## 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 language | English (US) |
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Pages (from-to) | 2063-2087 |

Number of pages | 25 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 51 |

Issue number | 4 |

DOIs | |

State | Published - 2013 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

## Keywords

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