An approximate block Newton method for coupled iterations of nonlinear solvers: Theory and conjugate heat transfer applications

Andrew Yeckel, Lisa Lun, Jeffrey J. Derby

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


A new, approximate block Newton (ABN) method is derived and tested for the coupled solution of nonlinear models, each of which is treated as a modular, black box. Such an approach is motivated by a desire to maintain software flexibility without sacrificing solution efficiency or robustness. Though block Newton methods of similar type have been proposed and studied, we present a unique derivation and use it to sort out some of the more confusing points in the literature. In particular, we show that our ABN method behaves like a Newton iteration preconditioned by an inexact Newton solver derived from subproblem Jacobians. The method is demonstrated on several conjugate heat transfer problems modeled after melt crystal growth processes. These problems are represented by partitioned spatial regions, each modeled by independent heat transfer codes and linked by temperature and flux matching conditions at the boundaries common to the partitions. Whereas a typical block Gauss-Seidel iteration fails about half the time for the model problem, quadratic convergence is achieved by the ABN method under all conditions studied here. Additional performance advantages over existing methods are demonstrated and discussed.

Original languageEnglish (US)
Pages (from-to)8566-8588
Number of pages23
JournalJournal of Computational Physics
Issue number23
StatePublished - Dec 10 2009

Bibliographical note

Funding Information:
This work has been supported in part by a seed grant from the Minnesota Supercomputing Institute and by the Department of Energy, National Nuclear Security Administration , under Award Numbers DE-FG52-06NA27498 and DE-FG52-08NA28768 , the content of which does not necessarily reflect the position or policy of the United States Government, and no official endorsement should be inferred. Andrew Yeckel gives thanks to Fraunhofer Gesellschaft for financial support through a PROF.X 2 scholarship, to Jochen Friedrich for hosting his stay at the Crystal Growth Laboratory at Fraunhofer IISB, and to Thomas Jung for advice on research direction at a critical juncture.


  • Approximate Newton methods
  • Block Gauss-Seidel methods
  • Crystal growth
  • Modular iterations
  • Multiphysics coupling
  • Multiscale coupling

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