A weak Galerkin finite element scheme with boundary continuity for second-order elliptic problems

A new weak Galerkin (WG) finite element method for solving the second-order elliptic problems on polygonal meshes by using polynomials of boundary continuity is introduced and analyzed. The WG method is utilizing weak functions and their weak derivatives which can be approximated by polynomials in different combination of polynomial spaces. Different combination gives rise to different weak Galerkin finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of certain combination of polynomial spaces that minimize the degree of freedom in the numerical scheme, yet without losing the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete H1 norm and the standard L2 norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show that the WG method achieves the optimal convergence order with a relatively low computational cost, which reveals a great promise of the flexibility and accuracy of the WG method.