We consider a singular perturbation problem which describes 2D Darcy-Stokes flow. An H（div）- conforming rectangular element, DS-R14, is proposed and analyzed first. This element has 14 degrees of freedom for velocit...We consider a singular perturbation problem which describes 2D Darcy-Stokes flow. An H（div）- conforming rectangular element, DS-R14, is proposed and analyzed first. This element has 14 degrees of freedom for velocity and is proved to be uniformly convergent with respect to perturbation constant. We then simplify this element to get another H（div）-conforming rectangular element, DS-R12, which has 12 degrees of freedom for velocity. The uniform convergence is also obtained for this element. Finally, we construct a de Rham complex corresponding to DS-R12 element.展开更多
Hierarchical bases of arbitrary order for H(div)-conforming triangular and tetrahedral elements are constructedwith the goal of improving the conditioning of the mass and stiffness matrices.For the basis with the tria...Hierarchical bases of arbitrary order for H(div)-conforming triangular and tetrahedral elements are constructedwith the goal of improving the conditioning of the mass and stiffness matrices.For the basis with the triangular element,it is found numerically that the conditioning is acceptable up to the approximation of order four,and is better than a corresponding basis in the dissertation by Sabine Zaglmayr[High Order Finite Element Methods for Electromagnetic Field Computation,Johannes Kepler Universit¨at,Linz,2006].The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four.The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four.For the tetrahedral element,it is identified numerically that the conditioning is acceptable only up to the approximation of order three.Compared with the newly constructed basis for the triangular element,the sparsity of the massmatrices fromthe basis for the tetrahedral element is relatively sparser.展开更多
Classical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in H(div)-norm for general unstructured meshes containing hexahedra and prisms....Classical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in H(div)-norm for general unstructured meshes containing hexahedra and prisms.We propose two new families of high-order elements for hexahedra,triangular prisms and pyramids that recover the optimal convergence.These elements have compatible restrictions with each other,such that they can be used directly on general hybrid meshes.Moreover the H(div)proposed spaces are completing the De Rham diagram with optimal elements previously constructed for H1 and H(curl)approximation.The obtained pyramidal elements are compared theoretically and numerically with other elements of the literature.Eventually,numerical results demonstrate the efficiency of the finite elements constructed.展开更多
In order to solve the magnetohydrodynamics(MHD)equations with a H(div)-conforming element,a novel approach is proposed to ensure the exact divergence-free condition on the magnetic field.The idea is to add on each ele...In order to solve the magnetohydrodynamics(MHD)equations with a H(div)-conforming element,a novel approach is proposed to ensure the exact divergence-free condition on the magnetic field.The idea is to add on each element an extra interior bubble function from a higher order hierarchicalH(div)-conforming basis.Four such hierarchical bases for theH(div)-conforming quadrilateral,triangular,hexahedral,and tetrahedral elements are either proposed(in the case of tetrahedral)or reviewed.Numerical results have been presented to show the linear independence of the basis functions for the two simplicial elements.Good matrix conditioning has been confirmed numerically up to the fourth order for the triangular element and up to the third order for the tetrahedral element.展开更多
We consider H（curl, Ω）-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral...We consider H（curl, Ω）-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H^1 （Ω）-context along with local discrete Helmholtz-type decompositions of the edge element space.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11071226)the Hong Kong Research Grants Council(Grant No.201112)
文摘We consider a singular perturbation problem which describes 2D Darcy-Stokes flow. An H（div）- conforming rectangular element, DS-R14, is proposed and analyzed first. This element has 14 degrees of freedom for velocity and is proved to be uniformly convergent with respect to perturbation constant. We then simplify this element to get another H（div）-conforming rectangular element, DS-R12, which has 12 degrees of freedom for velocity. The uniform convergence is also obtained for this element. Finally, we construct a de Rham complex corresponding to DS-R12 element.
基金supported in part by a DOE grant DEFG0205ER25678 and NSF grant DMS-1005441。
文摘Hierarchical bases of arbitrary order for H(div)-conforming triangular and tetrahedral elements are constructedwith the goal of improving the conditioning of the mass and stiffness matrices.For the basis with the triangular element,it is found numerically that the conditioning is acceptable up to the approximation of order four,and is better than a corresponding basis in the dissertation by Sabine Zaglmayr[High Order Finite Element Methods for Electromagnetic Field Computation,Johannes Kepler Universit¨at,Linz,2006].The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four.The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four.For the tetrahedral element,it is identified numerically that the conditioning is acceptable only up to the approximation of order three.Compared with the newly constructed basis for the triangular element,the sparsity of the massmatrices fromthe basis for the tetrahedral element is relatively sparser.
文摘Classical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in H(div)-norm for general unstructured meshes containing hexahedra and prisms.We propose two new families of high-order elements for hexahedra,triangular prisms and pyramids that recover the optimal convergence.These elements have compatible restrictions with each other,such that they can be used directly on general hybrid meshes.Moreover the H(div)proposed spaces are completing the De Rham diagram with optimal elements previously constructed for H1 and H(curl)approximation.The obtained pyramidal elements are compared theoretically and numerically with other elements of the literature.Eventually,numerical results demonstrate the efficiency of the finite elements constructed.
基金This research is supported in part by a DOE grant DEFG0205ER25678 and a NSF grant DMS-1005441.
文摘In order to solve the magnetohydrodynamics(MHD)equations with a H(div)-conforming element,a novel approach is proposed to ensure the exact divergence-free condition on the magnetic field.The idea is to add on each element an extra interior bubble function from a higher order hierarchicalH(div)-conforming basis.Four such hierarchical bases for theH(div)-conforming quadrilateral,triangular,hexahedral,and tetrahedral elements are either proposed(in the case of tetrahedral)or reviewed.Numerical results have been presented to show the linear independence of the basis functions for the two simplicial elements.Good matrix conditioning has been confirmed numerically up to the fourth order for the triangular element and up to the third order for the tetrahedral element.
基金supported in part by China NSF under the grant 60873177by the National Basic Research Project under the grant 2005CB321702
文摘We consider H（curl, Ω）-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H^1 （Ω）-context along with local discrete Helmholtz-type decompositions of the edge element space.
