The modified Poisson-Boltzmann(MPB)equations are often used to describe the equilibrium particle distribution of ionic systems.In this paper,we propose a fast algorithm to solve the MPB equations with the self Green’...The modified Poisson-Boltzmann(MPB)equations are often used to describe the equilibrium particle distribution of ionic systems.In this paper,we propose a fast algorithm to solve the MPB equations with the self Green’s function as the self-energy in three dimensions,where the solution of the self Green’s function poses a computational bottleneck due to the requirement of solving a high-dimensional partial differential equation.Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green’s function,building upon our previous result of two dimensions.This approach yields an algorithm with a complexity of O(N log N)by strategically leveraging the locality and low-rank characteristics of the corresponding operators.Additionally,the theoretical O(N)complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction.Extensive numerical results are conducted to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions.展开更多
In this paper, we first introduce interpolation operator of projection type in three dimen- sions, from which we derive weak estimates for tricubic block finite elements. Then using the estimate for the W 2, 1-seminor...In this paper, we first introduce interpolation operator of projection type in three dimen- sions, from which we derive weak estimates for tricubic block finite elements. Then using the estimate for the W 2, 1-seminorm of the discrete derivative Green's function and the weak estimates, we show that the tricubic block finite element solution uh and the tricubic interpolant of projection type Πh3u have superclose gradient in the pointwise sense of the L∞-norm. Finally, this supercloseness is applied to superconvergence analysis, and the global superconvergence of the finite element approximation is derived.展开更多
基金support from the National Natural Science Foundation of China(Grant Nos.12071288 and 12325113)the Science and Technology Commission of Shanghai Municipality of China(Grant No.21JC1403700)+1 种基金Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA25010403)support of the US National Science Foundation under awards DMS-2244988 and DMS-2206333.
文摘The modified Poisson-Boltzmann(MPB)equations are often used to describe the equilibrium particle distribution of ionic systems.In this paper,we propose a fast algorithm to solve the MPB equations with the self Green’s function as the self-energy in three dimensions,where the solution of the self Green’s function poses a computational bottleneck due to the requirement of solving a high-dimensional partial differential equation.Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green’s function,building upon our previous result of two dimensions.This approach yields an algorithm with a complexity of O(N log N)by strategically leveraging the locality and low-rank characteristics of the corresponding operators.Additionally,the theoretical O(N)complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction.Extensive numerical results are conducted to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions.
基金supported by Natural Science Foundation of Ningbo City (Grant No. 2008A610020)National Natural Science Foundation of China (Grant No. 10671065)the Scientific Research Fund of Hunan Provincial Education Department (Grant Nos. 07C576, 03C212)
文摘In this paper, we first introduce interpolation operator of projection type in three dimen- sions, from which we derive weak estimates for tricubic block finite elements. Then using the estimate for the W 2, 1-seminorm of the discrete derivative Green's function and the weak estimates, we show that the tricubic block finite element solution uh and the tricubic interpolant of projection type Πh3u have superclose gradient in the pointwise sense of the L∞-norm. Finally, this supercloseness is applied to superconvergence analysis, and the global superconvergence of the finite element approximation is derived.
