A $P_{k+2}$ polynomial lifting operator on polygons and polyhedrons

Abstract

A $P_{k+2}$ polynomial lifting operator is defined on polygons and polyhedrons. It lifts discontinuous polynomials inside the polygon/polyhedron and on the faces to a one-piece $P_{k+2}$ polynomial. With this lifting operator, we prove that the weak Galerkin finite element solution, after this lifting, converges at two orders higher than the optimal order, in both $L^2$ and $H^1$ norms. The theory is confirmed by numerical solutions of 2D and 3D Poisson equations.