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hanging nodes treatment

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 77 (2008) 117132

Arbitrary-level hanging nodes and automaticadaptivity in the hp-FEM

Pavel Soln a,b,, Jakub Cerveny a,b, Ivo Dolezel ba Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968-0514, USA

b Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejskova 5, Prague, CZ 10800

Received 4 June 2006; received in revised form 20 February 2007; accepted 21 February 2007Available online 2 March 2007

Abstract

In this paper we present a new automatic adaptivity algorithm for the hp-FEM which is based on arbitrary-level hanging nodesand local element projections. The method is very simple to implement compared to other existing hp-adaptive strategies, whileits performance is comparable or superior. This is demonstrated on several numerical examples which include the L-shape domainproblem, a problem with internal layer, and the Girkmann problem of linear elasticity. With appropriate simplifications, the proposedtechnique can be applied to standard lower-order and spectral finite element methods. 2007 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Constrained approximation; Hanging nodes; hp-FEM; Automatic hp-adaptivity

1. Introduction

The hp-FEM is a modern version of the Finite Element Method (FEM) which combines finite elements of variablesize (h) and polynomial degree (p) in order to obtain fast exponential convergence [37,12,14,21,23,24,27]. Theoutstanding performance of the hp-FEM as well as its superiority over standard (low-order) FEM has been provenin a number of convincing non-academic applications during the past decade [9,10,12,23,24,29], and the method isbecoming increasingly popular among practitioners.

However, there still are issues that need to be addressed before the hp-FEM can become standard in computationalengineering software. One of the well-known drawbacks of the method is its algorithmic complexity and relativelyhigh implementation cost. Current research in the hp-FEM covers a variety of topics, such as development of fastautomatic adaptive strategies and goal-oriented adaptivity [18,20,23,24], design of optimal higher-order shape func-tions for various types of continuous and vector-valued finite elements required by various types of PDEs [2,19,26],parallelization and automatic load balancing [9,10,15], application to problems with boundary layers [16], preservationof nonnegativity of physically nonnegative quantities (discrete maximum principles) [11,25,28], etc.

In this study we present a feasible algorithm for the treatment of multiple-level hanging nodes and show its positiveeffects on both the speed of convergence of the hp-FEM and simplification of automatic hp-adaptivity. The paper is

Corresponding author. Tel.: +1 915 760 5829.E-mail addresses: solin@utep.edu (P. Soln), jcerveny@utep.edu (J. Cerveny), dolezel@fel.cvut.cz (I. Dolezel).URLs: http://hpfem.math.utep.edu/ (P. Soln), http://hpfem.math.utep.edu/ (J. Cerveny).

0378-4754/$32.00 2007 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2007.02.011

mailto:solin@utep.edumailto:jcerveny@utep.edumailto:dolezel@fel.cvut.czdx.doi.org/10.1016/j.matcom.2007.02.011

118 P. Soln et al. / Mathematics and Computers in Simulation 77 (2008) 117132

organized as follows: In Section 2 we introduce irregularity rules and show their relation to the speed of convergence ofadaptive hp-FEM. The algorithmic treatment of arbitrary-level hanging nodes is described in Section 3. New automaticadaptive strategy for the hp-FEM based on arbitrary-level hanging nodes is presented in Section 4. Numerical examplesand comparisons follow in Section 5.

2. Hanging nodes and irregularity rules

When working with regular meshes (where two elements either share a common vertex, common edge, or theirintersection is empty), adaptivity often is done using the so-called redgreen refinement strategy [1]. This techniquefirst subdivides desired elements into geometrically convenient subelements with hanging nodes and then it eliminatesthe hanging nodes by forcing refinement of additional elements, as illustrated in Fig. 1.

This approach preserves the regularity of the mesh but at the same time it introduces elements with sharp angleswhich, in general, are not desirable in finite element analysis. Often this happens when repeated refinements occur insome part of the domain, e.g., toward a boundary layer or point singularity.

The green refinements can be avoided by introducing hanging nodes, i.e., by allowing irregular meshes whereelement vertices can lie in the interior of edges of other elements. To ease the computer implementation, most finiteelement codes working with hanging nodes limit the maximum difference of refinement levels of adjacent elementsto one (1-irregularity rule)see, e.g., Refs. [18,20,23]. In the following, by k-irregularity rule (or k-level hangingnodes) we mean this type of restriction where the maximum difference of refinement levels of adjacent elements is k.In this context, k= 0 corresponds to adaptivity with regular meshes and k= to adaptivity with arbitrary-level hangingnodes.

