Hyea Hyun Kim

Two-Level Overlapping Schwarz Algorithms for a Staggered Discontinuous Galerkin Method

Two overlapping Schwarz algorithms are developed for a discontinuous Galerkin finite element approximation of second order scalar elliptic problems in both two and three dimensions. The discontinuous Galerkin formulation is based on a staggered discretization introduced by Chung and Engquist [SIAM J. Numer. Anal., 47 (2009), pp. 3820--3848] for the acoustic wave equation. Two types of coarse problems are introduced for the two-level Schwarz algorithms. The first is built on a nonoverlapping subdomain partition, which allows quite general subdomain partitions, and the second on introducing an additional coarse triangulation that can also be quite independent of the fine triangulation. Condition number bounds are established and numerical results are presented.

A Preconditioner for the FETI-DP Formulation with Mortar Methods in Two Dimensions

In this paper, we consider a dual-primal FETI (FETI-DP) method for elliptic problems on nonmatching grids. The FETI-DP method is a domain decomposition method that uses Lagrange multipliers to match solutions continuously across subdomain boundaries in the sense of dual-primal variables. We use the mortar matching condition as the continuity constraints for the FETI-DP formulation. We construct a preconditioner for the FETI-DP operator and show that the condition number of the preconditioned FETI-DP operator is bounded by

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

C \max_{i=1,\ldots,N} \{ \left(1+\log\left( H_i / h_i \right)\right)^2 \},

A FETI-DP Formulation of Three Dimensional Elasticity Problems with Mortar Discretization

where

Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations

H_i

A Three-Level BDDC Algorithm for Mortar Discretizations

and

A Neumann--Dirichlet Preconditioner for a FETI-DP Formulation of the Two-Dimensional Stokes Problem with Mortar Methods

h_i

A BDDC Algorithm with Enriched Coarse Spaces for Two-Dimensional Elliptic Problems with Oscillatory and High Contrast Coefficients

are sizes of domain and mesh for each subdomain, respectively, and