| The second order elliptic equation, which is also known as the diffusion-convection equation, is of great interest in many branches of physics and industry. In this dissertation, we use the weak Galerkin finite element method to study the general elliptic equation with Dirichlet boundary condition and the elliptic equation with mixed boundary conditions. Weak Galerkin finite element schemes are proposed and analyzed. These schemes feature piecewise polynomials of degree k ≥ 1 on each element and piecewise polynomials of degree k -- 1 ≥ 0 on each edge or face of the element. Error estimates of optimal order are established in both discrete H1 and standard L2 norm. Numerical examples are investigated to support the theoretical results. |
