The Lower/Upper Bound Property of Approximate Eigenvalues by Nonconforming Finite Element Methods for Elliptic Operators

This paper is a complement of the work (Hu et al. in arXiv:1112.1145v1[math.NA], 2011), where a general theory is proposed to analyze the lower bound property of discrete eigenvalues of elliptic operators by nonconforming finite element methods. One main purpose of this paper is to propose a novel approach to analyze the lower bound property of discrete eigenvalues produced by the Crouzeix–Raviart element when exact eigenfunctions are smooth. In particular, under some conditions on the triangular mesh, it is proved that the Crouzeix–Raviart element method of the Laplace operator yields eigenvalues below exact ones. Such a theoretical result explains most of numerical results in the literature and also partially answers the problem of Boffi (Acta Numerica 1–120, 2010). This approach can be applied to the Crouzeix–Raviart element of the Stokes eigenvalue problem and the Morley element of the buckling eigenvalue problem of a plate. As a second main purpose, a new identity of the error of eigenvalues is introduced to study the upper bound property of eigenvalues by nonconforming finite element methods, which is successfully used to explain why eigenvalues produced by the rotated $Q_1$ element of second order elliptic operators (when eigenfunctions are smooth), the Adini element (when eigenfunctions are singular) and the new Zienkiewicz-type element of fourth order elliptic operators, are above exact ones.     

A High Accuracy Post-processing Algorithm for the Eigenvalues of Elliptic Operators

In a very recent paper (Hu et al., The lower bounds for eigenvalues of elliptic operators by nonconforming finite element methods, Preprint, 2010), we prove that the eigenvalues by the nonconforming finite element methods are smaller than the exact ones for the elliptic operators. It is well-known that the conforming finite element methods produce the eigenvalues above to the exact ones. In this paper, we combine these two aspects and derive a new post-processing algorithm to approximate the eigenvalues of elliptic operators. We implement this algorithm and find that it actually yields very high accuracy approximation on very coarser mesh. The numerical results demonstrate that the high accuracy herein is of two fold: the much higher accuracy approximation on the very coarser mesh and the much higher convergence rate than a single lower/upper bound approximation. Moreover, we propose some acceleration technique for the algorithm of the discrete eigenvalue problem based on the solution of the discrete eigenvalue problem which yields the upper bound of the eigenvalue. With this acceleration technique we only need several iterations (two iterations in our example) to find the numerical solution of the discrete eigenvalue problem which produces the lower bound of the eigenvalue. Therefore we only need to solve essentially one discrete eigenvalue problem.     

Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs

In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space $V({{\mathcal {T}_{H}}},\boldsymbol {P})$ . The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ ; thereby, only a linear system of equations is solved on the richer space $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ . In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces $V({{\mathcal {T}_{H}}},\boldsymbol {P})$ and $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ , respectively, in an automatic fashion. Numerical experiments are presented for both two- and three-dimensional problems; in each case, we demonstrate that the CPU time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method.     

A Posteriori Error Estimates of Two-Grid Finite Element Methods for Nonlinear Elliptic Problems

In this article, we study the residual-based a posteriori error estimates of the two-grid finite element methods for the second order nonlinear elliptic boundary value problems. Computable upper and lower bounds on the error in the \(H^1\)-norm are established. Numerical experiments are also provided to illustrate the performance of the proposed estimators.     

A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

This article provides a systematic study for the weak Galerkin (WG) finite element method for second order elliptic problems by exploring polynomial approximations with various degrees for each local element. A typical local WG element is of the form \(P_k(T)\times P_j(\partial T)\Vert P_\ell (T)^2\), where \(k\ge 1\) is the degree of polynomials in the interior of the element T, \(j\ge 0\) is the degree of polynomials on the boundary of T, and \(\ell \ge 0\) is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. A general framework of stability and error estimate is developed for the corresponding numerical solutions. Numerical results are presented to confirm the theoretical results. The work reveals some previously undiscovered strengths of the WG method for second order elliptic problems, and the results are expected to be generalizable to other type of partial differential equations.     

A Posteriori Error Estimates for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

A residual type a posteriori error estimator is presented and analyzed for Weak Galerkin finite element methods for second order elliptic problems. The error estimator is proved to be efficient and reliable through two estimates, one from below and the other from above, in terms of an $H^1$ -equivalent norm for the exact error. Two numerical experiments are conducted to demonstrate the effectiveness of adaptive mesh refinement guided by this estimator.     

Anisotropic Nonconforming { EQ}_1^{rot} Quadrilateral Finite Element Approximation to Second Order Elliptic Problems

The main aim of this paper is to study the nonconforming $EQ_1^{rot}$ quadrilateral finite element approximation to second order elliptic problems on anisotropic meshes. The optimal order error estimates in broken energy norm and $L^2$ -norm are obtained, and three numerical experiments are carried out to confirm the theoretical results.     

