Mesh-dependent stability for finite element approximations of parabolic equations with mass lumping

In this paper, based on some mesh-dependent estimates on the extreme eigenvalues of a general finite element system defined on a simplicial mesh, novel and sharp bounds on the permissible time step size are derived for the mass lumping finite element approximations of parabolic equations. The bounds are dependent not only on the mesh size but also on the mesh shape. These results provide guidance to the stability of numerical solutions of parabolic problems in relation to the unstructured geometric meshing. Numerical experiments on both uniform meshes and adaptive meshes are presented to validate the theoretical analysis.     

A constrained optimization approach to finite element mesh smoothing

The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the “smoothness” or “shape” of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a “smooth” final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method.     

The finite element method for parabolic equations

Summary In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.The work was partially supported by ONR Contract N00014-77-C-0623     

Anisotropic a posteriori error estimation for the mixed discontinuous Galerkin approximation of the Stokes problem

This article presents a posteriori error estimates for the mixed discontinuous Galerkin approximation of the stationary Stokes problem. We consider anisotropic finite element discretizations, i.e., elements with very large aspect ratio. Our analysis covers two‐ and three‐dimensional domains. Lower and upper error bounds are proved with minimal assumptions on the meshes. The lower error bound is uniform with respect to the mesh anisotropy. The upper error bound depends on a proper alignment of the anisotropy of the mesh, which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimator. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

高波数问题的超收敛性

 本文考虑求解Helmholtz方程的有限元方法的超逼近性质以及基于PPR后处理方法的超收敛性质.我们首先给出了矩形网格上的p-次元在收敛条件k（kh）^2p+1≤C₀下的有限元解和基于Lobatto点的有限元插值之间的超逼近以及重构的有限元梯度和精确解之间的超收敛分析.然后我们给出了四边形网格上的线性有限元方法的分析.这些估计都给出了与波数k和网格尺寸h的依赖关系.同时我们回顾了三角形网格上的线性有限元的超收敛结果.最后我们给出了数值实验并且结合Richardson外推进一步减少了误差.     

Graded Meshes for Higher Order FEM

A singularly perturbed one-dimensional convection-diffusion problem is solved numerically by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded (based on a mesh generating function) in the fine mesh region. Error estimates in the ε-weighted energy norm are proved. We derive an 'optimal' mesh generating function in order to minimize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving error estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results.     

Robust a Posteriori Error Estimation for a Singularly Perturbed Reaction–Diffusion Equation on Anisotropic Tetrahedral Meshes

We consider a singularly perturbed reaction–diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Interior maximum norm estimates for finite element discretizations of the Stokes equations

Interior estimates are proved in the L ^∞ norm for stable finite element discretizations of the Stokes equations on translation invariant meshes. These estimates yield information about the quality of the finite element solution in subdomains a positive distance from the boundary. While they have been established for second-order elliptic problems, these interior, or local, maximum norm estimates for the Stokes equations are new. By applying finite differenciation methods on a translation invariant mesh, we obtain optimal convergence rates in the mesh size h in the maximum norm. These results can be used for analyzing superconvergence in finite element methods for the Stokes equations.     

The asymptotically optimal meshsize function for bi-p degree interpolation over rectangular elements

A method is presented to recover near optimal interpolation on finite element meshes based on information in the approximation error on an initial mesh. Only a certain class of admissable meshes with rectangular elements in the computational domains are allowed. The method attempts to reach the optimal mesh in one step from the initial mesh, and is based on the notion of meshsize function components or mesh density functions. Asymptotical results showing the optimality of the recovered meshes are given, and extensive computational verification of the method in the special case of Lagrange polynomial interpolation is provided.     

Neumann–Neumann algorithms for a mortar Crouzeix–Raviart element for 2nd order elliptic problems

The paper proposes two scalable variants of the Neumann–Neumann algorithm for the lowest order Crouzeix–Raviart finite element or the nonconforming P ₁ finite element on nonmatching meshes. The overall discretization is done using a mortar technique which is based on the application of an approximate matching condition for the discrete functions, requiring function values only at the mesh interface nodes. The algorithms are analyzed using the abstract Schwarz framework, proving a convergence which is independent of the jumps in the coefficients of the problem and only depends logarithmically on the ratio between the subdomain size and the mesh size.     

