A stabilizer-free C0 weak Galerkin method for the biharmonic equations

In this article, we present and analyze a stabilizer-free C⁰ weak Galerkin (SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is formulated in terms of cell unknowns which are C⁰ continuous piecewise polynomials of degree k + 2 with k ≽ 0 and in terms of face unknowns which are discontinuous piecewise polynomials of degree k + 1. The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the C¹ conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete H²-like norm and the H¹ norm for k ≽ 0 are established for the corresponding WG finite element solutions. Error estimates in the L² norm are also derived with an optimal order of convergence for k > 0 and sub-optimal order of convergence for k = 0. Numerical experiments are shown to confirm the theoretical results.