A dual finite element complex on the barycentric refinement

Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex centered on Raviart-Thomas divergence conforming vector fields. It can be seen as a realization of the simplicial cochain complex. We construct a new complex of finite element spaces on the barycentric refinement of the mesh which can be seen as a realization of the simplicial chain complex on the original (unrefined) mesh, such that the duality is non-degenerate on for each . In particular is a space of -conforming vector fields which is dual to Raviart-Thomas -conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.

     

A regional mixed refinement procedure for finite element mesh design

A practical algorithm is developed for automated mesh design in finite element stress analysis. A regional mixed mesh improvement procedure is introduced. The error control, algorithm implementation, code development, and the solution accuracy are discussed. Numerical examples include automated mesh designs for plane elastic media with singularities. The efficiency of the procedure is demonstrated.     

A dual graph-norm refinement indicator for finite volume approximations of the Euler equations

Summary. We generalise and apply a refinement indicator of the type originally designed by Mackenzie, Süli and Warnecke in [15] and [16] for linear Friedrichs systems to the Euler equations of inviscid, compressible fluid flow. The Euler equations are symmetrized by means of entropy variables and locally linearized about a constant state to obtain a symmetric hyperbolic system to which an a posteriori error analysis of the type introduced in [15] can be applied. We discuss the details of the implementation of the refinement indicator into the DLR--Code which is based on a finite volume method of box type on an unstructured grid and present numerical results. Received May 15, 1995 / Revised version received April 17, 1996     

Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems

We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided the mesh size tends to zero.     

Selective finite element refinement in torsional problems based on the membrane analogy

This work presents a selective finite element refinement strategy based on the h-refinement type, in the context of a posteriori error estimates considerations (error computed after the application of the proposed refining scheme), based on a graphical procedure to determine progressively better estimates for the maximum shearing stress in prismatic torsional members. It is structured in an integrated FORTRAN code and DELPHI based environment to refine an initial arbitrary finite element mesh. The proposed procedure is founded on the membrane analogy that exists between membrane deflections and the torsion problem in the sense that the location of the membrane largest gradient drives the refining procedure. It is shown that multiple level application of the proposed method to two members with different cross sectional geometries with known analytic solutions leads to progressively more accurate estimates (< 1.0% error in most cases) for the maximum shearing stresses calculations. Finally, the proposed method is applied to the torsional analysis of an L section member, showing that for this practical case the procedure results in a very accurate calculation as well.     

Anisotropic mesh refinement for finite element methods based on error reduction

In this paper, we propose an anisotropic adaptive refinement algorithm based on the finite element methods for the numerical solution of partial differential equations. In 2-D, for a given triangular grid and finite element approximating space V, we obtain information on location and direction of refinement by estimating the reduction of the error if a single degree of freedom is added to V. For our model problem the algorithm fits highly stretched triangles along an interior layer, reducing the number of degrees of freedom that a standard h-type isotropic refinement algorithm would use.     

Schwarz methods with local refinement for the p-version finite element method

Summary. In some applications, the accuracy of the numerical solution of an elliptic problem needs to be increased only in certain parts of the domain. In this paper, local refinement is introduced for an overlapping additive Schwarz algorithm for the $-version finite element method. Both uniform and variable degree refinements are considered. The resulting algorithm is highly parallel and scalable. In two and three dimensions, we prove an optimal bound for the condition number of the iteration operator under certain hypotheses on the refinement region. This bound is independent of the degree $, the number of subdomains $ and the mesh size $. In the general two dimensional case, we prove an almost optimal bound with polylogarithmic growth in $. Received February 20, 1993 / Revised version received January 20, 1994     

A dynamic data structure suitable for adaptive mesh refinement in finite element method

This paper describes a dynamic data structure and its implementation, used for an optimum mesh generator. The implementation of this mesh generator was a part of a software package implemented to solve electromagnetic field problems using the finite element method. This mesh generator takes advantage of the Delaunay algorithm, which maximizes the summation of the smallest angles in all triangles and thus creates a mesh that is proved to be an optimum mesh for use in the finite element method. The dynamic data structure is explained and the source code is reviewed. The programs have been written in Pascal programming language.     

Convergence analysis of the adaptive finite element method with the red-green refinement

In this paper, we analyze the convergence of the adaptive conforming P 1 element method with the red-green refinement. Since the mesh after refining is not nested into the one before, the Galerkin-orthogonality does not hold for this case. To overcome such a difficulty, we prove some quasi-orthogonality instead under some mild condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive method by establishing the reduction of some total error. To weaken the condition on the...     

Domain-decomposition approach to local grid refinement in finite element collocation

Advection-dominated flows occur widely in the transport of groundwater contaminants, the movements of fluids in enhanced oil recovery projects, and many other contexts. In numerical models of such flows, adaptive local grid refinement is a conceptually attractive approach for resolving the sharp fronts or layers that tend to characterize the solutions. However, this approach can be difficult to implement in practice. A domain decomposition method developed by Bramble, Ewing, Pasciak, and Schatz, known as the BEPS method, overcomes many of the difficulties. We demonstrate the applicability of BEPS ideas to finite element collocation on trial spaces of piecewise Hermite cubics. The resulting scheme allows one to refine selected parts of a spatial grid without destroying algebraic efficiencies associated with the original coarse grid. We apply the method to steady-state problems with boundary and interior layers and a time-dependent advection-diffusion problem.     

