On the Rate of Convergence of Flux Reconstruction for Steady-State Problems

This paper derives analytical estimates for the rates of convergence of numerical first and second derivative operators involved in flux reconstruction (FR). These estimates yield the rate of convergence for steady-state advection-diffusion problems when error is measured in the vector seminorm induced by the advection-diffusion operator. This serves to rigorously quantify the effect of polynomial order, correction functions, and upwinding coefficients on the accuracy of FR schemes. We prove that for centered fluxes, the derivative operator exhibits superconvergence for a special class of correction functions, designated “SFR,” which includes the nodal DG scheme. Finally, we also show that the rate of convergence of the vector 2 norm of error for such steady-state problems is identical to the short-time rate of convergence for time-dependent problems derived in [K. Asthana et al., Analysis and Design of Optimal Discontinuous Finite Element Schemes, Ph.D. thesis, Stanford University, Stanford, CA, 2016].