Estimation of interpolation error constants for the P0 and P1 triangular finite elements

doi:10.1016/j.cma.2006.10.029

Computer Methods in Applied Mechanics and Engineering

Volume 196, Issues 37–40, 1 August 2007, Pages 3750-3758

Dedicated to Professor Ivo Babuška on the occasion of his 80th birthday.

https://doi.org/10.1016/j.cma.2006.10.029 Get rights and content

Abstract

We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but some results are also given for the non-conforming case. We obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles. In particular, we explicitly determine the Babuška–Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation $\sqrt{λ} + \tan \sqrt{λ} = 0$ , so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions.

Introduction

The finite element method (FEM) is now recognized as a powerful numerical method for wide classes of partial differential equations. Furthermore, it also has sound mathematical bases such as highly refined a priori and a posteriori error estimations. In the classical a priori error analysis of FEM, interpolation errors are essential to derive final error estimates in various norms [7], [8], [10]. In this process, there appear various positive constants besides the standard discretization parameter h and norms (or seminorms), but it has been very difficult to evaluate such constants explicitly. For quantitative purposes, however, it is indispensable to evaluate or bound them as accurately as possible, because sharper estimates enable more efficient finite element computations. Thus such an evaluation has become progressively more important and has been attempted especially for adaptive finite element calculations based on a posteriori error estimation as well as for numerical verification by FEM [1], [4], [6], [7], [13]. In this paper, we will give a few fundamental results on some interpolation error constants of the most popular triangular finite elements.

More specifically, we give some results on interpolation error constants appearing in the popular P₀ (piecewise constant) and P₁ (piecewise linear) triangular finite elements. Essentially based on the paper of Babuška–Aziz [3], we analyze the dependence of several constants on the geometric parameters such as the maximum interior angle and the minimum edge length of the triangle more quantitatively than in [3]. Above all, the optimal constant (C₃ in this paper) appearing in the H¹ error estimate of the P₁ interpolation of H² functions over the unit isosceles right triangle is essential and frequently used, and it was explicitly evaluated firstly by Natterer [15]. On the other hand, this constant was shown to be closely related to the one (C₁ in this paper) presented and effectively used by Babuška and Aziz in conjunction with the maximum angle condition [3]. More precisely, C₁ gives an upper bound quite close to the optimal constant C₃, and the relation between C₃ and C₁ was further discussed in [13], [18]. Thus a precise estimation of these two constants is very important, and a number of researchers have given bounds for these using various approximation methods including numerical verification, see e.g. [2], [11], [13], [14], [15], [18]. Furthermore, these constants can be also used to evaluate the interpolation error constants for the non-conforming P₁ triangle, as will be mentioned later.

For the above Babuška–Aziz constant, we have succeeded in obtaining a value which is in a sense optimal. That is, by analytically solving an eigenvalue problem for the 2D Laplacian over the above triangular domain, we can show that the constant can be easily determined from a solution of the simple transcendental equation $\sqrt{λ} + \tan \sqrt{λ} = 0$ . In this process, we use the reflection (or symmetry) method [16]. Moreover, we have obtained some explicit relations for the dependence of such constants on the geometry of triangles. It is to be emphasized that they are consistent with the maximum angle condition in [3]. We also give some numerical and analytical results, the latter of which are based on asymptotic analysis. Thus our results can be effectively used in the quantitative a priori and a posteriori error estimations of the finite element solutions by the P₁ triangular element and also those based on the P₀ triangle. The former is of course the most classical and fundamental one, but still in frequent use, while the latter appears in some mixed finite element methods and implicitly on various occasions. Moreover, we also give some results for the non-conforming P₁ triangle by using the constants for the P₀ and the conforming P₁ triangles.

Section snippets

Preliminaries

Let h, α and θ be positive constants such that $h > 0, 0 < α ⩽ 1, (\frac{π}{3} ⩽) \cos^{- 1} \frac{α}{2} ⩽ θ < π .$ Then we define the triangle $T_{α, θ, h}$ by $▵ OAB$ with three vertices $O (0, 0)$ , $A (h, 0)$ and $B (α h \cos θ, α h \sin θ)$ . From (1), AB is shown to be the edge of maximum length, i.e. $\bar{AB} ⩾ h ⩾ α h$ , so that $h = \bar{OA}$ here denotes the medium edge length, although the notation h is often used as the largest edge length. A point on the closure of $T_{α, θ, h}$ is denoted by $x = {x_{1}, x_{2}}$ . By an appropriate congruent transformation in R², we can configure any triangle as

Dependence of constants on $θ$

This section is devoted to analysis of the effects of the maximum interior angle θ on $C_{i} (α, θ)$ ’s for fixed α. For $C_{3} (α, θ)$ , the well-known maximum angle condition was derived in [3]. However, the results reported there are not fully quantitative, so that we give here more quantitative estimates for the constants including $C_{3} (α, θ)$ .

