Long Chen

Bio: Long Chen is an academic researcher from University of California, Irvine. The author has contributed to research in topics: Finite element method & Multigrid method. The author has an hindex of 27, co-authored 105 publications receiving 2332 citations. Previous affiliations of Long Chen include University of California, San Diego & Beijing University of Technology.

The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation

Abstract: A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson-Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson-Boltzmann equation based on certain quasi-uniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson-Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson-Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson-Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.

Optimal anisotropic meshes for minimizing interpolation errors in $L^p$-norm

Abstract: In this paper, we present a new optimal interpolation error estimate in L p norm (1 < p ≤ ∞) for finite element simplicial meshes in any spatial dimension. A sufficient condition for a mesh to be nearly optimal is that it is quasi-uniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the L p norm. Some numerical examples are also given to support the theoretical estimates.

Mesh Smoothing Schemes Based on Optimal Delaunay Triangulations.

Abstract: We present several mesh smoothing schemes based on the concept of optimal Delaunay triangulations. We define the optimal Delaunay triangulation (ODT) as the triangulation that minimizes the interpolation error among all triangulations with the same number of vertices. ODTs aim to equidistribute the edge length under a new metric related to the Hessian matrix of the approximated function. Therefore we define the interpolation error as the mesh quality and move each node to a new location, in its local patch, that reduces the interpolation error. With several formulas for the interpolation error, we derive a suitable set of mesh smoothers among which Laplacian smoothing is a special case. The computational cost of proposed new mesh smoothing schemes in the isotropic case is as low as Laplacian smoothing while the error-based mesh quality is provably improved. Our mesh smoothing schemes also work well in the anisotropic case.

TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator

Abstract: TetGen is a Cpp program for generating good quality tetrahedral meshes aimed to support numerical methods and scientific computing. The problem of quality tetrahedral mesh generation is challenged by many theoretical and practical issues. TetGen uses Delaunay-based algorithms which have theoretical guarantee of correctness. It can robustly handle arbitrary complex 3D geometries and is fast in practice. The source code of TetGen is freely available.This article presents the essential algorithms and techniques used to develop TetGen. The intended audience are researchers or developers in mesh generation or other related areas. It describes the key software components of TetGen, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations (combination of flips and edge removal), and filtered exact geometric predicates. The essential algorithms include incremental Delaunay algorithms for inserting vertices, constrained Delaunay algorithms for inserting constraints (edges and triangles), a new edge recovery algorithm for recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation. Experimental examples as well as comparisons with other softwares are presented.