An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

     

Steklov eigenvalue problem and its applications An accurate a posteriori error estimator for the

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine meshes and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.     

Adaptive FEM and a posteriori error control for stationary coupled bulk-surface PDEs

We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. The numerical error of the finite element solution can be controlled by a residual a posteriori error estimator which takes into account the approximation errors due to the discretisation in space as well as the polyhedral approximation of the surface. The estimators naturally lead to refinement indicators for an adaptive algorithm to control the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the adaptive algorithm. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

AN ADAPTIVE INVERSE ITERATION FEM FOR THE INHOMOGENEOUS DIELECTRIC WAVEGUIDES

We introduce an adaptive finite element method for computing electromagnetic guided waves in a closed, inhomogeneous, pillared three-dimensional waveguide at a given frequency based on the inverse iteration method. The problem is formulated as a generalized eigenvalue problems. By modifying the exact inverse iteration algorithm for the eigenvalue problem, we design a new adaptive inverse iteration finite element algorithm. Adaptive finite element methods based on a posteriori error estimate are known to be successful in resolving singularities of eigenfunctions which deteriorate the finite element convergence. We construct a posteriori error estimator for the electromagnetic guided waves problem. Numerical results are reported to illustrate the quasi-optimal performance of our adaptive inverse iteration finite element method.     

A posteriori error estimates for mixed finite element solutions of convex optimal control problems

In this paper, we present an a posteriori error analysis for mixed finite element approximation of convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates can be used to construct reliable adaptive mixed finite elements for the control problems.     

稳态Poisson-Nernst-Planck方程的残量型后验误差估计

针对稳态的Poisson-Nernst-Planck方程研究了一种残量型的后验误差估计子, 对方程的两个解-浓度和电势, 都分别给出了上界和下界估计. 数值实验表明, 基于这种后验误差估计子构造的自适应有限元算法对于稳态的Poisson-Nernst-Planck方程是有效的.     

Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.     

A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

In this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness of the algorithm.     

An adaptive stabilized finite element method for the generalized Stokes problem

In this work we present an adaptive strategy (based on an a posteriori error estimator) for a stabilized finite element method for the Stokes problem, with and without a reaction term. The hierarchical type estimator is based on the solution of local problems posed on appropriate finite dimensional spaces of bubble-like functions. An equivalence result between the norm of the finite element error and the estimator is given, where the dependence of the constants on the physics of the problem is explicited. Several numerical results confirming both the theoretical results and the good performance of the estimator are given.     

An adaptive time stepping method with efficient error control for second-order evolution problems

This work is concerned with time stepping fnite element methods for abstract second order evolution problems.We derive optimal order a posteriori error estimates and a posteriori nodal superconvergence error estimates using the energy approach and the duality argument.With the help of the a posteriori error estimator developed in this work,we will further propose an adaptive time stepping strategy.A number of numerical experiments are performed to illustrate the reliability and efciency of the a posteriori error estimates and to assess the efectiveness of the proposed adaptive time stepping method.     

A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.     

An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem

Based on a posteriori error estimator with hierarchical bases, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the elliptic problem with mixed boundary conditions. For the posteriori error estimator, we are only required to solve a linear algebraic system with diagonal entries corresponding to the degree of freedoms, which significantly reduces the computational cost. The upper and lower bounds of the error estimator are shown to addresses the reliability and efficiency of the adaptive approach. Numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed method.     

A posteriori error estimates for optimal control problems constrained by convection-diffusion equations

We propose a characteristic finite element discretization of evolutionary type convection-diffusion optimal control problems. Nondivergence-free velocity fields and bilateral inequality control constraints are handled. Then some residual type a posteriori error estimates are analyzed for the approximations of the control, the state, and the adjoint state. Based on the derived error estimators, we use them as error indicators in developing efficient multi-set adaptive meshes characteristic finite element algorithm for such optimal control problems. Finally, one numerical example is given to check the feasibility and validity of multi-set adaptive meshes refinements.     

A posteriori error estimates for optimal control problems governed by parabolic equations

Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem. Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002 RID="*" ID="*" Supported by EPSRC research grant GR/R31980     

Convergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems

Recently an adaptive nonconforming finite element method (ANFEM) has been developed by Carstensen and Hoppe (in Numer Math 103:251–266, 2006). In this paper, we extend the result to some nonsymmetric and indefinite problems. The main tools in our analysis are a posteriori error estimators and a quasi-orthogonality property. In this case, we need to overcome two main difficulties: one stems from the nonconformity of the finite element space, the other is how to handle the effect of a nonsymmetric and indefinite bilinear form. An appropriate adaptive nonconforming finite element method featuring a marking strategy based on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order Crouzeix–Raviart element. It is shown that the ANFEM is a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops. Moreover, quasi-optimality in the sense of quasi-optimal algorithmic complexity can be shown for the ANFEM. The results of numerical experiments confirm the theoretical findings.     

An a posteriori error estimator for anisotropic refinement

Summary. Besides an algorithm for local refinement, an a posteriori error estimator is the basic tool of every adaptive finite element method. Using information generated by such an error estimator the refinement of the grid is controlled. For 2nd order elliptic problems we present an error estimator for anisotropically refined grids, like -d cuboidal and 3-d prismatic grids, that gives correct information about the size of the error; additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way. Numerical examples are presented for 2-d rectangular and 3-d prismatic grids. Received March 15, 1994 / Revised version received June 3, 1994     

求解椭圆边值问题惩罚形式的间断有限元方法

本文提出了一个新的求解二阶椭圆边值问题的惩罚形式间断有限元方法并给出了稳定性和收敛性分析. 特别地,本文建立了间断有限元解的基于余量的后验误差估计,给出了求解间断有限元方程的自适应算法.       

An adaptive finite element method for the sparse optimal control of fractional diffusion

We propose and analyze an a posteriori error estimator for a partial differential equation (PDE)-constrained optimization problem involving a nondifferentiable cost functional, fractional diffusion, and control-constraints. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly PDE and propose an equivalent optimal control problem with a local state equation. For such an equivalent problem, we design an a posteriori error estimator which can be defined as the sum of four contributions: two contributions related to the approximation of the state and adjoint equations and two contributions that account for the discretization of the control variable and its associated subgradient. The contributions related to the discretization of the state and adjoint equations rely on anisotropic error estimators in weighted Sobolev spaces. We prove that the proposed a posteriori error estimator is locally efficient and, under suitable assumptions, reliable. We design an adaptive scheme that yields, for the examples that we perform, optimal experimental rates of convergence.     

A posteriori error estimates of finite element methods for nonlinear quadratic boundary optimal control problem

This paper is aimed at studying finite element discretization for a class of quadratic boundary optimal control problems governed by nonlinear elliptic equations. We derive a posteriori error estimates for the coupled state and control approximation. Such estimates can be used to construct a reliable adaptive finite element approximation for the boundary optimal control problem. Finally, we present a numerical example to confirm our theoretical results.     

Reliable a posteriori error estimation for state-constrained optimal control

We derive a reliable a posteriori error estimator for a state-constrained elliptic optimal control problem taking into account both regularisation and discretisation. The estimator is applicable to finite element discretisations of the problem with both discretised and non-discretised control. The performance of our estimator is illustrated by several numerical examples for which we also introduce an adaptation strategy for the regularisation parameter.