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AN ODYSSEY INTO LOCAL REFINEMENT AND MULTILEVEL PRECONDITIONING III: IMPLEMENTATION AND NUMERICAL
 

Summary: AN ODYSSEY INTO LOCAL REFINEMENT AND MULTILEVEL
PRECONDITIONING III: IMPLEMENTATION AND NUMERICAL
EXPERIMENTS
BURAK AKSOYLU, STEPHEN BOND, AND MICHAEL HOLST§
Abstract. In this paper, we examine a number of additive and multiplicative multilevel iterative
methods and preconditioners in the setting of two-dimensional local mesh refinement. While standard
multilevel methods are effective for uniform refinement-based discretizations of elliptic equations,
they tend to be less effective for algebraic systems which arise from discretizations on locally refined
meshes, losing their optimal behavior in both storage and computational complexity. Our primary
focus here is on BPX-style additive and multiplicative multilevel preconditioners, and on various
stabilizations of the additive and multiplicative hierarchical basis method (HB), and their use in
the local mesh refinement setting. In the first two papers of this trilogy, it was shown that both
BPX and wavelet stabilizations of HB have uniformly bounded condition numbers on several classes
of locally refined 2D and 3D meshes based on fairly standard (and easily implementable) red and
red-green mesh refinement algorithms. In this third article of the trilogy, we describe in detail the
implementation of these types of algorithms, including detailed discussions of the datastructures
and traversal algorithms we employ for obtaining optimal storage and computational complexity
in our implementations. We show how each of the algorithms can be implemented using standard
datatypes available in languages such as C and FORTRAN, so that the resulting algorithms have
optimal (linear) storage requirements, thereby the resulting multilevel method or preconditioner can

  

Source: Aksoylu, Burak - Center for Computation and Technology & Department of Mathematics, Louisiana State University

 

Collections: Computer Technologies and Information Sciences