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Research

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Publication details for Dr Stefano Giani

Antonietti, Paola, Giani, Stefano & Houston, Paul (2013). hp-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains. SIAM Journal on Scientific Computing 35(3): A1417-A1439.

Author(s) from Durham

Abstract

In this paper we introduce the hp-version discontinuous Galerkin composite finite
element method for the discretization of second-order elliptic partial differential equations. This class
of methods allows for the approximation of problems posed on computational domains which may
contain a huge number of local geometrical features, or microstructures. While standard numerical
methods can be devised for such problems, the computational effort may be extremely high, as
the minimal number of elements needed to represent the underlying domain can be very large. In
contrast, the minimal dimension of the underlying composite finite element space is independent of
the number of geometric features. The key idea in the construction of this latter class of methods is
that the computational domain Ω is no longer resolved by the mesh; instead, the finite element basis
(or shape) functions are adapted to the geometric details present in Ω. In this paper, we extend these
ideas to the discontinuous Galerkin setting, based on employing the hp-version of the finite element
method. Numerical experiments highlighting the practical application of the proposed numerical
scheme will be presented.