A time-domain homogenization technique for lamination stacks in dual finite element formulations

A time-domain homogenization technique is developed to take the eddy currents in lamination stacks into account with dual 3-D magnetodynamic b$\mathbf{b}$- and h$\mathbf{h}$-conform finite element formulations. The lamination stack is considered as a source region carrying predefined current density and magnetic flux density distributions describing the eddy currents and skin effect in each lamination. These distributions are related and are approximated with sub-basis functions. The stacked laminations are then converted into continuums with which terms are associated for considering the eddy current loops produced by parallel fluxes, through the homogenization of the sub-basis function contributions.     

A posteriori error estimation for the dual mixed finite element method of the Stokes problem

This Note presents an a posteriori error estimator of residual type for the stationary Stokes problem using the dual mixed FEM. We prove lower and upper error bounds with the explicit dependence of the viscosity parameter and without any regularity assumption on the solution. To cite this article: M. Farhloul et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Adaptive mesh refinement for the control of cost and quality in finite element analysis

Adaptive strategies are a necessary tool to make finite element analysis applicable to engineering practice. In this paper, attention is restricted to mesh adaptivity. Traditionally, the most common mesh adaptive strategies for linear problems are used to reach a prescribed accuracy. This goal is best met with an h-adaptive scheme in combination with an error estimator. In an industrial context, the aim of the mechanical simulations in engineering design is not only to obtain greatest quality but more often a compromise between the desired quality and the computation cost (CPU time, storage, software, competence, human cost, computer used). In this paper, we propose the use of alternative mesh refinement criteria with an h-adaptive procedure for 3D elastic problems. The alternative mesh refinement criteria (MR) are based on: prescribed number of elements with maximum accuracy, prescribed CPU time with maximum accuracy and prescribed memory size with maximum accuracy. These adaptive strategies are based on a technique of error in constitutive relation (the process could be used with other error estimators) and an efficient adaptive technique which automatically takes into account the steep gradient areas. This work proposes a 3D method of adaptivity with the latest version of the INRIA automatic mesh generator GAMHIC3D.     

A note on finite element wavelets

1. IotroductionThe study of mu1ti-wavelets was initiated by Goodman, Lee and Tang[2]. Later, acharacterization of sca1ing functions and wavelets was established by Goodman and Lee[3].In [4] GoodInan constructed multi-wavelets that satisfy certain interpolating conditions.Geronimo, Hardin and Massopust[5] used fractal interpolation to construct orthogonal scal-ing functions gh1 and gh2 (with multiplicity r = 2) that are both symmetric and colltinuousbut nondifferentiable. Strang and Strela…     

Sensitivities for topological graph changes in finite element meshes and application to adaptive refinement

We propose a novel approach to adaptivity in FEM based on local sensitivities for topological mesh changes. To this end, we consider refinement as a continuous operation on the edge graph of the finite element discretization, for instance by splitting nodes along edges and expanding edges to elements. Thereby, we introduce the concept of a topological mesh derivative for a given objective function that depends on the discrete solution of the underlying PDE. These sensitivities may in turn be used as refinement indicators within an adaptive algorithm. For their calculation, we rely on the first-order asymptotic expansion of the Galerkin solution with respect to the topological mesh change. As a proof of concept, we consider the total potential energy of a linear symmetric second-order elliptic PDE, minimization of which is known to decrease the approximation error in the energy norm. In this case, our approach yields local sensitivities that are closely related to the reduction of the energy error upon refinement and may therefore be used as refinement indicators in an adaptive algorithm. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the numerical solution of axisymmetric domain optimization problems by dual finite element method

Shape optimization of an axisymmetric three-dimensional domain with an elliptic boundary value state problem is solved. Since the cost functional is given in terms of the cogradient of the solution, a dual finite element method based on the minimum of complementary energy principle is used. © 1994 John Wiley & Sons, Inc.     

A posteriori error estimation for the dual mixed finite element method of the elasticity problem in a polygonal domain

In this article, we propose a residual based reliable and efficient error estimator for the new dual mixed finite element method of the elasticity problem in a polygonal domain, introduced by M. Farhloul and M. Fortin. With the help of a specific generalized Helmholtz decomposition of the error on the strain tensor and the classical decomposition of the error on the gradient of the displacements, we show that our global error estimator is reliable. Efficiency of our estimator follows by using classical inverse estimates. The lower and upper error bounds obtained are uniform with respect to the Lamé coefficient λ, in particular avoiding locking phenomena. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

In this paper we formulate and analyze a discretization method for a 2D linear singularly perturbed convection-diffusion problem with a singular perturbation parameter . The method is based on a nonconforming combination of the conventional Galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method, and on a mixture of triangular and rectangular elements. It is shown that the method is stable with respect to a semi-discrete energy norm and the approximation error in the semi-discrete energy norm is bounded by with independent of the mesh parameter , the diffusion coefficient and the exact solution of the problem.

     

A new cubic nonconforming finite element on rectangles

A new nonconforming rectangle element with cubic convergence for the energy norm is introduced. The degrees of freedom (DOFs) are defined by the 12 values at the three Gauss points on each of the four edges. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are 11. The nonconforming element consists of . We count the corresponding dimension for Dirichlet and Neumann boundary value problems of second‐order elliptic problems. We also present the optimal error estimates in both broken energy and norms. Finally, numerical examples match our theoretical results very well. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 691–705, 2015     

A review of parallel finite element methods on the DAP

This paper reviews the research work that has been done to implement the finite element method for solving partial differential equations on the ICL distributed array processor (DAP). A brief outline of the principle features of the method is given, followed by details of the novel techniques required for implementation on the highly parallel architecture. Various methods of solution of the finite element equations are discussed; both direct and iterative techniques are included. The current state-of-the-art favours the use of the preconditioned conjugate gradient method. Some suggestions for future research work on parallel finite element methods are made.