Data structures and concepts for adaptive finite element methods

The administration of strongly nonuniform, adaptively generated finite element meshes requires specialized techniques and data structures. A special data structure of this kind is described in this paper. It relies on points, edges and triangles as basic structures and is especially well suited for the realization of iterative solvers like the hierarchical basis or the multilevel nodal basis method.     

An adaptive finite element method for evolutionary convection dominated problems

We present an adaptive finite element method for evolutionary convection–diffusion problems. The algorithm is based on an a posteriori indicator of the size of the oscillations displayed by the finite element approximation. The procedure is able to refine or coarsen dynamically the mesh adjusting it automatically to evolving layers. The method produces nearly non-oscillatory approximations in the convection dominated regime. We check the performance of the adaptive method with some numerical experiments.     

The adaptive finite element method based on multi-scale discretizations for eigenvalue problems

In this paper, adaptive finite element methods for differential operator eigenvalue problems are discussed. For multi-scale discretization schemes based on Rayleigh quotient iteration (see Scheme 3 in [Y. Yang, H. Bi, A two-grid discretization scheme based on shifted-inverse power method, SIAM J. Numer. Anal. 49 (2011) 1602–1624]), a reliable and efficient a posteriori error indicator is given, in addition, a new adaptive algorithm based on the multi-scale discretizations is proposed, and we apply the algorithm to the Schrödinger equation for hydrogen atoms. The algorithm is performed under the package of Chen, and satisfactory numerical results are obtained.     

A class of asymmetric simplicial finite element methods for solving finite incompressible elasticity problems

We consider the numerical solution of finite incompressible elasticity problems for Mooney-Rivlin materials in n-dimensional space, n = 2, 3.In representing the displacement vector field with standard Lagrange finite elements, in general one faces the difficulty of the total number of constraints exceeding the total number of displacement degrees of freedom, in the discrete problem — if the nonlinear incompressibility condition is to be approximated within satisfactory accuracy.In this paper we introduce a new class of piecewise reduced quadratic, conforming, simplicial finite elements that eliminates the above inconvenience with almost no extra implementational cost, thereby allowing highly satisfactory approximate solutions.This statement is illustrated by numerical examples in which our method is combined with an algorithm of augmented Lagrangian type proposed by Glowinski and Le Tallec for the continuous problem.     

A review of some a posteriori error estimates for adaptive finite element methods

Recently, the adaptive finite element methods have gained a very important position among numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications. While the classical a posteriori error estimates are oriented to the use in h-methods the contemporary higher order hp-methods usually require new approaches in a posteriori error estimation.     

Convergence of an upwind control-volume mixed finite element method for convection–diffusion problems

Summary We consider a upwind control volume mixed finite element method for convection–diffusion problem on rectangular grids. These methods use the lowest order Raviart–Thomas mixed finite element space as the trial functional space and associate control-volumes, or covolumes, with the vector variable as well as the scalar variable. Chou et al. [6] established a one-half order convergence in discrete L ²-norms. In this paper, we establish a first order convergence for both the vector variable as well as the scalar variable in discrete L ²-norms.      

Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results.     

Stress based finite element methods for solving contact problems: Comparisons between various solution methods

This paper deals with numerical methods for solving unilateral contact problems with friction. Although these problems are usually defined in terms of the displacement, a stress based approach to the problem is developed here. The “equilibrium” finite elements method is therefore used. Using these elements make it possible to satisfy the local equilibrium condition a priori, but on the other hand, prescribed and contact forces have to be introduced using Lagrangian multipliers. The problem obtained is therefore a non-linear, constrained problem and the global system matrix is non-positive definite. Various solution algorithms are thus proposed and compared. Comparisons between the classical method and that developed here show that the stress formulation gives very satisfactory results in terms of the stresses.     

