Jung-Han Kimn's Publications

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    Thesis

  • Jung-Han Kimn
    Overlapping Schwarz Algorithms using Discontinuous Iterates for Poisson's Equation
    Ph.D. thesis, Technical Report TR2001-817, Department of Computer Science,
    Courant Institute, May 2001. pdf, abstract and PostScript files.

    Published

  • Jung-Han Kimn
    A Convergence Theory for An Overlapping Schwarz Algorithm using Discontinuous Iterates
    Numerische Mathematik, vol. 100 pp. 117-139, March, 2005. Journal version

    Abstract: A new type of overlapping Schwarz methods, using discontinuous iterates, is constructed by modifying the classical overlapping Schwarz algorithm. This new algorithm allows for discontinuous iterates across the artificial interface. For Poisson's equation, this algorithm can be considered as an overlapping version of Lions' Robin iteration method for which little is known concerning the rate of convergence. Since overlap improves the performance of the classical algorithms considerably, the existence of a uniform convergence factor is the fundamental question for our algorithm. A new theory using Lagrange multipliers is developed and conditions are found for the existence of an almost uniform convergence factor for the dual variables, which implies rapid convergence of the primal variables, in the two overlapping subdomain case. Our result also shows a relation between the boundary conditions of the given problem and the artificial interface condition. Numerical results for the general case with cross points are also presented. They indicate possible extensions of our results to this more general case.

  • Jung-Han Kimn and Marcus Sarkis
    OBDD: Overlapping Balancing Domain Decomposition Methods and Generalizations to the Helmholtz Equation
    In Olof B. Widlund and David E. Keyes, editors, Domain Decomposition Methods in Science and Engineering XVI, volume 55 of Lecture Notes in Computational Science and Engineering, Pages 309-316, Springer-Verlag, 2006 Prepublication version (Updated in Journal)

    Introduction: Balancing Domain Decomposition (BDD) methods belong to the family of preconditioners based on nonoverlapping decomposition of subregions and they have been tested successfully on several challenging large scale applications. Here we extend the BDD algorithms to the case of overlapping subregions and we name them Overlapping Balancing Domain Decomposition (OBDD) algorithms. Like the BDD methods, coarse space and weighting matrices play crucial roles in making both the proposed algorithms scalable with respect to the number of subdomains as well as making balanced the local Neumann subproblems on the overlapping subregions on each iteration of the preconditioned system. The OBDD algorithms also differ from the standard overlapping additive Schwarz method (ASM) of hybrid form since those are based on Dirichlet local problems on the overlapping subregions. This difference motivated us to generalize the OBDD algorithms to the Helmholtz equation where we use the Sommerfeld boundary condition for the local problems and a combination of partition of unity and plane waves for the coarse problem.

  • Jung-Han Kimn and Blaise Bourdin
    Numerical Implementation of Overlapping Balancing Domain Decomposition Methods on Unstructured Meshes
    In Olof B. Widlund and David E. Keyes, editors, Domain Decomposition Methods in Science and Engineering XVI, volume 55 of Lecture Notes in Computational Science and Engineering, Pages 301-307, Springer-Verlag, 2006 Prepublication version (Updated in Journal)

    Summary: The Overlapping Balancing Domain Decomposition (OBDD) methods can be considered as an extension of the Balancing Domain Decomposition (BDD) methods to the case of overlapping subdomains. This new approach, has been proposed and studied by Jung-Han Kimn and Marcus Sarkis. In this paper, we will discuss its practical parallel implementation and present numerical experiments on large unstructured meshes.

  • Jung-Han Kimn and Marcus Sarkis
    Restricted Overlapping Balancing Domain Decomposition Methods and Restricted Coarse Problem for the Helmholtz Equation
    Jung-Han Kimn and Marcus Sarkis, Computer Methods in Applied Mechanics and Engineering, 196 (2007) 1507-1514 Journal version

    Abstract: Overlapping balancing domain decomposition methods and their combination with restricted additive Schwarz methods are proposed for the Helmholtz equation. These new methods also extend previous works on nonoverlapping balancing domain decomposition methods toward simplifying their coarse problems and local solvers, and generalize restricted Schwarz methods limited to overlapping domain decomposition and Dirichlet local solvers, thus allowing nonoverlapping domain decomposition and/or Neumann and Sommerfeld local solvers. Finally, we introduce coarse spaces based on partitions of unity and planes waves and show how oblique projection coarse problems can be designed from restricted additive Schwarz methods. Numerical testes are presented.

    Submitted

  • Matthew Anderson and Jung-Han Kimn
    A Numerical Approach to Space-Time Finite Element Methods for Wave Equations
    Submitted to Journal of Computational Physics, 2006 arXiv.org Version

    Abstract: We study a space-time finite element approach for the nonhomogeneous wave equation using a continuous time Galerkin method. We present fully implicit examples in 1+1, 2+1, and 3+1 dimensions using linear quadrilateral, hexahedral, and tesseractic elements. Krylov solvers with additive Schwarz preconditioning are used for solving the linear system. We introduce a time decomposition strategy in preconditioning which significantly improves performance when compared with unpreconditioned cases.

    In Preparation

  • Jung-Han Kimn and Marcus Sarkis
    Theoretical Analysis Theory Overlapping Balancing Domain Decomposition Methods for Elliptic Problems
    In preparation

  • Jung-Han Kimn and Blaise Bourdin
    Parallel Implementation of Overlapping Balancing Domain Decomposition Methods on Unstructured Meshes
    In preparation

  • Suman Gupta, Jung-Han Kimn and Seung-Jong Park
    Fluid-based Simulation For Large Scale Networks: Numerical Analysis Framework
    In preparation