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Maximum-norm a posteriori error bounds for parabolic equations discretised by the extrapolated Euler method in time and FEM in space

Torsten Linß, Goran Radojev

Abstract

A class of linear parabolic equations is considered. We derive a framework for the a posteriori error analysis of time discretisations by Richardson extrapolation of arbitrary order combined with finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator. The crucial point in the analysis is the design of suitable polynomial reconstructions in time from approximations that are given only in the mesh points.

Maximum-norm a posteriori error bounds for parabolic equations discretised by the extrapolated Euler method in time and FEM in space

Abstract

A class of linear parabolic equations is considered. We derive a framework for the a posteriori error analysis of time discretisations by Richardson extrapolation of arbitrary order combined with finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator. The crucial point in the analysis is the design of suitable polynomial reconstructions in time from approximations that are given only in the mesh points.

Paper Structure

This paper contains 12 sections, 2 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Weak formulation
  3. Time discretisation
  4. A posteriori error analysis
  5. A posteriori error estimation for the elliptic problem
  6. Green's functions
  7. Elliptic reconstruction
  8. A posteriori error estimation for the parabolic problem
  9. General framework
  10. Temporal reconstructions
  11. Error contributions by the spatial discretisation
  12. A posteriori error bound

Key Result

Lemma 1

Let $v\in V$ be arbitrary. Then where the $\sigma_j$ have been defined in def:sigma_j and

Theorems & Definitions (6)

  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Theorem 1
  • Remark 3