A goal-oriented hp-adaptive finite element approach to radar scattering problems

The paper presents application of an hp-adaptive finite element method for scattering of electromagnetic waves. The main objective of the numerical analysis is to determine the characteristics of the scattered waves indicating the power being scattered at a given direction––i.e. the radar cross-section (RCS). This is achieved considering the scattered far-field which defines RCS and which is expressed as a linear functional of the solution. Techniques of error estimation for the far-field are considered and an h-adaptive strategy leading to the fast reduction of the error of the far-field is presented. The simulations are performed with a three-dimensional version of an hp-adaptive finite element method for electromagnetics based on the hexahedral edge elements combined with infinite elements for modeling the unbounded space surrounding the scattering object.     

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.     

An adaptive finite element scheme for transient problems in CFD

An adaptive finite element scheme for transient problems is presented. The classic h-enrichment / coarsening is employed in conjunction with a triangular finite element discretization in two dimensions. A mesh change is performed every n timesteps, depending on the Courant number employed and the number of ‘protective layers’ added ahead of the refined region. In order to simplify the refinement/ coarsening logic and to be as fast as possible, only one level of refinement/coarsening is allowed per mesh change. A high degree of vectorizability has been achieved on the CRAY XMP 12 at NRL. Several examples involving shock-shock interactions and the impact of shocks on structures demonstrate the performance of the method, indicating that considerable savings in CPU time and storage can be realized even for strongly unsteady flows.     

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.     

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.     

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.     

An adaptive grid method for degenerate semilinear quenching problems

This paper studies the structure and implementation of a specially formulated adaptive grid method for computing the numerical solution of a class of degenerate semilinear quenching problems. Modified arc-length procedures are used in both the space and time for robust adjustments of the spatial and temporal discretizations throughout the computation. A Milne-like device is introduced in the error estimation and control of the computational error. The numerical solution generated converges monotonically to the physical solution of the underlying problem up to the quenching time. Numerical examples are given for demonstrating our conclusions.     

Numerical evaluation of finite element methods¶in convection-diffusion problems

In this paper we propose an inf-sup test to identify and measure the stability of various finite element methods for the solution of multi-dimensional convection-diffusion problem. Received: December 1999 / Revised version: May 2000     

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.     

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.