New Cartesian grid methods for interface problems using the finite element formulation

Summary New finite element methods based on Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients. The triangulations in these methods do not need to fit the interfaces. The basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately. Both non-conforming and conforming finite element spaces are considered. Corresponding interpolation functions are proved to be second order accurate in the maximum norm. The conforming finite element method has been shown to be convergent. With Cartesian triangulations, these new methods can be used as finite difference methods. Numerical examples are provided to support the methods and the theoretical analysis. Mathematics Subject Classification (2000):65L10, 65L60, 65L70In this research, Zhilin Li is supported in part by USA ARO grants, 39676-MA and 43751-MA, USA NSF grants DMS-0073403 and DMS-0201094; USA North Carolina State University FR&PD grant; Tao Lin is supported in part a USA NSF grant DMS-97-04621. Special thanks to Thomas Hou for his participation and contribution to this project. We are also grateful to R. LeVeque, K. Bube, and T. Chan for useful discussions.     

A posteriori error analysis for elliptic pdes on domains with complicated structures

Summary. The discretisation of boundary value problems on complicated domains cannot resolve all geometric details such as small holes or pores. The model problem of this paper consists of a triangulated polygonal domain with holes of a size of the mesh-width at most and mixed boundary conditions for the Poisson equation. Reliable and efficient a posteriori error estimates are presented for a fully numerical discretisation with conforming piecewise affine finite elements. Emphasis is on technical difficulties with the numerical approximation of the domain and their influence on the constants in the reliability and efficiency estimates. Mathematics Subject Classification (2000):65N30, 65R20, 73C50Received: 28, June 2001     

Almost optimal interior penalty discontinuous approximations of symmetric elliptic problems on non-matching grids

Summary. We consider an interior penalty discontinuous approximation for symmetric elliptic problems of second order on non-matching grids in this paper. The main result is an almost optimal error estimate for the interior penalty approximation of the original problem based on partitioning of the domain into a finite number of subdomains. Further, an error analysis for the finite element approximation of the penalty formulation is given. Finally, numerical experiments on a series of model second order problems are presented. Mathematics Subject Classification (2000):65F10, 65N20, 65N30The work of the first and the second authors has been partially supported by the National Science Foundation under Grant DMS-9973328. The work of the last author was performed under the auspices of the U. S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract W-7405-Eng-48.     

Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis

Summary. The numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given Dirichlet data u_D by functions u_D,h in the trace space of a finite element space on _D. In this paper, quantitative a priori and a posteriori estimates are presented for two choices of u_D,h, namely the nodal interpolation and the orthogonal projection in L²(_D) onto the trace space. Two corresponding extension operators allow for an estimate of the boundary data approximation in global H¹ and L² a priori and a posteriori error estimates. The results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation.Mathematics Subject Classification (1991): 65N30, 65R20, 73C50This work was initiated while C. Carstensen was visiting the Max Planck Institute for Mathematics in the Sciences, Leipzig. S. Bartels acknowledges support by the German Research Foundation (DFG) within the Graduiertenkolleg Effiziente Algorithmen und Mehrskalenmethoden and the priority program Analysis, Modeling, and Simulation of Multiscale Problems. G. Dolzmann gratefully acknowledges partial support by the Max Planck Society and by the NSF through grant DMS0104118.     

An a priori error estimate for finite element discretizations in nonlinear elasticity for polyconvex materials under small loads

Summary. We prove optimal a priori error estimates in W^1,p for finite element minimizers of polyconvex energy functionals with small applied loads. The proof relies on a quantitative version of Zhangs stability estimate (K. Zhang, Arch. Rat. Mech. Anal. 114 (1991), 95-117).Mathematics Subject Classification (2000):35G25, 73G25, 65N12This project was initiated while GD visited the California Institute of Technology in Pasadena supported by AFOSR/MURI (F 49602-98-1-0433) in 98-99. A short visit of CC to the California Institute of Technology the was supported by the Powell Foundation.14 September 1999Revised: 20 April 2000     

Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains

Summary. In this paper, we study the spectral properties of Dirichlet problems for second order elliptic equation with rapidly oscillating coefficients in a perforated domain. The asymptotic expansions of eigenvalues and eigenfunctions for this kind of problem are obtained, and the multiscale finite element algorithms and numerical results are proposed. Mathematics Subject Classification (2000):65F10, 35P15This work is Supported by National Natural Science Foundation of China (grant # 19932030) and Special Funds for Major State Basic Research Projects (grant # TG2000067102)     

Analysis of a cell boundary element method

The CBEM (cell boundary element method) was proposed as a numerical method for second-order elliptic problems by the first author in the earlier paper [10]. In this paper we prove a quasi-optimal order of convergence of the method, O(h^1–) for >0 in H¹-norm for the triangular mesh; also a stability result is obtained. We provide numerical examples and it is observed that the method conserves flux exactly when a certain condition on meshes is satisfied. This work was supported by KOSEF 2000-1-10300-001-5.AMS subject classification 65N30, 65N38, 65N50     

Convergence Rates for an Adaptive Dual Weighted Residual Finite Element Algorithm

Basic convergence rates are established for an adaptive algorithm based on the dual weighted residual error representation,

Finite Order Rank-One Convex Envelopes and Computation of Microstructures with Laminates in Laminates

Finite order rank-one convex envelopes are introduced and it is shown that the i-th order laminated microstructures, or laminates in laminates, can be solved by any of the k-th order rank-one convex envelopes with k i. It is also shown that in finite element approximations of microstructures, replacing the non-quasiconvex potential energy density by its k-th order rank-one convex envelope, one can generally obtain sharper numerical results. Especially, for crystalline microstructures with laminates in laminates of order no greater than k + 1, numerical results with up to the computer precision can be obtained. Numerical examples on the first and second order rank-one convex envelopes for the Ericksen-James two-dimensional model for elastic crystals are given. A numerical example on finite element approximations of a crystalline microstructure by using the first order rank-one convex envelope and the periodic relaxation method is also presented. The methods turn out to be very successful for microstructures with laminates in laminates.     

Lax-type isospectral deformations on nilmanifolds

A one-parameter familyg(t) (0 t T) of Riemannian metric on a compact manifold is called an isospectral deformation of a metricg(0) if the Laplace-Beltrami operators associated to the metricsg(t) have the same spectra. Examples of non-trivial isospectral deformations were constructed on solvmanifolds for the first time by C.S. Gordon and E. Wilson on the basis of Kirillov theory. This paper considers the isospectral deformations on nilmanifolds from the dynamical point of view. First, we see for certain isospectral deformations that the associated Hamiltonian systems of geodesic flows are decomposed into a collection of reduced systems which are left invariant as Hamiltonian systems under the deformations. This fact is formulated by the classical Lax equations. Next, by using a quantization procedure, we attempt to obtain Lax equations for the reduced Laplacians from the classical Lax equations. As a result, we show that certain isospectral deformations by Gordon-Wilson are represented by the Lax equations.     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

High accuracy analysis of the characteristic‐nonconforming FEM for a convection‐dominated transport problem

A low order characteristic‐nonconforming finite element method is proposed for solving a two‐dimensional convection‐dominated transport problem. On the basis of the distinguish property of element, that is, the consistency error can be estimated as order O(h²), one order higher than that of its interpolation error, the superclose result in broken energy norm is derived for the fully discrete scheme. In the process, we use the interpolation operator instead of the so‐called elliptic projection, which is an indispensable tool in the traditional finite element analysis. Furthermore, the global superconvergence is obtained by using the interpolated postprocessing technique. Lastly, some numerical experiments are provided to verify our theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Uniform convergence and a posteriori error estimators for the enhanced strain finite element method

Summary. Enhanced strain elements, frequently employed in practice, are known to improve the approximation of standard (non-enhanced) displacement-based elements in finite element computations. The first contribution in this work towards a complete theoretical explanation for this observation is a proof of robust convergence of enhanced element schemes: it is shown that such schemes are locking-free in the incompressible limit, in the sense that the error bound in the a priori estimate is independent of the relevant Lamé constant. The second contribution is a residual-based a posteriori error estimate; the L ² norm of the stress error is estimated by a reliable and efficient estimator that can be computed from the residuals. Mathematics Subject Classification (2000):65N30     

Error analysis of a mixed finite element method for the Cahn-Hilliard equation

Summary. We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation u_t + (u–^–1f(u)) = 0, where > 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on only in some lower polynomial order for small . The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as 0 in [29].Mathematics Subject Classification (1991): 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99, 53A10Acknowledgments. The first author would like to thank Nicholas Alikakos for explaining all the fascinating properties of the Allen-Cahn and Cahn-Hilliard equations to him. He would also like to thank Nicholas Alikakos and Xinfu Chen for answering his questions regarding the spectrum estimate in Proposition 1. The second author gratefully acknowledges financial support by the DFG.     

