georgoulis. hp–Version discontinuous Galerkin methods on essentially arbitrarily-shaped elements. Submitted for publication. PDF

[2] A. Cangiani, Z. Dong, E. H. Georgoulis and P. Houston. hp–Version discontinuous Galerkin methods on polygonal and polyhedral meshes. SpringerBriefs in Mathematics (2017)

[3] A. Cangiani, E. H. Georgoulis, and Y. Sabawi. Adaptive discontinuous Galerkin methods for elliptic interface problems. Mathematics of Computation 87(314) pp. 2675 – 2707 (2018)

[4] Stephen A. Metcalfe. Adaptive discontinuous Galerkin methods for nonlinear parabolic problems. PhD Thesis, University of Leicester (2015).

[5] Younis A. Sabawi. Adaptive discontinuous Galerkin methods for interface problems. PhD Thesis, University of Leicester (2017).