Portal de Periódicos da CAPES

Additive Schur Complement Approximation and Application to Multilevel Preconditioning

2012; Society for Industrial and Applied Mathematics; Volume: 34; Issue: 6 Linguagem: Inglês

10.1137/110845082

1095-7197

Johannes Kraus,

Electromagnetic Simulation and Numerical Methods

This paper introduces an algorithm for additive Schur complement approximation (ASCA), which can be applied in various iterative methods for solving systems of linear algebraic equations arising from finite element (FE) discretization of partial differential equations (PDE). It is shown how the ASCA can be used to set up a nonlinear algebraic multilevel iteration (AMLI) method. This requires the construction of a linear (multiplicative) two-level preconditioner at each level. The latter is computed in the course of a simultaneous exact two-by-two block factorization of local (stiffness) matrices associated with a covering of the entire domain by overlapping subdomains. Unlike in Schwarz type domain decomposition methods, this method does not require a global coarse problem but instead uses local coarse problems to provide global communication. A robust condition number bound is proved for a particular partitioning of the nodes of a uniform grid. The presented numerical tests demonstrate that the ASCA, when combined with a proper AMLI-cycle, results in a multilevel method of optimal order of computational complexity.