Atle Loneland

Martin J. Gander, Atle Loneland, Talal Rahman

We propose a new, harmonically enriched multiscale coarse space (HEM) for domain decomposition methods. For a coercive high contrast model problem, we show how to enrich the coarse space so that the method is robust against any variations and discontinuities in the problem parameters both inside subdomains and across and along subdomain boundaries. We prove our results for an enrichment strategy based on solving simple, lower dimensional eigenvalue problems on the interfaces between subdomains, and we call the resulting coarse space the spectral harmonically enriched multiscale coarse space (SHEM). We then also give a variant that performs equally well in practice, and does not require the solve of eigenvalue problems, which we call non-spectral harmonically enriched multiscale coarse space (NSHEM). Our enrichment process naturally reaches the optimal coarse space represented by the full discrete harmonic space, which enables us to turn the method into a direct solver (OHEM). We also extensively test our new coarse spaces numerically, and the results confirm our analysis

Atle Loneland, Leszek Marcinkowski, Talal Rahman

In this paper we introduce an additive Schwarz method for a Crouzeix-Raviart Finite Volume Element (CRFVE) discretization of a second order elliptic problem with discontinuous coefficients, where the discontinuities are both inside the subdomains and across and along the subdomain boundaries. We show that, depending on the distribution of the coefficient in the model problem, the parameters describing the GMRES convergence rate of the proposed method depend linearly or quadratically on the mesh parameters $H/h$. Also, under certain restrictions on the distribution of the coefficient, the convergence of the GMRES method is independent of jumps in the coefficient.

Leszek Marcinkowski, Talal Rahman, Atle Loneland et al.

A symmetric and a nonsymmetric variant of the additive Schwarz preconditioner are proposed for the solution of a nonsymmetric system of algebraic equations arising from a general finite volume element discretization of symmetric elliptic problems with large jumps in the entries of the coefficient matrices across subdomains. It is shown in the analysis, that the convergence of the preconditioned GMRES iteration with the proposed preconditioners, depends polylogarithmically on the mesh parameters, in other words, the convergence is only weakly dependent on the mesh parameters, and it is robust with respect to the jumps in the coefficients.