Eun-Hee Park

A FETI-DP Formulation for the Stokes Problem without Primal Pressure Components

A scalable FETI-DP (dual-primal finite element tearing and interconnecting) algorithm for the Stokes problem that employs a lumped preconditioner is developed and analyzed. A pair of inf-sup stable velocity and pressure finite element spaces is used to obtain a discrete problem. Differently from previous approaches, no primal pressure unknowns are selected and only velocity primal unknowns at subdomain corners are selected. This leads to a symmetric and positive definite coarse problem matrix in the FETI-DP operator, while a larger and indefinite coarse problem appears in the previous approaches. In addition, its condition number bound is proved to be the same as the FETI-DP algorithm with a lumped preconditioner for elliptic problems. Numerical results are included.

A dual iterative substructuring method with a penalty term

An iterative substructuring method with Lagrange multipliers is considered for second order elliptic problems, which is a variant of the FETI-DP method. The standard FETI-DP formulation is associated with the saddle-point problem which is induced from the minimization problem with a constraint for imposing the continuity across the interface. Starting from the slightly changed saddle-point problem by addition of a penalty term with a positive penalization parameter η, we propose a dual substructuring method which is implemented iteratively by the conjugate gradient method. In spite of the absence of any preconditioners, it is shown that the proposed method is numerically scalable in the sense that for a large value of η, the condition number of the resultant dual problem is bounded by a constant independent of both the subdomain size H and the mesh size h. Computational issues and numerical results are presented.

A Domain Decomposition Method Based on Augmented Lagrangian with a Penalty Term

An iterative substructuring method with Lagrange multipliers is considered for the second order elliptic problem, which is a variant of the FETI-DP method. The standard FETI-DP formulation is associated with a saddle-point problem which is induced from the minimization problem with a constraint for imposing the continuity across the interface. Starting from the slightly changed saddle-point problem by addition of a penalty term with a positive penalization parameter η, we propose a dual substructuring method which is implemented iteratively by the conjugate gradient method. In spite of the absence of any preconditioners, it is shown that the proposed method is numerically scalable in the sense that for a large value of η, the condition number of the resultant dual problem is bounded by a constant independent of both the subdomain size H and the mesh size h. We discuss computational issues and present numerical results.

Recent Advances in Domain Decomposition Methods for the Stokes Problem

Domain decomposition methods for the Stokes problem are developed under a more general framework, which allows both continuous and discontinuous pressure functions and more flexibility in the construction of the coarse problem. For the case of discontinuous pressure functions, a coarse problem related to only primal velocity unknowns is shown to give scalability in both dual and primal types of domain decomposition methods. The two formulations are shown to have the same extreme eigenvalues and the ratio of the two extreme eigenvalues weakly depends on the local problem size. This property results in a good scalability in both the primal and dual formulations for the case with discontinuous pressure functions. The primal formulation can also be applied to the case with continuous pressure functions and various numerical experiments are carried out to present promising features of our approach.

Absolutely stable explicit schemes for reaction systems.

We introduce two numerical schemes for solving a system of ordinary differential equations which characterizes several kinds of linear reactions and diffusion from biochemistry, physiology, etc. The methods consist of sequential applications of the simple exact solver for a reversible reaction. We provide absolute stability and convergence of the proposed explicit methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)