A Coupled Multigrid Method for Nonconforming Finite Element Discretizations of the 2D-Stokes Equation

This paper investigates a multigrid method for the solution of the saddle point formulation of the discrete Stokes equation obtained with inf–sup stable nonconforming finite elements of lowest order. A smoother proposed by Braess and Sarazin (1997) is used and L ²-projection as well as simple averaging are considered as prolongation. The W-cycle convergence in the L ²-norm of the velocity with a rate independently of the level and linearly decreasing with increasing number of smoothing steps is proven. Numerical tests confirm the theoretically predicted results. Received January 19, 1999; revised September 13, 1999     

有限域与Ramsey数Rn(k)的下界

研究有限域GF(p')上的循环图的结构性质，给出一些图的团数的解析表达式，并给出计算Rarnsey数Rn(k)下界的一种算法，得到一个Ramsey数的新下界：R3(8)≥4111。     

Higher-Order Adaptive Finite Difference Methods for Fully Nonlinear Elliptic Equations

We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for adaptive meshes and complicated geometries, while still ensuring consistency, monotonicity, and convergence. We describe an algorithm for efficiently computing the non-traditional finite difference stencils. We also present a strategy for computing formally higher-order convergent methods. Computational examples demonstrate the efficiency, accuracy, and flexibility of the methods.     

Implementation of Generalized Finite Element Methods for Homogenization Problems

The implementation of a generalized Finite Element Method (FEM) for problems with coefficients or geometry that oscillate locally at a small length scale 1 is described. Two-scale FE-spaces are combined conformingly with standard FE. Numerical experiments show that the complexity of the algorithm is independent of the micro length scale .     

Adaptive Multilevel Correction Method for Finite Element Approximations of Elliptic Optimal Control Problems

In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which requires the solution of the optimization problem on new finite element space after each mesh refinement, with our approach we only need to solve two linear boundary value problems on current refined mesh and an optimization problem on a very low dimensional space. The linear boundary value problems can be solved with well-established multigrid method designed for elliptic equation and the optimization problems are of small scale corresponding to the space built with the coarsest space plus two enriched bases. Our approach can achieve the similar accuracy with standard AFEM but greatly reduces the computational cost. Numerical experiments demonstrate the efficiency of our proposed algorithm.     

Finite Element Methods for a System of Dispersive Equations

The present study is concerned with the numerical approximation of periodic solutions of systems of Korteweg–de Vries type, coupled through their nonlinear terms. We construct, analyze and numerically validate two types of schemes that differ in their treatment of the third derivatives appearing in the system. One approach preserves a certain important invariant of the system, up to round-off error, while the other, somewhat more standard method introduces a measure of dissipation. For both methods, we prove convergence of a semi-discrete approximation and highlight differences in the basic assumptions required for each. Numerical experiments are also conducted with the aim of ascertaining the accuracy of the two schemes when integrations are made over long time intervals.     

Fully Computable a Posteriori Error Bounds for Hybridizable Discontinuous Galerkin Finite Element Approximations

We introduce an entirely new class of high-order methods for computational fluid dynamics based on the Gaussian process (GP) family of stochastic functions. Our approach is to use kernel-based GP prediction methods to interpolate/reconstruct high-order approximations for solving hyperbolic PDEs. We present a new high-order formulation to solve (magneto)hydrodynamic equations using the GP approach that furnishes an alternative to conventional polynomial-based approaches.     

Isospectral Domains for Discrete Elliptic Operators

Concerning the Laplace operator with homogeneous Dirichlet boundary conditions, the classical notion of isospectrality assumes that two domains are related when they give rise to the same spectrum. In two dimensions, non isometric, isospectral domains exist. It is not known however if all the eigenvalues relative to a specific domain can be preserved under suitable continuous deformation of its geometry. We show that this is possible when the 2D Laplacian is replaced by a finite dimensional version and the geometry is modified by respecting certain constraints. The analysis is carried out in a very small finite dimensional space, but it can be extended to more accurate finite-dimensional representations of the 2D Laplacian, with an increase of computational complexity. The aim of this paper is to introduce the preliminary steps in view of more serious generalizations.     

Finite Difference/Finite Element Methods for Distributed-Order Time Fractional Diffusion Equations

In this paper, a class of distributed-order time fractional diffusion equations (DOFDEs) on bounded domains is considered. By L1 method in temporal direction, a semi-discrete variational formulation of DOFDEs is obtained firstly. The stability and convergence of this semi-discrete scheme are discussed, and the corresponding fully discrete finite element scheme is investigated. To improve the convergence rate in time, the weighted and shifted Grünwald difference method is used. By this method, another finite element scheme for DOFDEs is obtained, and the corresponding stability and convergence are considered. And then, as a supplement, a higher order finite difference scheme of the Caputo fractional derivative is developed. By this scheme, an approach is suggested to improve the time convergence rate of our methods. Finally, some numerical examples are given for verification of our theoretical analysis.