We develop a high order reconstructed discontinuous approximation(RDA)method for solving a mixed formulation of the quad-curl problem in two and three dimensions.This mixed formulation is established by adding an auxi...We develop a high order reconstructed discontinuous approximation(RDA)method for solving a mixed formulation of the quad-curl problem in two and three dimensions.This mixed formulation is established by adding an auxiliary variable to control the divergence of the field.The approximation space for the original variable is constructed by patch reconstruction with exactly one degree of freedom per element in each dimension and the auxiliary variable is approximated by the piecewise constant space.We prove the optimal convergence rate under the energy norm and also suboptimal L2 convergence using a duality approach.Numerical results are provided to verify the theoretical analysis.展开更多
We propose and analyze a new family of nonconforming finite elements for the three-dimensional quad-curl problem.The proposed finite element spaces are subspaces of H(curl),but not of H(grad curl),which are different ...We propose and analyze a new family of nonconforming finite elements for the three-dimensional quad-curl problem.The proposed finite element spaces are subspaces of H(curl),but not of H(grad curl),which are different from the existing nonconforming ones[10,12,13].The well-posedness of the discrete problem is proved and optimal error estimates in discrete H(grad curl)norm,H(curl)norm and L2 norm are derived.Numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.展开更多
Quad-curl equations with Navier type boundary conditions are studied in this paper.Stable order reduced formulations equivalent to the model problems are presented,and finite element discretizations are designed.Optim...Quad-curl equations with Navier type boundary conditions are studied in this paper.Stable order reduced formulations equivalent to the model problems are presented,and finite element discretizations are designed.Optimal convergence rates are proved.展开更多
Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming P_1-P_0 element for the Stokes equation in three dimensions are studied.Commutative dia...Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming P_1-P_0 element for the Stokes equation in three dimensions are studied.Commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators.The lower order H(gradcurl)-nonconforming finite element only has 14 degrees of freedom,whose basis functions are explicitly given in terms of the barycentric coordinates.The H(gradcurl)-nonconforming elements are applied to solve the quad-curl problem,and the optimal convergence is derived.By the nonconforming finite element Stokes complexes,the mixed finite element methods of the quad-curl problem are decoupled into two mixed methods of the Maxwell equation and the nonconforming P_1-P_0 element method for the Stokes equation,based on which a fast solver is discussed.Numerical results are provided to verify the theoretical convergence rates.展开更多
基金supported by National Natural Science Foundation of China(No.12288101)High-Performance Computing Platform of Peking University.
文摘We develop a high order reconstructed discontinuous approximation(RDA)method for solving a mixed formulation of the quad-curl problem in two and three dimensions.This mixed formulation is established by adding an auxiliary variable to control the divergence of the field.The approximation space for the original variable is constructed by patch reconstruction with exactly one degree of freedom per element in each dimension and the auxiliary variable is approximated by the piecewise constant space.We prove the optimal convergence rate under the energy norm and also suboptimal L2 convergence using a duality approach.Numerical results are provided to verify the theoretical analysis.
基金supported in part by the National Natural Science Foundation of China grant NSFC 12131005.
文摘We propose and analyze a new family of nonconforming finite elements for the three-dimensional quad-curl problem.The proposed finite element spaces are subspaces of H(curl),but not of H(grad curl),which are different from the existing nonconforming ones[10,12,13].The well-posedness of the discrete problem is proved and optimal error estimates in discrete H(grad curl)norm,H(curl)norm and L2 norm are derived.Numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.
基金The work of the first author was supported in part by Yunnan Provincial Science and Technology Department Research Award:Interdisciplinary Research in Computational Mathematics and Mechanics with Applications in Energy Engineering(Grant No.2009CI128)Yunnan Provincial Science and Technology Project(Grant No.2014RA071)The work of the second author was supported partially by the National Natural Science Foundation of China with Grant Nos 11471026 and 11871465 and National Centre for Mathematics and Interdisciplinary Sciences,Chinese Academy of Sciences.
文摘Quad-curl equations with Navier type boundary conditions are studied in this paper.Stable order reduced formulations equivalent to the model problems are presented,and finite element discretizations are designed.Optimal convergence rates are proved.
基金supported by National Natural Science Foundation of China (Grant Nos.12171300 and 11771338)the Natural Science Foundation of Shanghai (Grant No.21ZR1480500)the Fundamental Research Funds for the Central Universities (Grant No.2019110066)。
文摘Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming P_1-P_0 element for the Stokes equation in three dimensions are studied.Commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators.The lower order H(gradcurl)-nonconforming finite element only has 14 degrees of freedom,whose basis functions are explicitly given in terms of the barycentric coordinates.The H(gradcurl)-nonconforming elements are applied to solve the quad-curl problem,and the optimal convergence is derived.By the nonconforming finite element Stokes complexes,the mixed finite element methods of the quad-curl problem are decoupled into two mixed methods of the Maxwell equation and the nonconforming P_1-P_0 element method for the Stokes equation,based on which a fast solver is discussed.Numerical results are provided to verify the theoretical convergence rates.
