Isogeometric Analysis Discontinuous Galerkin discretizations for elliptic problems with discontinuous coefficients

Dr. Ioannis Toulopoulos

March 18, 2014, 2:30 p.m. S2 059

In this talk, Isogeometric Analysis (IGA) methods utilizing discontinuous approximations spaces for the solution of an elliptic problem with discontinuous coefficients will be presented.

The problem is set in a complex domain $\Omega \subset \mathbb{R}^d, d=2,3$, which is subdivided in a union of sub-domains, $\bar{\Omega}=\cup_{i=1}^N \bar{\Omega_i}$, with interior interfaces $\Gamma=\cup_{i=1}^N \partial \Omega_i \smallsetminus \partial \Omega$. The diffusion coefficients may have jump discontinuities only along the interior interfaces, $F \in \Gamma$. The solution of the problem is approximated in every sub-domain applying IGA methodology, without matching grid conditions along the $\partial \Omega_i$, as well without imposing continuity requirements for the approximation spaces on $\partial \Omega_i$. The numerical scheme is completed by applying Discontinuous Galerkin (DG) techniques. Numerical fluxes with interior penalty jump terms are used on the interfaces of the sub-domains.

In the first part of the talk, error estimates in the classical $|.|_{DG}$-norm (consisting of the broken gradient plus a jump term) will be shown under the usual regularity assumption, $u \in W^{s\geq 2,2}(\Omega)$. In the second part, we consider the model problem with low regularity solution $u \in W^{2,p \in(1,2)}(\Omega)$ and derive error estimates in the $|.|_{DG}$. These estimates are optimal with respect to space size discretization. The error analysis makes use of several auxiliary results of the finite element methods, e.g. trace inequalities, interpolation error estimates. These results will be again expressed and discussed in the IGA framework.