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.

中文翻译：

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一种 $P_{ķ+2}$ 多边形和多面体的多项式提升算子

一种 $P_{ķ+2}$在多边形和多面体上定义多项式提升算子。它将多边形/多面体内部以及面上的不连续多项式提升为一个$P_{ķ+2}$多项式。利用该提升算子，我们证明了弱的Galerkin有限元解在此提升之后收敛于两个比最优阶高的阶。${\mathrm{大号}}^2$ 和 $H^{\mathrm{1个}}$规范。该理论已通过2D和3D泊松方程的数值解得到证实。