Numerical comparison of preconditionings for large sparse finite element problems

A numerical study of the efficiency of the modified conjugate gradients (MCG) is performed using different preconditioning schemes. The MCG behavior is evaluated in connection with the solution of large linear sets of symmetric positive definite (p.d.) equations, arising from the finite element (f.e.) integration of partial differential equations of parabolic and elliptic type and the analysis of the leftmost eingenspectrum of the corresponding matrices. A simple incomplete Cholesky factorization ICCG(O) having the same sparsity pattern as the original problem is compared with a more complex technique ICAJ (Ψ) where the triangular factor is allowed to progressively fill in depending on a rejection parameter Ψ. The performance of the preconditioning algorithms is explored on finite element equations whose size N ranges between 150 and 2300. The results show that an optimal Ψ_opt may be found which minimizes the overall CPU time for the solution of both the linear system and the eigenproblem. The comparison indicates that ICAJ (Ψ_opt) is not significantly more efficient than ICCG(O), which therefore appears to be a simple, robust, and reliable method for the preconditioning of large sparse finite element models.     

A note on the normwise perturbation theory for the regular generalized eigenproblem

In this paper, we present a normwise perturbation theory for the regular generalized eigenproblem Ax = λBx, when λ is a semi-simple and finite eigenvalue, which departs from the classical analysis with the chordal norm [9]. A backward error and a condition number are derived for a choice of flexible measure to represent independent perturbations in the matrices A and B. The concept of optimal backward error associated with an eigenvalue only is also discussed. © 1998 John Wiley & Sons, Ltd.     

Superconvergence of finite volume element method for a nonlinear elliptic problem

In this article, we consider the finite volume element method for the second‐order nonlinear elliptic problem and obtain the H¹ and W^1,∞ superconvergence estimates between the solution of the finite volume element method and that of the finite element method, which reveal that the finite volume element method is in close relationship with the finite element method. With these superconvergence estimates, we establish the L^p and W^1,p (2 < p ≤ ∞) error estimates for the finite volume element method for the second‐order nonlinear elliptic problem. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

An Optimal L∞–error Estimate for an Approximation of a Parabolic Variational Inequality

In this article, an optimal error estimate for parabolic variational inequalities is studied. Existence and uniqueness of the solution is provided by the introduction of a constructive algorithm. An optimally L^∞-asymptotic behavior in uniform norm is proved using the semi-implicit time scheme combined with the finite element spatial approximation. The approach is based on the concept of subsolutions.     

On the Convergence of Finite Element Method for Second Order Elliptic Interface Problems

The purpose of this paper is to study the convergence of finite element approximation to the exact solution of general self-adjoint elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it is difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math. 1998; 79:175–202]. In this paper, an isoparametric type of discretization is used to prove optimal order error estimates in L ² and H ¹ norms when the global regularity of the solution is low. The interface is assumed to be of arbitrary shape and is smooth for our purpose. Further, for the purpose of numerical computations, we discuss the effect of numerical quadrature on finite element solution, and the related optimal order estimates are also established.     

Fully discrete T‐ψ finite element method to solve a nonlinear induction hardening problem

We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T ‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T ‐ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.     

Development of a P2 element with optimal L2 convergence for biharmonic equation

It is well known that convergence rate of finite element approximation is suboptimal in the L² norm for solving biharmonic equations when P₂ or Q₂ element is used. The goal of this paper is to derive a weak Galerkin (WG) P₂ element with the L² optimal convergence rate by assuming the exact solution sufficiently smooth. In addition, our new WG finite element method can be applied to general mesh such as hybrid mesh, polygonal mesh or mesh with hanging node. The numerical experiments have been conducted on different meshes including hybrid meshes with mixed of pentagon and rectangle and mixed of hexagon and triangle.     

Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids

In this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s ∈ [1,2] order O(h ^s )-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H ^1+s (Ω). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h ^3/2?ε) with ε > 0 if u ∈ H ³(Ω). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s ∈ {1,2}, the given error estimates are strictly local.     

A posteriori error estimate for the Stokes–Darcy system

We consider the a posteriori error estimates for finite element approximations of the Stokes–Darcy system. The finite element spaces adopted are the Hood–Taylor element for the velocity and the pressure in fluid region and conforming piecewise quadratic element for the pressure in porous media region. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. Copyright © 2011 John Wiley & Sons, Ltd.     

