A FETI method with a mesh independent condition number for the iteration matrix
The authors introduce a framework for FETI methods using ideas from the decomposition via Lagrange multipliers of \(H_{0}^{1}(\Omega)\) derived by \textit{P. A. Raviart} and \textit{J. M. Thomas} [Math. Comput. 31, 391--413 (1977; Zbl 0364.65082)] and complemented with the detailed work on polygonal domains developed by \textit{P. Grisvard} [Recherches en Mathématiques Appliquées. 22. Paris: Masson. Berlin: Springer-Verlag. (1992; Zbl 0766.35001)]. They consider the following model problem \[ \begin{aligned} -\Delta u + u & = f \quad \text{in }\Omega,\\ u & = 0 \quad \text{on }\partial\Omega. \end{aligned} \] The authors compute the action of the Lagrange multipliers using the natural \(H_{00}^{1/2}\) scalar product, therefore no consistency error appears. Their analysis allows to deal with cross points and floating subdomains in a natural manner. As a byproduct, they obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning. This result improves the standard asymptotic bound for this condition number given by \((1 + \log (H/h))^{2}\), where \(H\) and \(h\) are the characteristic subdomain size and element size respectively, shown by \textit{J. Mandel} and \textit{R. Tezaur} [Numer. Math. 73, No.~4, 473--487 (1996; Zbl 0880.65087)]. Finally, numerical results that confirm the presented theoretical analysis are available.
finite element tearing and interconnecting (FETI) methods
scientific article; zbMATH DE number 5546679
scientific article