一类抛物型方程有限元算法的计算准则

用有限元法分析瞬态温度场,很有可能得到“振荡”和“超界”的计算结果.这两种现象不符合热传导规律.为解决此问题,我们提出时间单调性和空间单调性的概念,推导出三维无源热传导方程的数值解的时间单调性的几组充分条件.对某些特殊边值问题,使用规则单元网格,可以得到合理结果时Δt/Δx2的上下界公式.文中还研究了空间单调性.最后我们还讨论了集中质量阵的算法.针对以热传导方程为代表的这一类抛物型方程的有限元算法,我们创造性地给出几组计算准则.     

有限元网格的超收敛测度及其应用

给出线性有限元求解二阶椭圆问题的有限元网格超收敛测度及其应用.有限元超收敛经常是在具有一定结构的特殊网格条件下讨论的,而本文从一般网格出发,导出一种网格的范数用来描述超收敛所需要的网格条件以及超收敛的程度.并且通过对这种网格范数性质的考察,可以证明对于通常考虑的一些特殊网格的超收敛的存在性.更进一步,我们可以通过正则细分的方式在一般区域上也可以自动获得超收敛网格.最后给出相关的数值结果来验证本文的理论分析.     

On using meshes of mixed element types in adaptive finite element analysis

Four different automatic mesh generators capable of generating either triangular meshes or hybrid meshes of mixed element types have been used in the mesh generation process. The performance of these mesh generators were tested by applying them to the adaptive finite element refinement procedure. It is found that by carefully controlling the quality and grading of the quadrilateral elements, an increase in efficiency over pure triangular meshes can be achieved. Furthermore, if linear elements are employed, an optimal hybrid mesh can be obtained most economically by a combined use of the mesh coring technique suggested by Lo and Lau and a selective removal of diagonals from the triangular element mesh. On the other hand, if quadratic elements are used, it is preferable to generate a pure triangular mesh first, and then obtain a hybrid mesh by merging of triangles.     

A posteriori error estimators for optimal distributed control governed by the first‐order linear hyperbolic equation: DG method

In this article, We analyze the ‐version of the discontinuous Galerkin finite element method (DGFEM) for the distributed first‐order linear hyperbolic optimal control problems. We derive a posteriori error estimators on general finite element meshes which are sharp in the mesh‐width . These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems. For the DGFEM we admit very general irregular meshes. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

The influence of boundary approximation on transfer operators between non-nested meshes

Finite element methods with non-matching meshes can offer increased flexibility in many applications. Although the specific reasons for the use of non-matching meshes are apparently diverse, the common difficulty in all these numerical methods is the transfer of finite element approximation associated with one mesh to finite element approximation associated with another mesh. This paper complements previous quantitative studies of transfer operators between finite element spaces associated with unrelated meshes (T. Dickopf, R. Krause, Evaluating local approximations of the L²-orthogonal projection between non-nested finite element spaces, Tech. Rep. 2012-01, Institute of Computational Science, Università della Svizzera italiana, 2012). We study the important use case in which functions are mapped between a regular background mesh and an unstructured mesh of a complex geometry. Here, the former does not approximate the latter. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

A Study on the Conditioning of Finite Element Equations with Arbitrary Anisotropic Meshes via a Density Function Approach

The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix is investigated using a density function approach proposed by Fried in 1973. It is shown that the approach can be made mathematically rigorous for general domains and used to develop bounds on the smallest eigenvalue and the condition number that are sharper than existing estimates in one and two dimensions and comparable in three and higher dimensions. The new results reveal that the mesh concentration near the boundary has less influence on the condition number than the mesh concentration in the interior of the domain. This is especially true for the Jacobi preconditioned system where the former has little or almost no influence on the condition number. Numerical examples are presented.     

An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems

An efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. An adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the final iteration of the adaptive procedure, allowing only mesh and time-step size refinements before. It is proved that at each time step the adaptive algorithm is able to reduce the error indicators (and thus the error) below any given tolerance within a finite number of iteration steps. The key ingredient in the analysis is a new coarsening strategy. Numerical results are presented to show the competitive behavior of the proposed adaptive algorithm.