Global existence and nonexistence of solutions for the nonlinear pseudo‐parabolic equation with a memory term

In this paper, we study the initial boundary value problem of nonlinear pseudo‐parabolic equation with a memory term with initial conditions and Dirichlet boundary conditions. By the combination of the Galerkin method and Potential well theory, the existence of global solutions is derived. Moreover, not only the finite time blow up of solutions with the negative initial energy (E(0) < 0) but also the finite time blow up results with the nonnegative initial energy (0≤E(0) < d_k) are obtained by using Concavity method and Potential well theory. Copyright © 2014 John Wiley & Sons, Ltd.     

A unified framework for a posteriori error estimation for the Stokes problem

In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on $[H^1_0(\Omega )]^d$ -conforming velocity reconstruction and $\underline{\varvec{H}}(\mathrm{div},\Omega )$ -conforming, locally conservative flux (stress) reconstruction. It?gives guaranteed, fully computable global upper bounds as well as local lower bounds on the energy error. In order to apply this framework to a given numerical method, two simple conditions need to be checked. We show how to do this for various conforming and conforming stabilized finite element methods, the discontinuous Galerkin method, the Crouzeix–Raviart nonconforming finite element method, the mixed finite element method, and a general class of finite volume methods. The tools developed and used include a new simple equilibration on dual meshes and the solution of local Poisson-type Neumann problems by the mixed finite element method. Numerical experiments illustrate the theoretical developments.     

Order one invariants of immersions of surfaces into 3-space

We classify all order one invariants of immersions of a closed orientable surface F into ³, with values in an arbitrary Abelian group . We show that for any F and and any regular homotopy class of immersions of F into ³, the group of all order one invariants on is isomorphic to is the group of all functions from a set of cardinality . Our work includes foundations for the study of finite order invariants of immersions of a closed orientable surface into ³, analogous to chord diagrams and the 1-term and 4-term relations of knot theory.Partially supported by the Minerva FoundationMathamatics Subject Classification (2000):57M, 57R42     

On the stability of <Emphasis Type="Italic">BDMS</Emphasis> and <Emphasis Type="Italic">PEERS</Emphasis> elements

Summary. Using Fortin operators we give a new proof of stability for Stenbergs family of BDMS elements in linear elasticity. Our approach yields the inf-sup condition with respect to the standard norms, which is indispensable for a posteriori error analysis. Furthermore our technique allows the construction of another family of finite elements strongly related to the classical PEERS element. The given estimates are robust for nearly incompressible materials.Mathematics Subject Classification (2000): 65N30, 65N15, 65F15     

An adaptive finite element method for singular parabolic equations

Summary. In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented. Mathematics Subject Classification (1991):65N22, 65F10, 65N30Supported by the Austrian Science Foundation (FWF) under the grant SFB F013}\and Walter Zulehner     

The optimal rate of convergence of the <Emphasis Type="Italic">p</Emphasis>-version of the boundary element method in two dimensions

Summary. We introduce the Jacobi-weighted Besov and Sobolev spaces in the one-dimensional setting. In the framework of these spaces, we analyze lower and upper bounds for approximation errors in the p-version of the boundary element method for hypersingular and weakly singular integral operators on polygons. We prove the optimal rate of convergence for the p-version in the energy norms of and respectively.Mathematics Subject Classification (2000): 65N38This author is supported by NSERC of Canada under Grant OGP0046726 and partially supported by the FONDAP Program (Chile) on Numerical Analysis during his visit of the Universidad de Concepción in 2001.This author is supported by Fondecyt project no. 1010220 and by the FONDAP Program (Chile) on Numerical Analysis.Revised version received January 28, 2004