A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk, and Winther includes a well-developed theory of finite element methods for Hodge–Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of a residual type for Arnold–Falk–Winther mixed finite element methods for Hodge–de Rham–Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a unified treatment of the various Hodge–Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge–Laplacian.     

Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation

This paper discusses convergence and complexity of arbitrary,but fixed,order adaptive mixed element methods for the Poisson equation in two and three dimensions.The two main ingredients in the analysis,namely the quasi-orthogonality and the discrete reliability,are achieved by use of a discrete Helmholtz decomposition and a discrete inf-sup condition.The adaptive algorithms are shown to be contractive for the sum of the error of flux in L2-norm and the scaled error estimator after each step of mesh refinement and to be quasi-optimal with respect to the number of elements of underlying partitions.The methods do not require a separate treatment for the data oscillation.     

Adaptive finite elements for exterior domain problems

Summary. We present an adaptive finite element method for solving elliptic problems in exterior domains, that is for problems in the exterior of a bounded closed domain in , . We describe a procedure to generate a sequence of bounded computational domains , , more precisely, a sequence of successively finer and larger grids, until the desired accuracy of the solution is reached. To this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm by means of a residual based error estimator. Furthermore we prove convergence of the adaptive algorithm. Numerical examples show the optimal order of convergence. Received July 8, 1997 /Revised version received October 23, 1997     

Mixed finite element methods for linear elasticity with weakly imposed symmetry

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

     

SUPERCONVERGENCE ANALYSIS OF THE POLYNOMIAL PRESERVING RECOVERY FOR ELLIPTIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS

We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery(PPR)for Robin boundary elliptic problems on triangulartions.First,we improve the convergence rate between the finite element solution and the linear interpolation under the H1-norm by introducing a class of meshes satisfying the Condition(α,σ,μ).Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator.Finally,numerical tests are provided to verify the theoretical findings.     

The Picard–Lefschetz formula for p-adic cohomology

In this paper, we discuss a p-adic analogue of the Picard–Lefschetz formula. For a family with ordinary double points over a complete discrete valuation ring of mixed characteristic (0,p), we construct vanishing cycle modules which measure the difference between the rigid cohomology groups of the special fiber and the de Rham cohomology groups of the generic fiber. Furthermore, the monodromy operators on the de Rham cohomology groups of the generic fiber are described by the canonical generators of the vanishing cycle modules in the same way as in the case of the ℓ-adic (or classical) Picard–Lefschetz formula. For the construction and the proof, we use the logarithmic de Rham–Witt complexes and those weight filtrations investigated by Mokrane (Duke Math. J. 72(2):301–337, 1993).      

A POSTERIORI ERROR ESTIMATES IN ADINI FINITE ELEMENT FOR EIGENVALUE PROBLEMS

In this paper, we discuss a posteriori error estimates of the eigenvalue $\lambda_h$ given by Adini nonconforming finite element. We give an asymptotically exact error estimator of the $\lambda_h$. We prove that the order of convergence of the $\lambda_h$ is just 2 and the $\lambda_h$ converge from below for sufficiently small $h$.     

Convergence of adaptive FEM for some elliptic obstacle problem

In this work, we treat the convergence of adaptive lowest-order FEM for some elliptic obstacle problem with affine obstacle. For error estimation, we use a residual error estimator from [D. Braess, C. Carstensen, and R. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007), pp. 455–471]. We extend recent ideas from [J. Cascon, C. Kreuzer, R. Nochetto, and K. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), pp. 2524–2550] for the unrestricted variational problem to overcome the lack of Galerkin orthogonality. The main result states that an appropriately weighted sum of energy error, edge residuals and data oscillations satisfies a contraction property within each step of the adaptive feedback loop. This result is superior to a prior result from Braess et al. (2007) in two ways: first, it is unnecessary to control the decay of the data oscillations explicitly; second, our analysis avoids the use of some discrete local efficiency estimate so that the local mesh-refinement is fairly arbitrary.     

Error reduction and convergence for an adaptive mixed finite element method

An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart-Thomas finite element method with a reduction factor uniformly for the norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does not rely on duality or on regularity.

     

Convergence analysis of an adaptive edge element method for Maxwell's equations

An efficient and reliable a posteriori error estimate is derived for solving three-dimensional static Maxwell's equations by using the edge elements of first family. Based on the a posteriori error estimates, an adaptive finite element method is constructed and its convergence is established. Compared with the existing results, an important advantage of the new theory lies in its feature that the usual marking of elements based on the oscillation is not needed in our adaptive algorithm, while the linear convergence of the algorithm can be still demonstrated in terms of the reduction of the energy-norm error and the oscillation. Numerical examples are provided which demonstrate the effectiveness and robustness of the adaptive methods.     

