Christos Xenophontos

The hp finite element method for problems in mechanics with boundary layers

Abstract We consider the numerical approximation of boundary layer phenomena occurring in many singularly perturbed problems in mechanics, such as plate and shell problems. We present guidelines for the effective resolution of such layers in the context of existing, commercial p and hp finite element (FE) version codes. We show that if high order, ‘spectral’ elements are available, then just two elements are sufficient to approximate these layers at a near-exponential rate, independently of the problem parameters thickness or Reynolds number. We present hp mesh design principles for situations where both corner singularities and boundary layers are present.

Solving Laplacian problems with boundary singularities: a comparison of a singular function boundary integral method with the p/hp version of the finite element method

We solve a Laplacian problem over an L-shaped domain using a singular function boundary integral method as well as the p/hp finite element method. In the former method, the solution is approximated by the leading terms of the local asymptotic solution expansion, and the unknown singular coefficients are calculated directly. In the latter method, these coefficients are computed by post-processing the finite element solution. The predictions of the two methods are discussed and compared with recent numerical results in the literature.

A numerical study on the finite element solution of singularly perturbed systems of reaction-diffusion problems

We consider the approximation of singularly perturbed systems of reaction-diffusion problems, with the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. In this article we conduct a numerical study of several finite element methods applied to a model problem, having as our goal their assessment and the identification of a high order scheme which approximates the solution at an exponential rate of convergence, independently of the singular perturbation parameters.

A Singular Function Boundary Integral Method for Laplacian Problems with Boundary Singularities

A singular function boundary integral method for Laplacian problems with boundary singularities is analyzed. In this method, the solution is approximated by the truncated asymptotic expansion for the solution near the singular point and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. The resulting discrete problem is posed and solved on the boundary of the domain, away from the point of singularity. The main result of this paper is the proof of convergence of the method; in particular, we show that the method approximates the generalized stress intensity factors, i.e., the coefficients in the asymptotic expansion, at an exponential rate. A numerical example illustrating the convergence of the method is also presented.

Special boundary approximation methods for laplace equation problems with boundary singularities - Applications to the motz problem

Abstract We investigate the convergence of special boundary approximation methods (BAMs) used for the solution of Laplace problems with a boundary singularity. In these methods, the solution is approximated in terms of the leading terms of the asymptotic solution around the singularity. Since the approximation of the solution satisfies identically the governing equation and the boundary conditions along the segments causing the singularity, only the boundary conditions along the rest of the boundary need to be enforced. Four methods of imposing the essential boundary conditions are considered: the penalty, hybrid, and penalty/hybrid BAMs and the BAM with Lagrange multipliers. A priori error analyses and numerical experiments are carried out for the case of the Motz problem, and comparisons between all methods are made.

Application of the p-version of the finite element method to elastoplasticity with localization of deformation

In this paper we discuss the use of the p-version of the finite element method applied to elastoplastic problems that exhibit sharp (but continuous) deformation gradients. The deformation theory of deviatoric, linearly hardening elastoplasticity with an iterative, displacement based finite element framework is used. The focus of this work is on assessing the applicability of the p-version to the analysis of localized deformation with continuous strain and displacement fields. Presented examples demonstrate that the method can be used reliably with a proper finite element mesh design. Possible extensions of the work are also discussed.

Robust Approximation of Singularly Perturbed Delay Differential Equations by the hp Finite Element Method

Abstract. We consider the finite element approximation of the solution to a singularly perturbed second order differential equation with a constant delay. The boundary value problem can be cast as a singularly perturbed transmission problem, whose solution may be decomposed into a smooth part, a boundary layer part, an interior/interface layer part and a remainder. Upon discussing the regularity of each component, we show that under the assumption of analytic input data, the hp version of the finite element method on an appropriately designed mesh yields robust exponential convergence rates. Numerical results illustrating the theory are also included.

The hp finite element method for singularly perturbed problems in nonsmooth domains

We consider the numerical approximation of singularly perturbed elliptic boundary value problems over nonsmooth domains. We use a decomposition of the solution that contains a smooth part, a corner layer part and a boundary layer part. Explicit guidelines for choosing mesh-degree combinations are given that yield finite element spaces with robust approximation properties. In particular, we construct an hp finite element space that approximates all components uniformly, at a near exponential rate. c

THE hp FINITE ELEMENT METHOD FOR SINGULARLY PERTURBED PROBLEMS IN SMOOTH DOMAINS

We consider the numerical approximation of singularly perturbed elliptic problems in smooth domains. The solution to such problems can be decomposed into a smooth part and a boundary layer part. We present guidelines for the effective resolution of boundary layers in the context of the hp finite element method and we construct tensor product spaces that approximate these layers uniformly at a near-exponential rate.

Finite element computations for the Reissner–Mindlin plate model

SUMMARY We consider the finite element (FE) approximation of the Reissner‐Mindlin (RM) plate model, and indicate how to design meshes that yield accurate results when the p/hp version of the standard FE method is used. These guidelines allow quantities of engineering interest to be predicted numerically with great confidence near the boundary. We illustrate this through numerical computations in the case when both boundary layers and corner singularities are present. #1998 John Wiley & Sons, Ltd.