Efficient rigorous numerics for higher-dimensional PDEs via one-dimensional estimates

Marcio Gameiro, Jean Philippe Lessard

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

We present an efficient rigorous computational method which is an extension of the work Analytic Estimates and Rigorous Continuation for Equilibria of Higher-Dimensional PDEs (M. Gameiro and J.-P. Lessard, J. Differential Equations, 249 (2010), pp. 2237-2268). The idea is to generate sharp one-dimensional estimates using interval arithmetic which are then used to produce high-dimensional estimates. These estimates are used to construct the radii polynomials which provide an efficient way of determining a domain on which the contraction mapping theorem is applicable. Computing the equilibria using a finite-dimensional projection, the method verifies that the numerically produced equilibrium for the projection can be used to explicitly define a set which contains a unique equilibrium for the PDE. A new construction of the polynomials is presented where the nonlinearities are bounded by products of one-dimensional estimates as opposed to using FFT with large inputs. It is demonstrated that with this approach it is much cheaper to prove that the numerical output is correct than to recompute at a finer resolution. We apply this method to PDEs defined on three- and four-dimensional spatial domains.

Original languageEnglish (US)
Pages (from-to)2063-2087
Number of pages25
JournalSIAM Journal on Numerical Analysis
Volume51
Issue number4
DOIs
StatePublished - 2013
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Computer assisted proofs
  • Equilibria of PDEs
  • Higher-dimensional PDEs
  • Rigorous numerics

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