Regular assumption of finite element meshes is a basic condition of most analysis of finite element approximations both for conventional conforming elements and nonconforming elements. The aim of this paper is to pres...Regular assumption of finite element meshes is a basic condition of most analysis of finite element approximations both for conventional conforming elements and nonconforming elements. The aim of this paper is to present a novel approach of dealing with the approximation of a four-degree nonconforming finite element for the second order elliptic problems on the anisotropic meshes. The optimal error estimates of energy norm and L^2-norm without the regular assumption or quasi-uniform assumption are obtained based on some new special features of this element discovered herein. Numerical results are given to demonstrate validity of our theoretical analysis.展开更多
A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is d...A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.展开更多
We propose in this paper a robust surface mesh denoising method that can effectively remove mesh noise while faithfully preserving sharp features. This method utilizes surface fitting and projection techniques. Sharp ...We propose in this paper a robust surface mesh denoising method that can effectively remove mesh noise while faithfully preserving sharp features. This method utilizes surface fitting and projection techniques. Sharp features are preserved in the surface fitting algorithm by considering an anisotropic neighborhood of each vertex detected by the normal-weighted distance. In addition, to handle the mesh with a high level of noise, we perform a pre-filtering of surface normals prior to the neighborhood searching. A number of experimental results and comparisons demonstrate the excellent performance of our method in preserving important surface geometries while filtering mesh noise.展开更多
This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this elem...This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.展开更多
Anisotropic meshes are known to be well-suited for problems which exhibit anisotropic solution features. Defining an appropriate metric tensor and designing an efficient algorithm for anisotropic mesh gen- eration are...Anisotropic meshes are known to be well-suited for problems which exhibit anisotropic solution features. Defining an appropriate metric tensor and designing an efficient algorithm for anisotropic mesh gen- eration are two important aspects of the anisotropic mesh methodology. In this paper, we are concerned with the natural metric tensor for use in anisotropic mesh generation for anisotropic elliptic problems. We provide an algorithm to generate anisotropic meshes under the given metric tensor. We show that the inverse of the anisotropic diffusion matrix of the anisotropic elliptic problem is a natural metric tensor for the anisotropic mesh generation in three aspects： better discrete algebraic systems, more accurate finite element solution and superconvergence on the mesh nodes. Various numerical examples demonstrating the effectiveness are presented.展开更多
The nonconforming Crouzeix-Raviart type linear triangular finite element approximate to second-order elliptic problems is studied on anisotropic general triangular meshes in 2D satisfying the maximal angle condition a...The nonconforming Crouzeix-Raviart type linear triangular finite element approximate to second-order elliptic problems is studied on anisotropic general triangular meshes in 2D satisfying the maximal angle condition and the coordinate system condition. The optimal-order error estimates of the broken energy norm and L2-norm are obtained.展开更多
Mesh adaptation is studied from the mesh control point of view.Two principles,equidistribution and alignment,are obtained and found to be necessary and sufficient for a complete control of the size,shape,and orientati...Mesh adaptation is studied from the mesh control point of view.Two principles,equidistribution and alignment,are obtained and found to be necessary and sufficient for a complete control of the size,shape,and orientation of mesh elements.A key component in these principles is the monitor function,a symmetric and positive definite matrix used for specifying the mesh information.A monitor function is defined based on interpolation error in a way with which an error bound is minimized on a mesh satisfying the equidistribution and alignment conditions.Algorithms for generating meshes satisfying the conditions are developed and two-dimensional numerical results are presented.展开更多
Composite penalty method of a low order anisotropic nonconforming quadrilateral finite element for the Stokes problem is presented. This method with a large penalty parameter can achieve the same accuracy as the stand...Composite penalty method of a low order anisotropic nonconforming quadrilateral finite element for the Stokes problem is presented. This method with a large penalty parameter can achieve the same accuracy as the stand method with a small penalty parameter and the convergence rate of this method is two times as that of the standard method under the condition of the same order penalty parameter. The superconvergence for velocity is established as well. The results of this paper are also valid to the most of the known nonconforming finite element methods.展开更多
We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous Galerkin(DG)method with focus on the goal-oriented error estimates and adaptivity.We analyse the adjoint consis...We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous Galerkin(DG)method with focus on the goal-oriented error estimates and adaptivity.We analyse the adjoint consistency of the DG scheme where the adjoint problem is not formulated by the differentiation of the DG form and the target functional but using a suitable linearization of the nonlinear forms.Furthermore,we present the goal-oriented anisotropic hp-mesh adaptation method for the Euler equations.The theoretical results are supported by numerical experiments.展开更多
文摘Regular assumption of finite element meshes is a basic condition of most analysis of finite element approximations both for conventional conforming elements and nonconforming elements. The aim of this paper is to present a novel approach of dealing with the approximation of a four-degree nonconforming finite element for the second order elliptic problems on the anisotropic meshes. The optimal error estimates of energy norm and L^2-norm without the regular assumption or quasi-uniform assumption are obtained based on some new special features of this element discovered herein. Numerical results are given to demonstrate validity of our theoretical analysis.
