A partially orthonormal basis is constructed with better conditioning properties for tetrahedral H（curl）-conforming Nedelec elements. The shape functions are classified into several categories with respect to their ...A partially orthonormal basis is constructed with better conditioning properties for tetrahedral H（curl）-conforming Nedelec elements. The shape functions are classified into several categories with respect to their topological entities on the reference 3-simplex. The basis functions in each category are constructed to achieve maximum orthogonaiity. The numerical study on the matrix conditioning shows that for the mass and quasi-stiffness matrices, and in a logarithmic scale the condition number grows linearly vs. order of approximation up to order three. For each order of approximation, the condition number of the quasi-stiffness matrix is about one order less than the corresponding one for the mass matrix. Also, up to order six of approximation the conditioning of the mass and quasi- stiffness matrices with the proposed basis is better than the corresponding one with the Ainsworth-Coyle basis Internat. J. Numer. Methods. Engrg., 58：2103-2130, 2003. except for order four with the quasi-stiffness matrix. Moreover, with the new basis the composite matrix μM ＋ S has better conditioning than the Ainsworth-Coyle basis for a wide range of the parameter μ.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
文摘A partially orthonormal basis is constructed with better conditioning properties for tetrahedral H（curl）-conforming Nedelec elements. The shape functions are classified into several categories with respect to their topological entities on the reference 3-simplex. The basis functions in each category are constructed to achieve maximum orthogonaiity. The numerical study on the matrix conditioning shows that for the mass and quasi-stiffness matrices, and in a logarithmic scale the condition number grows linearly vs. order of approximation up to order three. For each order of approximation, the condition number of the quasi-stiffness matrix is about one order less than the corresponding one for the mass matrix. Also, up to order six of approximation the conditioning of the mass and quasi- stiffness matrices with the proposed basis is better than the corresponding one with the Ainsworth-Coyle basis Internat. J. Numer. Methods. Engrg., 58：2103-2130, 2003. except for order four with the quasi-stiffness matrix. Moreover, with the new basis the composite matrix μM ＋ S has better conditioning than the Ainsworth-Coyle basis for a wide range of the parameter μ.
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.
