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

TL;DR

This paper introduces a $P_{k+2}$ polynomial lifting operator for polygons and polyhedrons, enhancing weak Galerkin finite element solutions to achieve higher convergence rates, validated through numerical experiments in 2D and 3D.

Contribution

It develops a novel polynomial lifting operator that improves the convergence order of weak Galerkin finite element methods on polygons and polyhedrons.

Findings

Lifting operator increases convergence order by two in $L^2$ and $H^1$ norms.

Numerical experiments confirm theoretical convergence improvements.

Applicable to 2D and 3D Poisson problems.

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.