Project:

Iterative solvers for chimera finite element schemes

Researcher(s)	:	Prof. Dr. R. Hiptmair, SAM, ETH Zürich
	:	Prof. Dr. J. Zou, The Chinese University of Hong Kong
Funding	:	no external funding
Duration	:	from mid 2005

A chimera mesh arises from overlaying two completely unrelated, usually (semi-)structured, finite element meshes [3], see Fig. 1. Body fitted chimera meshes are very well suited for the resolution of boundary layers [1]. This makes them very popular in computational fluid dynamics.

Figure 1: Example of a chimera mesh comprising two unrelated triangulations

We consider a second order elliptic boundary value problem on a domain that is equipped with a chimera mesh composed of two conforming simplicial finite element triangulations. On each we have a $H^{1}(\Omega)$-conforming finite element space, call them $V_{1}$ and $V_{2}$. We tacitly assume zero boundary conditions at internal edges of one of the meshes.

It is natural to use the sum space $V_{N}=V_{1}+V_{2}\subset H^{1}(\Omega)$ as a finite element space for the Galerkin discretization of the boundary value problem. However, the resulting linear system of equation may be very ill-conditioned due to near linear dependence of basis functions on the two meshes.

This raises the issue of fast iterative solution of the resulting linear system of equations. One idea is the chimera Schwarz iteration, which treats the two meshes in turns in a block-Gauss-Seidel manner [1,2]. However, this method will become very slow asymptotically.

We investigate the use of local ill-conditioned problems as building blocks for a faster iterative solver. The idea is borrowed from multigrid methods.