Long-time L∞(L2) a posteriori error estimates for fully discrete parabolic problems

Abstract

Computable estimates for the error of finite element discretisations of parabolic problems in the L ∞ (0, T ; L 2 (Ω)) 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.

Convergence Analysis and Approximate Optimal Temporal Step Sizes for Some Finite Difference Methods Discretising Fisher's Equation

“…DEFINITION 2. Sutton [33]. Suppose {t n } N 0 forms a partition of [0, T], with t n = n t for n = 0, •••, N, where t = T/N.…”

Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods

“…Parabolic a posteriori error estimates. We recall some results from [40] on exponentially weighted time accumulations in Lemma 3.11. The key appeal of these is to enable L ∞ (L 2 ) error estimates in which the estimator terms accumulate through time in the minimal…”

“…The effectivities of such estimators can therefore become constant with t (since the error and estimator may both accumulate in the L ∞ ([0, t]) norm), rather than growing like t or t 1/2 as they would if only L 1 or L 2 accumulations were used respectively (see [40] for details). In the statement of the lemma, the term F represents an estimator term accumulating with simulation time, and ξ represents the error to be estimated.…”

“…1. Comparable numerical results for an analogous estimator using a fixed computational mesh are given in [40,Section 7].…”

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