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We consider the a posteriori error analysis of approximations of parabolicproblems based on arbitrarily high-order conforming Galerkin spatialdiscretizations and arbitrarily high-order discontinuous Galerkin temporaldiscretizations. Using equilibrated flux reconstructions, we present aposteriori error estimates for a norm composed of the $L^2(H^1)\capH^1(H^{-1})$-norm of the error and the temporal jumps of the numericalsolution. The estimators provide guaranteed upper bounds for this norm, withoutunknown constants. Furthermore, the efficiency of the estimators with respectto this norm is local in both space and time, with constants that are robustwith respect to the mesh-size, time-step size, and the spatial and temporalpolynomial degrees. We further show that this norm, which is key for localspace-time efficiency, is globally equivalent to the $L^2(H^1)\capH^1(H^{-1})$-norm of the error, with polynomial-degree robust constants. Theproposed estimators also have the practical advantage of allowing for verygeneral refinement and coarsening between the timesteps.

siam journal on numerical analysis

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