High order continuous local-conserving fluxes and finite-volume-like finite element solutions for elliptic equations

We derive a high order globally continuous and locally conservative flux field and a high order finite-volume-like solution from the continuous Galerkin (CG) finite element solution. The main idea is to postprocess the CG solution by solving a small linear algebraic system on each element of the underlying mesh. Both the postprocessed flux field and the finite-volume-like solution satisfy the conservation law on each control volume of the dual mesh. Moreover, both the postprocessed flux field and the gradient of finite-volume-like solution converge to the exact flux with optimal convergence rates. Our theoretical findings are validated by our numerical experiments.