Over-penalized Weak Galerkin Methods For Second-order Elliptic Problems

The weak Galerkin(WG)finite element method was first introduced by Wang and Ye for solving second order elliptic equations,with the use of weak functions and their weak gradients.The basic function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements,which make the WG methods flexible and robust in many applications.Jumps in discontinuous Galerkin(DG)methods result from the restriction of piecewise polynomials on edges,while the weak Galerkin methods originally use a single-valued function on the interior edges.In this article,we define weak jump by double-valued functions on interior edges shared by two elements rather than a limit of function defined in element passaging to its edge.Naturally,a weak jump comes from the difference between two weak functions defined on the same edge.We introduce an over-penalized weak Galerkin(OPWG)method,by adding a penalty term to control the weak jumps on the interior edges to reach some weak continuity.In this article,we apply the over-penalized weak Galerkin method for solving the second-order elliptic problems.Optimal priori error estimates in H1-and L2-norms are carefully analyzed for polynomials(Pk,Pk,RTk)(k ? 0),and some numerical experiments are given to validate theoretical results.We also present another type of overpenalized WG with adding a stabilizer,i.e.,with polynomials(Pk,Pk,[Pk-1]2)(k ? 1)or(Pk,Pk-1,[Pk-1]2)(k ? 1),in this case,the finite element partitions triangulations consist arbitrary shapes.Based on our analysis,the new OPWG methods have many advantages,such as more simple shape functions,and easy-to-realize stiff matrix,etc.Although at present the OPWG methods for the second-order elliptic problems are given,it is promising to apply it for other PDEs,such as elliptic interface problems,Stokes equations,and div-curl systems equations.