基金Supported by the National Natural Science Foundation of China (10671184)
文摘A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.
基金supported in part by the National Institutes of Health of USA under Grant No. R15HL103497 from the National Heart, Lung, and Blood Institute (NHLBI)a subcontract of NIH Award under Grant No. P41RR08605 from the National Biomedical Computation Resource
文摘We propose in this paper a robust surface mesh denoising method that can effectively remove mesh noise while faithfully preserving sharp features. This method utilizes surface fitting and projection techniques. Sharp features are preserved in the surface fitting algorithm by considering an anisotropic neighborhood of each vertex detected by the normal-weighted distance. In addition, to handle the mesh with a high level of noise, we perform a pre-filtering of surface normals prior to the neighborhood searching. A number of experimental results and comparisons demonstrate the excellent performance of our method in preserving important surface geometries while filtering mesh noise.
基金Supported by the National Natural Science Foundation of China(10371113,10671184)
文摘This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.
基金supported by National Natural Science Foundation of China(Grant Nos.11031006 and 11201397)Program for Changjiang Scholars and Innovative Research Team in University(Grant No.IRT1179)+2 种基金International Science and Technology Cooperation Program of China(Grant No.2010DFR00700)Hunan Education Department Project(Grant No.12B127)Hunan Provincial National Science Foundation Project(Grant No.12JJ4004)
文摘Anisotropic meshes are known to be well-suited for problems which exhibit anisotropic solution features. Defining an appropriate metric tensor and designing an efficient algorithm for anisotropic mesh gen- eration are two important aspects of the anisotropic mesh methodology. In this paper, we are concerned with the natural metric tensor for use in anisotropic mesh generation for anisotropic elliptic problems. We provide an algorithm to generate anisotropic meshes under the given metric tensor. We show that the inverse of the anisotropic diffusion matrix of the anisotropic elliptic problem is a natural metric tensor for the anisotropic mesh generation in three aspects： better discrete algebraic systems, more accurate finite element solution and superconvergence on the mesh nodes. Various numerical examples demonstrating the effectiveness are presented.
基金supported by the National Natural Science Foundation of China (No. 10971203)
文摘The nonconforming Crouzeix-Raviart type linear triangular finite element approximate to second-order elliptic problems is studied on anisotropic general triangular meshes in 2D satisfying the maximal angle condition and the coordinate system condition. The optimal-order error estimates of the broken energy norm and L2-norm are obtained.
基金supported in part by the NSF under grant DMS-0410545.
文摘Mesh adaptation is studied from the mesh control point of view.Two principles,equidistribution and alignment,are obtained and found to be necessary and sufficient for a complete control of the size,shape,and orientation of mesh elements.A key component in these principles is the monitor function,a symmetric and positive definite matrix used for specifying the mesh information.A monitor function is defined based on interpolation error in a way with which an error bound is minimized on a mesh satisfying the equidistribution and alignment conditions.Algorithms for generating meshes satisfying the conditions are developed and two-dimensional numerical results are presented.
基金Supported by the National Natural Science Foundation of China (10791203, 11271340)the Natural Science Foundation of Henan Province (112300410109)
文摘Composite penalty method of a low order anisotropic nonconforming quadrilateral finite element for the Stokes problem is presented. This method with a large penalty parameter can achieve the same accuracy as the stand method with a small penalty parameter and the convergence rate of this method is two times as that of the standard method under the condition of the same order penalty parameter. The superconvergence for velocity is established as well. The results of this paper are also valid to the most of the known nonconforming finite element methods.
基金Grant no.20-01074S of the Czech Science Foundation.
文摘We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous Galerkin(DG)method with focus on the goal-oriented error estimates and adaptivity.We analyse the adjoint consistency of the DG scheme where the adjoint problem is not formulated by the differentiation of the DG form and the target functional but using a suitable linearization of the nonlinear forms.Furthermore,we present the goal-oriented anisotropic hp-mesh adaptation method for the Euler equations.The theoretical results are supported by numerical experiments.
