ChenLong.bib

Long Chen and Michael Holst and Jinchao Xu. Convergence and Optimality of Adaptive Mixed Finite Element Methods. Preprint, :, 2006.

Long Chen and Michael J. Holst. Mesh adaptation based on {Optimal Delaunay} triangulations. Preprint, :, 2006.

Long Chen and Pengtao Sun and Jinchao Xu. Multilevel Homotopic Adaptive Finite Element Methods for Convection Dominated Problems. The Proceedings for 15th Conferences for Domain Decomposition Methods, :459--468, 2004.

Chen.L;Sun.P;Xu.J2004

ABSTRACT: A multilevel homotopic adaptive methods is presented in this paper for convection dominated problems. By the homotopy method with respect to the diffusion parameter, the grid are iteratively adapted to better approximate the solution. Some new theoretic results and practical techniques for the grid adaptation are presented. Numerical experiments show that a standard finite element scheme based on this properly adapted grid works in a robust and efficient manner.

Long Chen and Pengtao Sun and Jinchao Xu. Numerical Study of Finite Element Methods for the Convectional Dominated Problems. Preprint, :, 2005.

Long Chen and Pengtao Sun and Jinchao Xu. Optimal anisotropic simplicial meshes for minimizing interpolation errors in ${L}^p$-norm. Accepted by Mathematics of Computation, :, 2006.

Chen.L;Sun.P;Xu.J2006

ABSTRACT: In this paper, we present a new optimal interpolation error estimate in $L^p$ norm ($1\leq p\leq \infty$) 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 view points of minimizing interpolation error in the $L^p$ norm. Some numerical examples are also given to support the theoretical estimates.

Long Chen and Jinchao Xu. Optimal {Delaunay} triangulations. Journal of Computational Mathematics, 22(2):299-308, 2004.

Chen.L;Xu.J2004

ABSTRACT: The Delaunay triangulation, in both classic and more generalized sense, is studied in this paper for minimizing the linear interpolation error (measure in $L^p$-norm) for a given function. The classic Delaunay triangulation can then be characterized as an optimal triangulation that minimizes the interpolation error for the isotropic function $\|\mathbf x\|^2$ among all the triangulations with a given set of vertices. For a more general function, a function-dependent Delaunay triangulation is then defined to be an optimal triangulation that minimizes the interpolation error for this function and its construction can be obtained by a simple lifting and projection procedure. The optimal Delaunay triangulation is the one that minimizes the interpolation error among all triangulations with the same number of vertices, i.e. the distribution of vertices are optimized in order to minimize the interpolation error. Such a function-dependent optimal Delaunay triangulation is proved to exist for any given convex continuous function. On an optimal Delaunay triangulation associated with $f$, it is proved that $\nabla f$ at the interior vertices can be exactly recovered by the function values on its neighboring vertices. Since the optimal Delaunay triangulation is difficult to obtain in practice, the concept of nearly optimal triangulation is introduced and two sufficient conditions are presented for a triangulation to be nearly optimal.

Long Chen and Jinchao Xu. An Optimal Streamline Diffusion Finite Element Method for a Singularly Perturbed Problem. AMS Contemporary Mathematics Series: Recent Advances in Adaptive Computation, 383:236--246, 2005.

Chen.L;Xu.J2005a

ABSTRACT: The stability and accuracy of a streamline diffusion finite element method (SDFEM) on arbitrary grids applied to a linear 1-d singularly perturbed problem are studied in this paper. With a special choice of the stabilization quadratic bubble function, the SDFEM is shown to have an optimal second order in the sense that $\|u-u_{h}\|_{\infty}\leq C\inf_{v_{h}\in V^{h}}\|u-v_{h}\|_{\infty},$ where $u_{h}$ is the SDFEM approximation of the exact solution $u$ and $V_{h}$ is the linear finite element space. With the quasi-optimal interpolation error estimate, quasi-optimal convergence results for the SDFEM are obtained. As a consequence, an open question about the optimal choice of the monitor function for a second order scheme in the moving mesh method is answered.

Long Chen and Jinchao Xu. Stability and accuracy of adapted finite element methods for singularly perturbed problems. Technique Report, Department of Mathematics, The Pennsylvania State University, :, 2005.

Chen.L;Xu.J2005b

ABSTRACT: The stability and accuracy of a standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) are studied in this paper for a one dimensional singularly perturbed connvection-diffusion problem discretized on arbitrary grids. Both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity (often in the form of boundary layers) of the solution. Surprisingly the accuracy of the standard FEM is shown to depend crucially on the uniformity of the grid away from the singularity. In other words, the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but, in contrast, it is robust with respect to perturbation of properly adapted grid inside the boundary layer. Motivated by this discovery, a new SDFEM is developed based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property in the sense that $\|u-u_{N}\|_{\infty}\leq C\inf_{v_{N}\in V^{N}}\|u-v_{N}\|_{\infty},$ where $u_{N}$ is the SDFEM approximation in linear finite element space $V^{N}$ of the exact solution $u$. Finally several optimal convergence results for the standard FEM and the new SDFEM are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered.

Long Chen and Jingchao Xu. Topics on Adaptive Finite Element Methods. Book Chapter (to appear), :, 2006.

Long Chen and Jinchao Xu. Multiscale Adaptive Finite Element Methods. , :, 2006.

Long Chen. A Degree one map between two bridge knots. Master Thesis, Peking University, :, 2000.

Long Chen. Mesh smoothing schemes based on optimal {Delaunay} triangulations. 13th International Meshing Roundtable, :109-120, 2004.

Chen.L2004

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.

Long Chen. New Analysis of the Sphere Covering Problems and Optimal Polytope Approximation of Convex Bodies. Journal of Approximation Theory, 133:134-145, 2005.

Chen.L2005

ABSTRACT: In this paper, we show that both sphere covering problems and optimal polytope approximation of convex bodies are related to optimal Delaunay triangulations, which are the triangulations minimizing the interpolation error between function $x^2$ and its linear interpolant based on the underline triangulations. We then develop a new analysis based on the estimate of the interpolation error to get the Coxeter-Few-Rogers lower bound for the thickness in the sphere covering problem and a new estimate of the constant $del_n$ appeared in the optimal polytope approximation of convex bodies.

Long Chen. Robust and Accurate Algorithms for Solving Anisotropic Singularities. , :, 2005.

Long Chen. Superconvergence of tetrahedral linear finite elements. International Journal of Numerical Analysis and Modeling, 3:273--282, 2006.

Chen.L2006

ABSTRACT: In this paper, we show that the piecewise linear finite element solution $u_{h}$ and the linear interpolation $u_{I}$ have superclose gradient for tetrahedral meshes, where most elements are obtained by dividing approximate parallelepiped into six tetrahedra. We then analyze a post-processing gradient recovery scheme, showing that the global $L^2$ projection of $\nabla u_h$ is a superconvergent gradient approximation to $\nabla u$.