On verified computations of solutions for nonlinear parabolic problems

Abstract

Abstract:We consider the methods for guaranteed computations of solutions for nonlinear parabolic initial-boundary value problems. First, in order to make the basic principle clear, we briefly introduce the numerical verification methods of solutions for elliptic problems which we have developed up to now. Next, under some fundamental procedures of verification for parabolic problems based on the fixed point theorem with Newton's method, we describe a summary of our methods including additional new technique which could yield some improvements. The main contents of the paper consist of the guaranteed a posteriori estimates for the linearized inverse operators of parabolic type. In order to confirm the effectiveness of our methods, we give some numerical examples for the guaranteed bounds of iverse operators as well as give some prototype results for numerical verification of solutions of nonlinear parabolic problems. Moreover, we will mention an extention of the present technique to the verification of time-periodic solutions.

Rigorous FEM for 1D Burgers equation

“…The method has been used successfully to verify the solutions for elliptic (using FEM or Fourier basis) [10,11,13,19,20,29] and parabolic [21,[23][24][25][26]35] PDEs (but this mainly for periodic boundary conditions). The up to date information about these techniques can be found in the recent monograph [27].…”

Rigorous numerical computations for 1D advection equations with variable coefficients

“…Verified numerical computations for PDEs have been established by Nakao [19] and Plum [24] independently. These have been developed in the last three decades by their collaborators and many researchers in the field of dynamical systems (see, e.g., [5,17,21,22,25,27,32,34] and references therein). Recently, verified numerical computations enable us to understand traveling-waves, periodic solutions, invariant objects (including stationary solutions) of parabolic/elliptic PDEs, etc.…”

Nakao’s method and an interval difference scheme of second order for solving the elliptic BVP

“…The first approach to solve elliptic boundary value problems in interval arithmetic has been done by M.T. Nakao in 1988 [6] and extended in the following years [7,8]. His method is based on Galerkin's approximation and finite elements methods known from conventional theory for solving elliptical problems (see, e.g., [9] and [10]).…”