Approximate Error Bounds on Solutions of Nonlinearly Preconditioned PDEs

Lulu Liu, David E. Keyes

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


In many multiphysics applications, an ultimate quantity of interest can be written as a linear functional of the solution to the discretized governing nonlinear partial differential equations and finding a sufficiently accurate pointwise solution may be regarded as a step toward that end. In this paper, we derive a posteriori approximate error bounds for linear functionals corresponding to quantities of interest using two kinds of nonlinear preconditioning techniques. Nonlinear preconditioning, such as the inexact Newton with backtracking and nonlinear elimination algorithm and the multiplicative Schwarz preconditioned inexact Newton algorithm, may be effective in improving global convergence for Newton's method. It may prevent stagnation of the nonlinear residual norm and reduce the number of solutions of large ill-conditioned linear systems involving a global Jacobian required at each nonlinear iteration. We illustrate the effectiveness of the new bounds using canonical nonlinear PDE models: a flame sheet model and a nonlinear coupled lid-driven cavity problem.
Original languageEnglish (US)
Pages (from-to)A2526-A2554
Number of pages1
JournalSIAM Journal on Scientific Computing
Issue number4
StatePublished - Jul 15 2021

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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