This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jac...This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a C1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.展开更多
In this article,a family of H^2-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3 D.In the family,the Pl polynomial space is enriched by some high order polynom...In this article,a family of H^2-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3 D.In the family,the Pl polynomial space is enriched by some high order polynomials for all l≥3 and the corresponding finite element solution converges at the order l-1 in H2 norm.Moreover,the result is improved for two low order cases by using P6 and P7 polynomials to enrich P4 and P5 polynomial spaces,respectively.The error estimate is proved.The numerical results are.provided to confirm the theoretical findings.展开更多
Mixed triangular spectral element method using nodal basis on unstructured meshes is investigated in this paper.The method is based on equivalent first order system of the elliptic problem and rectangle-triangle trans...Mixed triangular spectral element method using nodal basis on unstructured meshes is investigated in this paper.The method is based on equivalent first order system of the elliptic problem and rectangle-triangle transforms.It fully enjoys the ten-sorial structure and flexibility in handling complex domains by using nodal basis and unstructured triangular mesh.Different from the usual Galerkin formulation,the mixed form is particularly advantageous in this context,since it can avoid the singularity in-duced by the rectangle-triangle transform in the calculation of the matrices,and does not require the evaluation of the stiffness matrix.An hp a priori error estimate is pres-ented for the proposed method.The implementation details and some numerical exam-ples are provided to validate the accuracy and flexibility of the method.展开更多
This study aimed to specialise a directional H^(2)(DH^(2))compression to matrices arising from the discontinuous Galerkin(DG)discretisation of the hypersingular equation in acoustics.The significantfinding is an algor...This study aimed to specialise a directional H^(2)(DH^(2))compression to matrices arising from the discontinuous Galerkin(DG)discretisation of the hypersingular equation in acoustics.The significantfinding is an algorithm that takes a DG stiffness matrix andfinds a near-optimal DH^(2) approximation for low and high-frequency problems.We introduced the necessary special optimisations to make this algorithm more efficient in the case of a DG stiffness matrix.Moreover,an automatic parameter tuning strategy makes it easy to use and versatile.Numerical comparisons with a classical Boundary Element Method(BEM)show that a DG scheme combined with a DH^(2) gives better computational efficiency than a classical BEM in the case of high-order finite elements and hp heterogeneous meshes.The results indicate that DG is suitable for an auto-adaptive context in integral equations.展开更多
文摘This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a C1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.
基金supported by National Natural Science Foundation of China(Grant Nos.11625101 and 11421101)。
文摘In this article,a family of H^2-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3 D.In the family,the Pl polynomial space is enriched by some high order polynomials for all l≥3 and the corresponding finite element solution converges at the order l-1 in H2 norm.Moreover,the result is improved for two low order cases by using P6 and P7 polynomials to enrich P4 and P5 polynomial spaces,respectively.The error estimate is proved.The numerical results are.provided to confirm the theoretical findings.
基金The first and second authors gratefully acknowledge the financial support provided by NSFC(grant 11771137)。
文摘Mixed triangular spectral element method using nodal basis on unstructured meshes is investigated in this paper.The method is based on equivalent first order system of the elliptic problem and rectangle-triangle transforms.It fully enjoys the ten-sorial structure and flexibility in handling complex domains by using nodal basis and unstructured triangular mesh.Different from the usual Galerkin formulation,the mixed form is particularly advantageous in this context,since it can avoid the singularity in-duced by the rectangle-triangle transform in the calculation of the matrices,and does not require the evaluation of the stiffness matrix.An hp a priori error estimate is pres-ented for the proposed method.The implementation details and some numerical exam-ples are provided to validate the accuracy and flexibility of the method.
文摘This study aimed to specialise a directional H^(2)(DH^(2))compression to matrices arising from the discontinuous Galerkin(DG)discretisation of the hypersingular equation in acoustics.The significantfinding is an algorithm that takes a DG stiffness matrix andfinds a near-optimal DH^(2) approximation for low and high-frequency problems.We introduced the necessary special optimisations to make this algorithm more efficient in the case of a DG stiffness matrix.Moreover,an automatic parameter tuning strategy makes it easy to use and versatile.Numerical comparisons with a classical Boundary Element Method(BEM)show that a DG scheme combined with a DH^(2) gives better computational efficiency than a classical BEM in the case of high-order finite elements and hp heterogeneous meshes.The results indicate that DG is suitable for an auto-adaptive context in integral equations.
