Long time $L^\infty(L^2)$ a posteriori error estimates for fully discrete parabolic problems

TL;DR

This paper develops new computable a posteriori error estimates for fully discrete parabolic problems in the $L^ Infty(0,T; L^2)$ norm, with constant effectivities suitable for long-time simulations, using energy techniques based on elliptic reconstructions.

Contribution

It introduces a family of optimal order error estimates with constant effectivities for long-time parabolic problem simulations, outperforming previous duality-based methods.

Findings

Error estimates with constant effectivity for long-time simulations

Numerical demonstration of effectivity growth in previous methods

New energy-based estimates suitable for both short and long-time problems

Abstract

Computable estimates for the error of finite element discretisations of parabolic problems in the $L^\infty(0,T; L^2)$ norm are developed, which exhibit constant effectivities (the ratio of the estimated error to the true error) with respect to the simulation time. These estimates, which are of optimal order, represent a significant advantage for long-time simulations, and are derived using energy techniques based on elliptic reconstructions. The effectivities of previous optimal order error estimates in this norm derived using energy techniques are shown numerically to grow either in proportion to the simulation duration or its square root, a key disadvantage compared with earlier estimators derived using parabolic duality arguments. The new estimates form a continuous family, almost all of which are new, reproducing certain familiar energy-based estimates well suited for short-time simulations and not available through the parabolic duality framework. For clarity, we demonstrate the technique applied to a linear parabolic problem discretised using standard conforming finite element methods in space coupled with backward Euler and Crank-Nicolson time discretisations, although it can be applied much more widely.