Mesh-uniform second-order midpoint error bound

Establish a mesh-uniform second-order-in-time error bound for the implicit midpoint discretization of the hybridizable discontinuous Galerkin approximation of time-modulated Drude–Lorentz Maxwell systems.

Background

The paper proves second-order temporal convergence for implicit midpoint on each fixed spatial mesh, but the associated regularity quantity may depend on the mesh size. Consequently, the analysis does not establish a second-order bound whose constant remains uniform as the HDG mesh is refined. The open problem is to control the relevant semidiscrete time-regularity quantities uniformly in the spatial discretization parameter and thereby prove a mesh-uniform second-order estimate.