Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems

TL;DR

This paper develops guaranteed, locally efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems, ensuring reliable error control in complex adaptive simulations.

Contribution

It introduces a posteriori error estimators that are guaranteed, locally efficient, and robust with respect to polynomial degrees for high-order space-time discretizations of parabolic problems.

Findings

Estimators provide guaranteed upper bounds without unknown constants.

Efficiency constants are robust with respect to mesh size, time step, and polynomial degrees.

Norms used are globally equivalent with polynomial-degree robust constants.

Abstract

We consider the a posteriori error analysis of approximations of parabolic problems based on arbitrarily high-order conforming Galerkin spatial discretizations and arbitrarily high-order discontinuous Galerkin temporal discretizations. Using equilibrated flux reconstructions, we present a posteriori error estimates for a norm composed of the $L^2(H^1)\cap H^1(H^{-1})$-norm of the error and the temporal jumps of the numerical solution. The estimators provide guaranteed upper bounds for this norm, without unknown constants. Furthermore, the efficiency of the estimators with respect to this norm is local in both space and time, with constants that are robust with respect to the mesh-size, time-step size, and the spatial and temporal polynomial degrees. We further show that this norm, which is key for local space-time efficiency, is globally equivalent to the $L^2(H^1)\cap H^1(H^{-1})$-norm of the error, with polynomial-degree robust constants. The proposed estimators also have the practical advantage of allowing for very general refinement and coarsening between the timesteps.