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 basis function spaces depend...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 basis function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements, which makes the WG methods flexible and robust in many applications. Different from the definition of jump in discontinuous Galerkin （DG） methods, we can define a new weaker jump from weak functions defined on edges. Those functions have double values on the interior edges shared by two elements rather than a limit of functions defined in an element tending to its edge. Naturally, the weak jump comes from the difference between two weak flmctions defined on the same edge. We introduce an over-penalized weak Galerkin （OPWG） method, which has two sets of edge-wise and element-wise shape functions, and adds a penalty term to control weak jumps on the interior edges. Furthermore, optimal a priori error estimates in H1 and L2 norms are established for the finite element （Pk（K）, Pk（e）, RTk（K））. In addition, some numerical experiments are given to validate theoretical results, and an incomplete LU decomposition has been used as a preconditioner to reduce iterations from the GMRES, CO, and BICGSTAB iterative methods.展开更多
In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a resi...In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.展开更多
基金The research of the second author is supported in part by the Natural Science Foundation of Gansu Province, China （Grant 18JR3RA290）, and Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund （the second phase）.
文摘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 basis function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements, which makes the WG methods flexible and robust in many applications. Different from the definition of jump in discontinuous Galerkin （DG） methods, we can define a new weaker jump from weak functions defined on edges. Those functions have double values on the interior edges shared by two elements rather than a limit of functions defined in an element tending to its edge. Naturally, the weak jump comes from the difference between two weak flmctions defined on the same edge. We introduce an over-penalized weak Galerkin （OPWG） method, which has two sets of edge-wise and element-wise shape functions, and adds a penalty term to control weak jumps on the interior edges. Furthermore, optimal a priori error estimates in H1 and L2 norms are established for the finite element （Pk（K）, Pk（e）, RTk（K））. In addition, some numerical experiments are given to validate theoretical results, and an incomplete LU decomposition has been used as a preconditioner to reduce iterations from the GMRES, CO, and BICGSTAB iterative methods.
基金We thank the anonymous referees for their valuable comments and suggestions which lead to an improved presentation of this paper. This work was supported by NSFC under the grant 11371199, 11226334 and 11301275, the Jiangsu Provincial 2011 Program （Collaborative Innovation Center of Climate Change）, the Program of Natural Science Research of Jiangsu Higher Education Institutions of China （Grant No. 12KJB110013）, Natural Science Foundation of Guangdong Province of China （Grant No. S2012040007993） and Educational Commission of Guangdong Province of China （Grant No. 2012LYM0122）.
文摘In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.