An optimally convergent adaptive mixed finite element method

We prove convergence and optimal complexity of an adaptive mixed finite element algorithm, based on the lowest-order Raviart–Thomas finite element space. In each step of the algorithm, the local refinement is either performed using simple edge residuals or a data oscillation term, depending on an adaptive marking strategy. The inexact solution of the discrete system is controlled by an adaptive stopping criterion related to the estimator.     

一类Kirchhoff 板弯问题自适应高阶混合元方法及理论分析

本文针对Kirchhoff 板弯问题提出了一个基于高阶Hellan-Herrmann-Johnson (简记为H-H-J)方法的自适应有限元算法, 分析了它的收敛性和计算复杂度. 证明了算法在执行过程中, 相应的拟能量误差会以几何级数单调衰减, 从而得到收敛性. 利用此单调下降性质, 进一步给出了算法的计算复杂度. 推导过程中的一个关键步骤是建立基于平衡方程的单元误差表示(error indicator) 与平衡方程右端载荷震荡项(data oscillation) 的局部等价关系.     

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 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.     

Quasi-Optimal Convergence Rate of an AFEM for Quasi-Linear Problems of Monotone Type

We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for a class of nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and Dörfler's strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, analogous to the one used by Diening and Kreuzer (2008) and equivalent to the total error defined by Cascón et. al. (2008). This contraction implies linear convergence of the discrete solutions to the exact solution in the usual $H^1$ Sobolev norm. Secondly, we use this contraction to derive the optimal complexity of the AFEM. The results are based on ideas from Diening and Kreuzer and extend the theory from Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.     

Convergence of an adaptive mixed finite element method for convection-diffusion-reaction equations

We prove the convergence of an adaptive mixed finite element method (AMFEM) for (nonsymmetric) convection-diffusion-reaction equations. The convergence result holds for the cases where convection or reaction is not present in convection- or reaction-dominated problems. A novel technique of analysis is developed by using the superconvergence of the scalar displacement variable instead of the quasi-orthogonality for the stress and displacement variables, and without marking the oscillation dependent on discrete solutions and data. We show that AMFEM is a contraction of the error of the stress and displacement variables plus some quantity. Numerical experiments confirm the theoretical results.     

New adaptive mixed finite element method (AMFEM)

The need to develop reliable and efficient adaptive algorithms using mixed finite element methods arises from various applications in fluid dynamics and computational continuum mechanics. In order to save degrees of freedom, not all but just some selected set of finite element domains are refined and hence the fundamental question of convergence requires a new mathematical argument as well as the question of optimality. We will present a new adaptive algorithm for mixed finite element methods to solve the model Poisson problem, for which optimal convergence can be proved. The a posteriori error control of mixed finite element methods dates back to Alonso (1996) Error estimators for a mixed method. and Carstensen (1997) A posteriori error estimate for the mixed finite element method. The error reduction and convergence for adaptive mixed finite element methods has already been proven by Carstensen and Hoppe (2006) Error Reduction and Convergence for an Adaptive Mixed Finite Element Method, Convergence analysis of an adaptive nonconforming finite element methods. Recently, Chen, Holst and Xu (2008) Convergence and Optimality of Adaptive Mixed Finite Element Methods. presented convergence and optimality for adaptive mixed finite element methods following arguments of Rob Stevenson for the conforming finite element method. Their algorithm reduces oscillations, before applying and a standard adaptive algorithm based on usual error estimation. The proposed algorithm does this in a natural way, by switching between the reduction of either the estimated error or oscillations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A posteriori error estimator for expanded mixed hybrid methods

In this article, we construct an a posteriori error estimator for expanded mixed hybrid finite‐element methods for second‐order elliptic problems. An a posteriori error analysis yields reliable and efficient estimate based on residuals. Several numerical examples are presented to show the effectivity of our error indicators. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 330–349, 2007     

Local problems on stars: A posteriori error estimators, convergence, and performance

A new computable a posteriori error estimator is introduced, which relies on the solution of small discrete problems on stars. It exhibits built-in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption. A simple adaptive strategy is designed, which simultaneously reduces error and data oscillation, and is shown to converge without mesh pre-adaptation nor explicit knowledge of constants. Numerical experiments reveal a competitive performance, show extremely good effectivity indices, and yield quasi-optimal meshes.

     

Adaptive control of local errors for elliptic problems using weighted Sobolev norms

We develop an a posteriori error estimator which focuses on the local H¹ error on a region of interest. The estimator bounds a weighted Sobolev norm of the error and is efficient up to oscillation terms. The new idea is very simple and applies to a large class of problems. An adaptive method guided by this estimator is implemented and compared to other local estimators, showing an excellent performance. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1266–1282, 2017