# A note on the optimal degree of the weak gradient of the stabilizer free weak Galerkin finite element method

## TL;DR

This paper investigates the optimal polynomial degree for the stabilizer free weak Galerkin method to balance convergence rate and computational efficiency, providing rigorous proof for the optimal choice.

## Contribution

It determines the optimal polynomial degree j0 for the stabilizer free weak Galerkin method with mathematical validation, enhancing its practical implementation.

## Key findings

- Optimal degree j0 ensures convergence without numerical locking
- Mathematical proof of the optimal degree j0
- Improved efficiency of the weak Galerkin method

## Abstract

Recently, a new stabilizer free weak Galerkin method (SFWG) is proposed, which is easier to implement. The idea is to raise the degree of polynomials j for computing weak gradient. It is shown that if j>=j0 for some j0, then SFWG achieves the optimal rate of convergence. However, large j will cause some numerical difficulties. To improve the efficiency of SFWG and avoid numerical locking, in this note, we provide the optimal j0 with rigorous mathematical proof.