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A $P_{k+2}$ polynomial lifting operator is defined on polygons andpolyhedrons. It lifts discontinuous polynomials inside the polygon/polyhedronand on the faces to a one-piece $P_{k+2}$ polynomial. With this liftingoperator, we prove that the weak Galerkin finite element solution, after thislifting, 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 3DPoisson equations.

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