Hyea Hyun Kim

Discontinuous Galerkin (DG) methods are a class of efficient tools for solving fluid flow problems. There are in the literature many greatly successful DG methods. In this paper, a new staggered DG method for the Stokes system is developed and analyzed. The key feature of our method is that the discrete system preserves the structures of the continuous problem, which results from the use of our new staggered DG spaces. This also provides local and global conservation properties, which are desira

A balancing domain decomposition by constraints (BDDC) algorithm with enriched coarse spaces is developed and analyzed for two-dimensional elliptic problems with oscillatory and high contrast coefficients. To obtain a robust algorithm based on the classical BDDC framework for conforming finite element methods, a set of enriched primal unknowns is constructed. The enriched component of the primal unknowns is chosen to reflect the local structures of the coefficient by solving two types of general

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

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 FET

A BDDC (balancing domain decomposition by constraints) method is developed for elliptic equations, with discontinuous coefficients, discretized by mortar finite element methods for geometrically nonconforming partitions in both two and three space dimensions. The coarse component of the preconditioner is defined in terms of one mortar constraint for each edge/face, which is the intersection of the boundaries of a pair of subdomains. A condition number bound of the form $C\max_i\{(1+\log(H_i/h_i)

In this paper, a FETI-DP formulation for three dimensional elasticity on nonmatching grids over geometrically nonconforming subdomain partitions is considered. To resolve the nonconformity of the finite elements, a mortar matching condition is imposed on the subdomain interfaces (faces). A FETI-DP algorithm is then built by enforcing the mortar matching condition in dual and primal ways. In order to make the FETI-DP algorithm scalable, a set of primal constraints, which include average and momen

In this paper, a three-level balancing domain decomposition by constraints (BDDC) algorithm is developed for the solutions of large sparse algebraic linear systems arising from the mortar discretization of elliptic boundary value problems. The mortar discretization is considered on geometrically nonconforming subdomain partitions. In two-level BDDC algorithms, the coarse problem needs to be solved exactly. However, its size will increase with the increase of the number of the subdomains. To over

A FETI-DP (dual-primal finite element tearing and interconnecting) formulation for the two-dimensional Stokes problem with mortar methods is considered. Separate sets of unknowns are used for velocity on interfaces, and the mortar constraints are enforced on the velocity unknowns by Lagrange multipliers. Average constraints on edges are further introduced as primal constraints to solvethe Stokes problem correctly and to obtain a scalable FETI-DP algorithm. A Neumann--Dirichlet preconditioner is

A balancing domain decomposition by constraints (BDDC) algorithm is developed for compressible elasticity problems in three dimensions with mortar discretization on geometrically nonconforming subdomain partitions. Material parameters of the elasticity problems may have jump across the subdomain interface. Coarse basis functions in the BDDC algorithm are constructed from primal constraints on faces, which are similar to the average matching condition and the moment matching condition considered

A FETI–DP (dual-primal finite element tearing and interconnecting) algorithm for the three-dimensional Stokes problem is developed and analyzed. This is an extension of the previous work for the two-dimensional problem in [H. H. Kim, C.-O. Lee, and E.-H. Park, SIAM J. Numer. Anal., 47 (2010), pp. 4142–4162]. Advantages of this approach are the coarse problem without primal pressure unknowns and the use of a computationally cheap lumped preconditioner. Especially in three dimensions, these advant

SUMMARY Convergence theories and a deluxe dual and primal finite element tearing and interconnecting algorithm are developed for a hybrid staggered DG finite element approximation of H(curl) elliptic problems in two dimensions. In addition to the advantages of staggered DG methods, the basis functions of the new hybrid staggered DG method are all locally supported in the triangular elements, and a Lagrange multiplier approach is applied to enforce the global connections of these basis functions.

Preconditioned conjugate gradient methods based on two‐level overlapping Schwarz methods often perform quite well. Such a preconditioner combines a coarse space solver with local components which are defined in terms of subregions that form an overlapping covering of the region on which the elliptic problem is defined. Precise bounds on the rate of convergence of such iterative methods have previously been obtained in the case of conforming lower order and spectral finite elements as well as in