It is illustrated in Fig. 2 that even the 1-irregularity rule does not avoid all forced refinements.The amount of forced refinements in the mesh depends strongly on the level of hanging nodes.

Fig. 1. Redgreen refinement.

Fig. 2. Refinement with 1-irregularity rule.

P. Soln et al. / Mathematics and Computers in Simulation 77 (2008) 117132 119

Fig. 3. Example of a mesh with one-level hanging nodes.

3. Arbitrary-level hanging nodes

The requirement of approximation continuity across an edge with hanging nodes is equivalent to a set of linear alge-braic relations between the coefficients of the constraining and constrained shape functions. In Ref. [24], these relationswere derived explicitly using transition matrices between pairs of constraining and constrained polynomial bases. Thisapproach is very efficient as long as the maximum number of constraint levels is small (say, less than five). When thenumber of constraint levels becomes larger, the number of various geometrical situations on a constraining edge growsrapidly, and thus it becomes more efficient to calculate the constraining-constrained relations in specific situations on-demand. Therefore, we propose a new algorithm which is presented in Section 3.3. However, before discussing arbitrary-level hanging nodes, first let us explain the structure of one-level and multiple-level constraints in Sections 3.1 and 3.2.

3.1. One-level constraints

Assume a mesh edge AB containing one-level hanging nodes, as shown in Fig. 3. Let the polynomial degree of thisedge be pAB 1.

There are pAB + 1 constraining shape functions on the edge AB which induce three different situations:

I. Vertex function on K1 corresponding to the vertex A. This shape function constrains the vertex functions on theelements K2, K3, K4 corresponding to the vertex C.

II. Vertex function on K1 corresponding to the vertex B. This shape function also constrains the vertex functions onthe elements K2, K3, K4 corresponding to the vertex C.

III. pAB 1 edge functions of polynomial degrees p= 2, 3, . . ., pAB onK1 corresponding to the edge AB.Edge functionof degree p constrains(a) p 1 edge functions on the element K2 corresponding to the edge AC,(b) vertex functions on the elements K2, K3, and K4 corresponding to the vertex C,(c) p 1 edge functions on the element K4 corresponding to the edge CB.

Notice that interior shape functions (bubble functions) which vanish on element boundaries are neither constrainingnor constrained. Fig. 4 illustrates the simplest situation with pAB = 2.

3.2. Multiple-level constraints

The case of multiple-level constraints is more interesting. Consider, for example, a mesh with four-level hangingnodes shown in Fig. 5.

Now there are pAB + 1 constraining shape functions on K1 associated with the edge AB. Each of them induces twotypes of constraints:

Direct constraints. These include constrained vertex functions associated with all vertices in the interior of the edgeAB (vertices C1, C4, C7, C2) and pAB 1 constrained edge functions for each subedge of AB (edges AC1, C1C4,C4C7, C7C2, C2B).

120 P. Soln et al. / Mathematics and Computers in Simulation 77 (2008) 117132

Fig. 4. Constraining and constrained shape functions on a quadratic edge AB with one-level hanging nodes.

Implied constraints. Constrained vertex functions may further constrain vertex functions associated with verticeswhich do not lie on the edge AB (vertices C3, C5, C6, C8). Vertex functions subject to implied constraints mayproduce further implied constraints.

Note that constrained edge functions never imply additional constraints and that edge functions never are subjectto implied constraints.

3.3. Algorithmic treatment

Arbitrary-level hanging nodes can be handled via a simple extension of standard connectivity arrays for element-by-element assembling procedures [24].

3.3.1. Extension of standard connectivity arraysRecall that the element-by-element assembling procedure requires a unique enumeration of the basis functions

v1, v2, . . . , vN of the global finite element space Vh,p, dim(Vh,p) =N as well as a unique local enumeration of shapefunctions on the reference domain K. On a reference triangle, for example, by vk , k= 1, 2, 3 we may denote the vertexfunctions and by ek2 ,

ek3 , . . . ,

ekpk

, k= 1, 2, 3 the edge functions associated with edge ek of polynomial degree pk.Then every element Ki in a regular mesh contains pointers to six nodes:

Fig. 5. Exampl