A priori error analysis of the mortar finite element method for variational inequalities

The mortar finite element method is a special domain decomposition method, which can handle the situation where meshes on different subdomains need not align across the interface. In this article, we will apply the mortar element method to general variational inequalities of free boundary type, such as free seepage flow, which may show different behaviors in different regions. We prove that if the solution of the original variational inequality belongs to H²(D), then the mortar element solution can achieve the same order error estimate as the conforming P₁ finite element solution. Application of the mortar element method to a free surface seepage problem and an obstacle problem verifies not only its convergence property but also its great computational efficiency. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A novel modeling method for in-plane eigenproblem estimation of the cable-stayed bridges

A novel modeling method is proposed and used to overcome the in-plane eigenproblem of cable-stayed bridges (CSBs). The modeling method is divided into three steps. Firstly, according to the multi-tower configuration and mechanical characteristics of the CSB, the entire CSB is divided into multiple substructures, namely, a single-tower CSB. Secondly, the substructure is treated by a novel method to make it be a chain-like system and the dynamic triple-beam model with discrete springs is developed. The eigenproblem of the substructure is solved by transfer matrix method (TMM). Then, the entire multi-beam model with discrete springs of the CSB can be obtained by assembling all substructures with consideration of the matching conditions, and the eigenvalue and eigenvector of the CSB can be solved by TMM considering the boundary conditions. The above method is demonstrated by four examples of two kinds of CSBs, namely, semi-floating and rigid-frame CSBs, which are also verified by finite element method (FEM). The proposed model and solution method can be used to calculate the natural frequency and mode shape and evaluate the vertical bending stiffness of the CSB.     

Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P₀, P₀, RT₀) . The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.     

A posteriori error analysis for linearization of nonlinear elliptic problems and their discretizations

The paper is devoted to a posteriori quantitative analysis for errors caused by linearization of non-linear elliptic boundary value problems and their finite element realizations. We employ duality theory in convex analysis to derive computable bounds on the difference between the solution of a non-linear problem and the solution of the linearized problem, by using the solution of the linearized problem only. We also derive computable bounds on differences between finite element solutions of the nonlinear problem and finite element solutions of the linearized problem, by using finite element solutions of the linearized problem only. Numerical experiments show that our a posteriori error bounds are efficient.     

Error analysis of a weighted least-squares finite element method for 2-D incompressible flows in velocity–stress–pressure formulation

In this paper we are concerned with a weighted least-squares finite element method for approximating the solution of boundary value problems for 2-D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second-order problem as a Petrovski-type first-order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert-type boundary conditions. A weighted least-squares finite element method is proposed for solving this extended first-order problem. The finite element approximations are defined to be the minimizers of a weighted least-squares functional over the finite element subspaces of the H¹ product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L²-norm and in the H¹-norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Pointwise superconvergence of the gradient for the linear tetrahedral element

We consider the finite element approximation to the solution of a self-adjoint, second-order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise linear tetrahedral elements. Although the resulting approximation to the gradient is optimal for functions from the approximating space, it is, however, only O(h). We show how this can be improved by the recovery, from the finite element solution, of an approximation to the gradient, which is pointwise of a higher order of accuracy than that of the gradient of the finite element approximation. This approximation, termed a recovered gradient function, is, thus, superconvergent. The major task of our analysis is the establishing of an (almost) constant bound on the W seminorm of the finite element approximation to a smoothed derivative Green's function. © 1994 John Wiley & Sons, Inc.     

Two-sided Grassmann–Rayleigh quotient iteration

The two-sided Rayleigh quotient iteration proposed by Ostrowski computes a pair of corresponding left–right eigenvectors of a matrix C. We propose a Grassmannian version of this iteration, i.e., its iterates are pairs of p-dimensional subspaces instead of one-dimensional subspaces in the classical case. The new iteration generically converges locally cubically to the pairs of left–right p-dimensional invariant subspaces of C. Moreover, Grassmannian versions of the Rayleigh quotient iteration are given for the generalized Hermitian eigenproblem, the Hamiltonian eigenproblem and the skew-Hamiltonian eigenproblem.     

A posteriori finite element error estimators for indefinite elliptic boundary value problems ∗

A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach.     

Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L₂ and H 1 . The convergence rate in the L_∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P₁ ? P₀ element, the H¹ error estimate of optimal order for finite volume solution (u_h,p_h) is analyzed. And, a uniform H¹ error estimate of optimal order for finite volume solution (u_h, p_h) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers

Considering a singularly perturbed problem with exponential and characteristic layers, we show convergence for non-standard higher-order finite elements using the streamline diffusion finite element method (SDFEM). Moreover, for the standard higher-order space Q_p\mathcal {Q}_{p} supercloseness of the numerical solution w.r.t. an interpolation of the exact solution in the streamline diffusion norm of order p+1/2 is proved.