This paper is concerned with a multi-domain spectral method, based on an interior penalty discontinuous Galerkin （IPDG） formulation, for the exterior Helmholtz problem truncated via an exact circular or spherical Di...This paper is concerned with a multi-domain spectral method, based on an interior penalty discontinuous Galerkin （IPDG） formulation, for the exterior Helmholtz problem truncated via an exact circular or spherical Dirichlet-to-Neumann （DtN） boundary con- dition. An effective iterative approach is proposed to localize the global DtN boundary condition, which facilitates the implementation of multi-domain methods, and the treat- ment for complex geometry of the scatterers. Under a discontinuous Galerkin formulation, the proposed method allows to use polynomial basis functions of different degree on dif- ferent subdomains, and more importantly, explicit wave number dependence estimates of the spectral scheme can be derived, which is somehow implausible for a multi-domain continuous Galerkin formulation.展开更多
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
基金The work of the first author was partially supported by the National Natural Science Foundation of China （11026065 and 11101196）. This work was largely done when this author worked as a Research Fellow in Nanyang Technological University. The work of the second author was supported by the National Natural Science Foundation of China （11201166）. The work of the third author was supported by Singapore MOE Tier 1 Grant （2013-2016）, and Singapore A＊STAR-SERC-PSF Grant 122-PSF-0007.
文摘This paper is concerned with a multi-domain spectral method, based on an interior penalty discontinuous Galerkin （IPDG） formulation, for the exterior Helmholtz problem truncated via an exact circular or spherical Dirichlet-to-Neumann （DtN） boundary con- dition. An effective iterative approach is proposed to localize the global DtN boundary condition, which facilitates the implementation of multi-domain methods, and the treat- ment for complex geometry of the scatterers. Under a discontinuous Galerkin formulation, the proposed method allows to use polynomial basis functions of different degree on dif- ferent subdomains, and more importantly, explicit wave number dependence estimates of the spectral scheme can be derived, which is somehow implausible for a multi-domain continuous Galerkin formulation.
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.
