Multispace and multilevel BDDC

Authors:
Jan Mandel;Bedřich Sousedík;Clark R. Dohrmann
Affiliations:
University of Colorado Denver, Department of Mathematical and Statistical Sciences, P.O. Box 173364, Campus Box 170, 80217, Denver, CO, USA;Univ. of Colorado Denver, Dept. of Math. and Stat. Sci., P.O. Box 173364, Campus Box 170, 80217, Denver, CO, USA and Acad. of Sci. of the Czech Republic, Inst. of Thermomechanics, Dolejškov ...;Sandia National Laboratories, Structural Dynamics Research Department, 87185-0847, Albuquerque, NM, USA
Venue:
Computing
Year:
2008

Citing 5
Cited 5

Domain decomposition: parallel multilevel methods for elliptic partial differential equations

Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions

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A Preconditioner for Substructuring Based on Constrained Energy Minimization

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An algebraic theory for primal and dual substructuring methods by constraints

Applied Numerical Mathematics - 6th IMACS International symposium on iterative methods in scientific computing
Three-Level BDDC in Three Dimensions

SIAM Journal on Scientific Computing

BDDC Preconditioners for Spectral Element Discretizations of Almost Incompressible Elasticity in Three Dimensions

SIAM Journal on Scientific Computing
Original article: Preconditioning of the coarse problem in the method of balanced domain decomposition by constraints

Mathematics and Computers in Simulation
Original article: Face-based selection of corners in 3D substructuring

Mathematics and Computers in Simulation
Original Article: Adaptive BDDC in three dimensions

Mathematics and Computers in Simulation
Adaptive-Multilevel BDDC and its parallel implementation

Computing

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Abstract

The Balancing Domain Decomposition by Constraints (BDDC) method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory.