Superconvergence Analysis Of Polynomial Preserving Recovery For Over-penalized Weak Galerkin Methods And Its Applications

The polynomial preserving recovery?PPR?technique was first introduced by Zhang and Naga by using the fitted solution values to recover the gradient and a posteriori error estimates in the energy norm was constructed.And then,polynomial preserving recovery of an over-penalized symmetric interior penalty Galerkin method[12]was presented for elliptic problems.In this work,we propose the PPR for an over-penalized weak Galerkin method,which combines weak Galerkin stabilizer with interior penalty terms,to solve second-order elliptic problems.The discontinuous Galerkin numerical solutions are used to recover the gradient.Optimal a priori error estimates in energy norm and a posteriori error estimates in L²-norms are carefully analyzed for polynomials?P_k,P_k,RT_k??k?0?with a posteriori error estimator presented,and some numerical experiments are given to validate theoretical results.In numerical experiments,we also present another type of over-penalized WG with adding a stabilizer,i.e.,elements(P_k,P_k,[P_k-1]^d)?k?1?.In addition,for uniform triangular meshes?Regular pattern?and Chevron pattern,the recovered gradients are superconvergent under the regular pattern and under the chevron pattern,respectively.Next,in order to apply the PPR method to the vector-valued function space,we take the classical weak Galerkin finite element method for solving the Stokes equation as an example,to realize the PPR method on the weak Galerkin solutions.Furthermore,the corresponding superconvergence analysis is given,while vector-valued weak Galerkin solutions of velocity are used to its gradient recovery.Optimal a priori error estimates in energy norm and a posteriori error estimates in L²-norms are carefully analyzed.