Manil Suri

The P and H-P versions of the finite element method, basic principles and properties

In the classical form of the finite element method called the hversion, piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy. In this paper, we discuss the fundamental theoretical ideas behind the relatively recent p version and h-p version. In the p version, a fixed mesh is used and p is allowed to increase. The h-p version combines both approaches. The authors describe and explain the basic properties and characteristics of these newer versions, especially in areas where their behavior is significantly different from that of the h version. Simplified proofs of key concepts are included and computational illustrations of several results are provided. A benchmark comparison between the various versions in included.

The optimal convergence rate of the p-version of the finite element method

Optimal error estimates for the p-version of the finite element method in two dimensions are proven for the case when

The p - and h-p version of the finite element method, an overview

u \in H^k (\Omega )

On locking and robustness in the finite element method

or u has singularities induced by the corners of the domain. The case of nonhomogenous essential boundary conditions is also analyzed.

Locking effects in the finite element approximation of elasticity problems

We survey the advances in the p- and the h-p versions of the finite element method. An up-to-date list of references related to these methods is provided.

The p and hp versions of the finite element method for problems with boundary layers

A numerical scheme for the approximation of a parameter-dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value. A robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter. Precise mathematical definitions for these terms are developed, their quantitative characterization is given, and some general theorems involving locking and robustness are proven. A model problem involving heat transfer is analyzed in detail using this mathematical framework, and various related computational results are described. Applications to some different problems involving locking are presented.

Analytical and computational assessment of locking in the hp finite element method

SummaryWe consider the finite element approximation of the 2D elasticity problem when the Poisson ratiov is close to 0.5. It is well-known that the performance of certain commonly used finite elements deteriorates asv→0, a phenomenon calledlocking. We analyze this phenomenon and characterize the strength of the locking androbustness of varioush-version schemes using triangular and rectangular elements. We prove that thep-andh-p versions are free of locking with respect to the error in the energy norm. A generalization of our theory to the 3D problem is also discussed.

Uniform hp convergence results for the mortar finite element method

We study the uniform approximation of boundary layer functions exp(-x/d) for x ∈ (0,1), d ∈ (0,1], by the p and hp versions of the finite element method. For the p version (with fixed mesh), we prove super-exponential convergence in the range p + 1/2 > e/(2d). We also establish, for this version, an overall convergence rate of O(p -1 √ln p) in the energy norm error which is uniform in d, and show that this rate is sharp (up to the √ln p term) when robust estimates uniform in d ∈ (0,1] are considered. For the p version with variable mesh (i.e., the hp version), we show that exponential convergence, uniform in d ∈ (0,1], is achieved by taking the first element at the boundary layer to be of size O(pd). Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when d is as small as, e.g., 10 -8 . They also illustrate the superiority of the hp approach over other methods, including a low-order h version with optimal exponential mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.

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

Locking is the phenomenon by which the numerical approximation of parameter-dependent problems deteriorates for values of the parameter close to a limiting value. In this paper, we give a definition of locking and develop precise computable and analytic ways to quantify it. Using the example of nearly incompressible elasticity, we show by means of computational and theoretical results, the difference between the h version and php version in combatting locking. Our results establish the superiority of high-order elements (both h, p and hp) when the standard variational form is used. We also discuss other issues such as curved elements, mixed methods, and locking phenomena for problems over anisotropic materials and over thin domains.

An hp Error Analysis of MITC Plate Elements

The mortar finite element is an example of a non-conforming method which can be used to decompose and re-compose a domain into sub-domains without requiring compatibility between the meshes on the separate components. We obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used, without assuming quasiuniformity for the meshes. Our results establish that the method is optimal when non-quasiuniform h or hp methods are used. Such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain. We also give an error estimate for when the p version is used. Numerical results for hp and hp mortar FEMs show that these methods behave as well as conforming FEMs. An hp extension theorem is also proved.