Section: New Results

Locally space-time efficient estimates for parabolic problems

Participants : Martin Vohralík, Alexandre Ern, Iain Smears.

Grants: GATIPOR 8.3.1

Publications: [33]

In [33], we derive for the first time a posteriori error estimates for parabolic problems which are both globally reliable and locally space-time efficient. By this, one means that the error between a known approximate numerical solution and the unknown exact solution of a model parabolic PDE (the heat equation) is bounded from above on the whole space-time domain by a fully computable estimator, while this estimator does not overestimate significantly the error and localizes it both in space and in time. More precisely, the estimator also gives lower bounds on the error, up to a generic constant, and this on each time interval and in a small neighborhood of each space mesh element. We consider arbitrarily high-order conforming Galerkin spatial discretizations and arbitrarily high-order discontinuous Galerkin temporal discretizations, and the error is measured in a norm composed of the $L^2\left(H^1\right)\cap H^1\left(H^{-1}\right)$-norm augmented by the temporal jumps of the numerical solution. The efficiency constant is robust with respect to (independent of) any mesh-size, time-step size, and the spatial and temporal polynomial degrees. The proposed estimators also have the practical advantage of not imposing any requirement on coarsening between the consecutive time steps.