On theories and applications of mixed finite element methods for linear boundary-value problems

This paper is devoted to a study of mathematical properties of certain mixed finite element approximations of linear boundary-value problems, and the application of such methods to simple representative problems which are designed to test the validity of the theory. In particular, the Oden—Reddy theory[1,2] is studied in some depth. An alternate approach to convergence questions, suggested by certain theorems of Babu ka[3], is also devised, and predictions of the two theories are briefly compared. As a result of this investigation, a number of criteria for using mixed methods in practical problems are identified, and it is shown that these criteria are supported by both theoretical arguments and by numerical experiments.     

An h-p-n adaptive finite element scheme for shell problems

We propose an adaptive finite element algorithm for shells which, in addition to the usual h-p adaption also shows adaptivity with respect to the order n of the dimension reduction. The idea of the algorithm is to adaptively capture and resolve the various length scales that may occur in shells. The algorithm presented in the paper is limited to axisymmetric problems, which reduces the h-p part of the problem to one dimension only. The performance of the algorithm is tested in some example cases where the shell is cylindrical. For comparison, we test the algorithm also when n is limited so that n k, where K = 1, 2 or 3. Choosing k = 2 essentially corresponds to the classical shell models.     

Mixed finite element method for contact problems

In the paper the model of finite elements for elastic contact problems was used. Real structures are modeled by finite elements and rigid finite elements. We calculate stresses in the surface of two substructures using Coulomb model of friction.The method given here is an iterative procedure which is planed to incorporate this technique in the system allowing for incremental elastic solution. The computer program is adapted to solving spatial problems.     

A posteriori error estimates of two-grid finite volume element methods for nonlinear elliptic problems

In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in $H^1$-norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator.     

An adaptive finite element scheme to solve fluid-structure vibration problems on non-matching grids

This paper deals with the computation of the vibration modes of a system consisting of a linear elastic solid interacting with an acoustic fluid. A finite element method based on meshes for each medium not matching on the fluid-solid interface is analyzed. Optimal order of convergence is proved for the approximation of the eigenfunctions, as well as a double order for the eigenvalues. Numerical tests confirming the theoretical results and showing the advantage of using non-matching grids are reported. Finally, an a posteriori error estimator for this method is introduced and combined with a mesh refinement strategy. The efficiency of this adaptive technique is tested with further numerical experiments. Received: 30 January 2001 / Accepted: 30 May 2001     

An adaptive finite element technique for structural dynamic analysis

An adaptive finite element discretization technique, which utilizes specially derived Ritz vectors, is presented for solving structural dynamics problems. The special Ritz vectors are applied as the bases of transformation in geometric coordinates for mode superposition dynamic analysis. To capture the low frequency response and the high frequency response using multigrid principles, a hierarchical formulation for the formation of the coefficient matrices is proposed and it is utilized in the framework of the adaptive h-refinement. Assuming that the solution can be resolved into a set of orthogonal vectors and the refined mesh which passes the refinement criteria for all the vectors can satisfy the refinement criteria for the solution, the Ritz vectors are used as sources to discretize the continuous spatial domain. An a posteriori energy norm of residual error serves as the error measure. Finally, the performance and the efficiency of the proposed technique is demonstrated by solving several examples.     

Convergence of iterative methods for stabilized saddle-point problems

The convergence of two classes of iterative methods for stabilized saddle-point problems (inexact Uzawa algorithms and a class of methods with symmetric preconditioners) are studied using Zulerhner’s unified approach.     

Multi-region adaptive finite element-boundary element method for elasto-plastic analysis

In this paper, a multi-region adaptive finite element–boundary element (FEM-BEM) method for elasto-plastic analysis is presented. The method is suitable for several plasticity models, i.e. von-Mises, Tresca, Mohr–Coulomb, Drucker–Prager, Hill, and Hoffman yield criteria. The domain of the original problem is divided into the finite element and the boundary element sub-domains: the FEM is utilized in regions where plastic material behaviour is expected to develop, whereas the complementary linear elastic region is approximated using the symmetric Galerkin BEM. The adaptive method estimates the finite element and boundary element sub-domains, automatically generates the corresponding meshes and adapts the sub-domains, according to the state of computation.