Weak Finite Element Method For Fractional Stationary Equation

A new weak Galerkin finite element method for solving the integral differential equations has been introduced and analyzed,such as second-order elliptic problems on polygonal meshes.For example,the discrete weak gradient operator was employed as a building block to approximate the solution of a model second order elliptic problem.It can be seen that the weak Galerkin method allowed the use of totally discontinuous function in the finite element procedure.The WG method was utilizing weak function and their weak derivatives which can be approximated by polynomials in different combination of polynomial spaces.Different combination gave rise to different weak Galerkin finite element methods,which made WG methods highly flexible and efficient in practical computation.In this paper,we used the weak Galerkin finite element method for fractional differential equations.We defined the weak derivatives in the fractional differential equations.We can divide v?x? into two party v?x?= {v₀,v_b}such that v₀? L²???and v_b? H^1/2???.And the continuity was compensated by the stabilizer s??,??through a suitable boundary integral defined on the boundary of elements.We proofed the existence and uniqueness of the fractional stationary equation.Error estimates of optimal order were established for the corresponding WG approximations.In addition,the paper also presented some numerical experiments to demonstrate the power of the WG method.