To this end, let us introduce the following simple affine transformation between $x = {x_{1}, x_{2}} \in T_{α, θ}$ and $ξ = {ξ_{1}, ξ_{2}} \in T_{α}$ : $ξ_{1} = x_{1} - x_{2} / \tan θ, ξ_{2} = x_{2} / \sin θ .$ This transformation is a bit

Dependence of constants on $α$

Up to now, we have given some basic results for dependence of error constants on h and θ. In this section, we will consider the dependence of such constants on α when $θ = π / 2$ . With this regard, we owe much the following results to the analysis by Babuška and Aziz [3]. In particular, the estimation $C_{3} (α) ⩽ C_{1}$ below is an important consequence derived in [3] and also in [13], [18], and so we here call C₁ the Babuška–Aziz constant.

Theorem 2

For h = 1 and $θ = π / 2$ , $C_{i} (α)$ $(0 ⩽ i ⩽ 4)$ are continuous positive-valued

Determination of C₀ and C₁

First let us determine C₀ exactly. Actually, its exact value is already known, see e.g. [13], [14]. However, we here state the results with a proof, since the underlying idea is somewhat common to the more complicated case of C₁.

Theorem 3

With regard to C₀, i.e., $C_{0} (α, θ)$ for α = 1 and $θ = π / 2$ , it holds that $C_{0} = 1 / π$ .

Proof

We will prove in two steps, each of which is based on rather well-known arguments and techniques. The triangular domain to be considered here is T.

(1) Let Ω be a unit square domain: $Ω = {x = {x_{1}, x_{2}} \in R^{2};$

Asymptotic behaviors of constants as $α \to + 0$

Moreover, we can analyze the asymptotic behaviors of the constants $C_{i} (α)$ ’s as $α \to + 0$ , cf. [12]. In particular, the right limit values $C_{i} (+ 0)$ ’s are given by zeros of certain transcendental equations (derived from eigenvalue problems of ordinary differential equations, ODE’s) in terms of the hypergeometric functions [20]. For example, $C_{2} (+ 0)^{- 1}$ is equal to the first positive zero of the Bessel function $J_{0} (z)$ .

For the analysis, we use various techniques including compactness arguments. We will publish

Nonconforming P₁ triangle

We have mainly considered the conforming P₁ triangle, which can naturally construct subspaces of H¹ space over the entire domain. But there also exists a non-conforming counterpart, which is also based on $P_{1}$ but uses as nodes the midpoints of edges or edges themselves [19]. Analysis of such an element is more complicated, since we must additionally evaluate the errors induced by the interelement discontinuity of the approximate functions. Still we can obtain some results for the interpolation

Numerical results

We performed numerical computations to see the actual dependence of various constants on α and θ. Here, we just show the results for $C_{1} (α)$ , $C_{2} (α)$ and $C_{3} (α)$ by the P₁ FEM with the uniform triangulation of the domain T_α. In such calculations, T_α is subdivided into a number of small congruent triangles $T_{α, π / 2, h}$ with $h = 1 / 20$ . The penalty method in [18] was also adopted to calculate $C_{3} (α)$ approximately. The resulting approximate problems are matrix eigenvalue ones, and can be solved numerically if

Concluding remarks

We have obtained some explicit relations for the dependence of a few interpolation error constants on geometric parameters of triangular finite elements. In particular, we have succeeded in determining the Babuška–Aziz constant from a very simple equation. We can effectively utilize these results to give upper bounds of the a priori and a posteriori error estimates of finite element solutions based on the P₁ and/or P₀ approximate functions. To obtain more clear picture for the dependence of the

Acknowledgements

The authors would like to express their deepest appreciation to Prof. M.T. Nakao of Graduate School of Mathematical Sciences, Kyushu University and Prof. N. Yamamoto of Department of Computer Science, The University of Electro-Communication for acquainting them with the importance of the present problem and a number of references. The estimation $C_{4} ⩽ \sqrt{12}$ was obtained through fruitful discussion with Prof. A. Kaneko of Ochanomizu University, who presented an estimation method in his textbook:

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Estimation of interpolation error constants for the P0 and P1 triangular finite elements

Abstract

Introduction

Section snippets

Preliminaries

Dependence of constants on θ

Dependence of constants on α

Determination of C0 and C1

Asymptotic behaviors of constants as α→+0

Nonconforming P1 triangle

Numerical results

Concluding remarks

Acknowledgements

A Posteriori Error Estimation in Finite Element Analysis

Computable finite element error bounds for Poisson’s equation

IMA J. Numer. Anal.

On the angle condition in the finite element method

SIAM J. Numer. Anal.

The Finite Element Method and Its Reliability

A comparison of finite element error bounds for Poisson’s equation

IMA J. Numer. Anal.

Adaptive Finite Element Methods for Differential Equations

The Mathematical Theory of Finite Element Methods

The Finite Element Method for Elliptic Problems

Determination of the Babuška–Aziz constant for the linear triangular finite element

Japan J. Ind. Appl. Math.

Numerical Methods for Elliptic and Parabolic Partial Differential Equations

Estimation of interpolation error constants for the P₀ and P₁ triangular finite elements

Dependence of constants on $θ$

Dependence of constants on $α$

Determination of C₀ and C₁

Asymptotic behaviors of constants as $α \to + 0$

Nonconforming P₁ triangle