Boundary element approximation for Maxwell's eigenvalue problem

We introduce a new method for computing eigenvalues of the Maxwell operator with boundary finite elements. On bounded domains with piecewise constant material coefficients, the Maxwell solution for fixed wave number can be represented by boundary integrals, which allows to reduce the eigenvalue problem to a nonlinear problem for determining the wave number along with boundary and interface traces. A Galerkin discretization yields a smooth nonlinear matrix eigenvalue problem that is solved by Newton's method or, alternatively, the contour integral method. Several numerical results including an application to the band structure computation of a photonic crystal illustrate the efficiency of this approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite element approximation of the Neumann eigenvalue problem in domains with multiple cracks

^** Email: belhach{at}poncelet.univ-metz.fr^*** Email: bucur{at}math.univ-metz.fr^**** Email: jmse{at}math.univ-metz.fr We study the Neumann–Laplacian eigenvalue problem in domainswith multiple cracks. We derive a mixed variational formulationwhich holds on the whole geometric domain (including the cracks)and implements efficient finite-element discretizations forthe computation of eigenvalues. Optimal error estimates aregiven and several numerical examples are presented, confirmingthe efficiency of the method. As applications, we numericallyinvestigate the behaviour of the low eigenvalues in domainswith a large number of cracks.     

Finite element approximation with numerical integration for differential eigenvalue problems

Error estimates of the finite element method with numerical integration for differential eigenvalue problems are presented. More specifically, refined results on the eigenvalue dependence for the eigenvalue and eigenfunction errors are proved. The theoretical results are illustrated by numerical experiments for a model problem.     

Finite element approximation to nonlinear coupled thermal problem

A nonlinear coupled elliptic system modelling a large class of engineering problems was discussed in [A.F.D. Loula, J. Zhu, Finite element analysis of a coupled nonlinear system, Comp. Appl. Math. 20 (3) (2001) 321–339; J. Zhu, A.F.D. Loula, Mixed finite element analysis of a thermally nonlinear coupled problem, Numer. Methods Partial Differential Equations 22 (1) (2006) 180–196]. The convergence analysis of iterative finite element approximation to the solution was done under an assumption of ‘small’ solution or source data which guarantees the uniqueness of the nonlinear coupled system. Generally, a nonlinear system may have multiple solutions. In this work, the regularity of the weak solutions is further studied. The nonlinear finite element approximations to the nonsingular solutions are then proposed and analyzed. Finally, the optimal order error estimates in H¹$H^1$-norm and L²$L^2$-norm as well as in W^1,p$W^{1,p}$-norm and L^p$L^p$-norm are obtained.     

Finite element approximation of the volume-matching problem

Summary Optimal orderH ¹ andL error bounds are obtained for a continuous piecewise linear finite element approximation of the volume matching problem. This problem consists of minimising |v| _1, ² overvH ¹() subject to the inequality constraintv0 and a number of linear equality constraints. The presence of the equality constraints leads to Lagrange multipliers, which in turn lead to complications with the standard error analysis for variational inequalities. Finally we consider an algorithm for solving the resulting algebraic problem.Supported by a SERC research studentship     

混合网格下的有限元特征值重构

利用有限元后处理技术在混合网格上重构了线性有限元解,使其梯度具有超收敛性,在此基础上利用Rayleigh商重构特征值,获得了线元特征值的四阶超收敛结果.     

Finite element approximation of nonlocal parabolic problem

In this article, we present a finite element scheme combined with backward Euler method to solve a nonlocal parabolic problem. An important issue in the numerical solution of nonlocal problems while using Newton's method is related to its structure. In fact differently from the local case where the Jacobian matrix is sparse and banded, in the nonlocal case the Jacobian matrix is dense and computations are much more onerous compared to that for differential equations. In order to avoid this difficulty, we use the technique given by Gudi (SIAM J Numer Anal 50 (2012), 657–668) for elliptic nonlocal problem of Kirchhoff type. We discuss the well‐posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for semidiscrete and fully discrete formulations in L² and H¹ norms. Results based on the usual finite element method are provided to confirm the theoretical estimates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 786–813, 2017     

Finite element approximation to a contact problem in linear thermoelasticity

A finite element approximation to the solution of a one-dimensional linear thermoelastic problem with unilateral contact of the Signorini type and heat flux is proposed. An error bound is derived and some numerical experiments are performed.

     

Finite element approximation to a contact problem for a nonlinear thermoviscoelastic beam

In this paper we propose and analyze a finite element method to the solution of a quasi-static contact problem between a nonlinear beam and a rigid obstacle. Error estimates and energy decay are obtained and some numerical simulations described.     

Finite element approximation of a free boundary plasma problem

In this article, we study a finite element approximation for a model free boundary plasma problem. Using a mixed approach (which resembles an optimal control problem with control constraints), we formulate a weak formulation and study the existence and uniqueness of a solution to the continuous model problem. Using the same setting, we formulate and analyze the discrete problem. We derive optimal order energy norm a priori error estimates proving the convergence of the method. Further, we derive a reliable and efficient a posteriori error estimator for the adaptive mesh refinement algorithm. Finally, we illustrate the theoretical results by some numerical examples.     

Finite element approximation

We study a discretization procedure, using finite elements, for a class of non-isotropic free-discontinuity problems based on the non-local approximation proposed in [18]     

Finite element analysis of a coupling eigenvalue problem on overlapping domains

In this paper, we consider a nonstandard elliptic eigenvalue problem on a rectangular domain, consisting of two overlapping rectangles, where the interaction between the subdomains is expressed through an integral coupling condition on their intersection. For this problem we set up finite element (FE) approximations, without and with numerical quadrature. The involved error analysis is affected by the nonlocal coupling condition, which requires the introduction and error estimation of a suitably modified vector Lagrange interpolant on the overall FE mesh. As a consequence, the resulting error estimates are sub-optimal, as compared to the ones established, e.g., in Vanmaele and van Keer (RAIRO – Math. Mod. Num. Anal 29(3) (1995) 339–365) for classical eigenvalue problems with local boundary or transition conditions.     

Finite element approximation of the elasticity spectral problem on curved domains

We analyze the finite element approximation of the spectral problem for the linear elasticity equation with mixed boundary conditions on a curved non-convex domain. In the framework of the abstract spectral approximation theory, we obtain optimal order error estimates for the approximation of eigenvalues and eigenvectors. Two kinds of problems are considered: the discrete domain does not coincide with the real one and mixed boundary conditions are imposed. Some numerical results are presented.     

Finite element approximation of a shape optimization problem for von karman system

An optimal shape design problem of an elastic body described by a system of two nonlinear elliptic equations is considered. The problem is to find the boundary of the domain occupied by the body in such a way that the stiffnes of the system in the equilibrium state is minimized.

It is assumed that the volume of the body is constant. Moreover, the function describing the boundary of the domain and its gradient are bounded.     

Finite element approximation of the Dirichlet problem using the boundary penalty method

Numerische Mathematik - This paper considers a finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a region Ω ⊂ ℝn (n=2...     

Steklov特征值问题的一种有效的Legendre-Galerkin谱逼近

本文给出Steklov特征值问题基于Legendre-Galerkin逼近的一种有效的谱方法.首先利用Legendre多项式构造了一组适当的基函数使得离散变分形式中的矩阵是稀疏的,然后推导了2维及3维情形下离散变分形式基于张量积的矩阵形式,由此可以快速地计算出离散的特征值和特征向量.文章还给出了误差分析和数值试验,数值结果表明本文提出的方法是稳定和有效的.