Efficiency of the residual-based a posteriori error estimator

Establish a lower bound (efficiency) for the residual-based a posteriori error estimator derived for fully discrete Runge–Kutta discontinuous Galerkin approximations of nonlinear convection–diffusion systems with convex entropy on the torus that use tensor-product SIAC filtering in space and temporal Hermite interpolation to reconstruct a space–time approximation; specifically, prove that the estimator controls from below the error between the entropy weak solution and the space–time reconstruction in the sup-in-time L2 norm together with the epsilon-weighted time-integrated H1 seminorm.

Background

The paper develops reliable a posteriori error estimators for fully discrete Runge–Kutta discontinuous Galerkin methods applied to nonlinear convection–diffusion systems with convex entropy in multiple dimensions. The approach combines a spatial reconstruction via tensor-product SIAC filtering with a temporal Hermite reconstruction and uses a relative entropy framework together with a residual splitting into hyperbolic and parabolic parts to obtain upper bounds that are robust in the vanishing-viscosity limit.

Although the authors prove reliability (upper bounds) of the estimator, they explicitly state that they cannot prove efficiency, i.e., a lower bound showing that the estimator controls the error from below. Their numerical experiments indicate that the residuals scale with the same order as the error, but a rigorous proof of efficiency remains unresolved.