UTMS 2006–17

August 4, 2006

Estimation of interpolation error constantsfor the P0 and P1 triangular finite elements
byFumio Kikuchi and Xuefeng LiuÌ

UNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPANEstimation of Interpolation Error Constantsfor the ¼ and ½ Triangular Finite Elements
Dedicated to Professor Ivo Babuška on the occasion of his 80th birthday

Fumio KIKUCHI , Xuefeng LIU
Graduate School of Mathematical Sciences, University of Tokyo
3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan

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, wemainly analyze the conforming case, but some results are also given for the non-conformingcase. 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  ,so that the solution can be numerically obtained with desired accuracy and verification. Suchhighly 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 andnumerical verification of finite element solutions.
Keywords : FEM, error estimates, triangular element, Babuška-Aziz constant, interpolation
error constants.
Mathematical Subject Classification 2000 : 65N15, 65N30

1 Introduction
The finite element method (FEM) is now recognized as a powerful numerical method for wideclasses 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  and norms (or seminorms), but it has been very difficult to evaluate
such constants explicitly. For quantitative purposes, however, it is indispensable to evaluate orbound them as accurately as possible, because sharper estimates enable more efficient finite
element computations. Thus such an evaluation has become progressively more important andhas 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 Corresponding author. Tel.: +81 3 5465 8345; fax: +81 3 5465 7011; e-mail: kikuchi@ms.u-tokyo.ac.jp

1

will give a few fundamental results on some interpolation error constants of the most populartriangular finite elements.
More specifically, we give some results on interpolation error constants appearing in the
popular  (piecewise constant) and  (piecewise linear) triangular finite elements. Essentially based on the paper of Babuška-Aziz [3], we analyze the dependence of several constantson the geometric parameters such as the maximum interior angle and the minimum edge lengthof the triangle more quantitatively than in [3]. Above all, the optimal constant (  in this paper) appearing in the   error estimate of the  interpolation of   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( in this paper) presented and effectively used by Babuška and Aziz in conjunction with the
maximum angle condition [3]. More precisely,   gives an upper bound quite close to the optimal constant  , and the relation between  and  was further discussed in [13, 18]. Thusa precise estimation of these two constants is very important, and a number of researchers havegiven 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 theinterpolation error constants for the non-conforming   triangle, as will be mentioned later.
For the above Babuska-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 Laplacianover the above triangular domain, we can show that the constant can be easily determined froma solution of the simple transcendental equation  . In this process, we use thereflection (or symmetry) method [16]. Moreover, we have obtained some explicit relations forthe 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 andanalytical results, the latter of which are based on asymptotic analysis. Thus our results canbe effectively used in the quantitative a priori and a posteriori error estimations of the finite
element solutions by the   triangular element and also those based on the  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   triangle by using the constantsfor the  and the conforming  triangles.

2 Preliminaries
Let ,  and  be positive constants such that







 







 

(1)

Then we define the triangle  by  with three vertices    ,   and      . From (1),  is shown to be the edge of maximum length, i. e.  , so that    here denotes the medium edge length, although the notation  is often
used as the largest edge length. A point on the closure of  is denoted by     . Byan appropriate congruent transformation in  , we can configure any triangle as  . As theusage in [3], we will use abbreviated notations    ,  and   (Fig. 1).
,
where
the
subscript
Let us denote the norm of    by 
 is often omitted.







Ê


2



             

   ½

  

 



 





 ½

 ¾
  



Figure 1: Notations for triangles :  

 ,

 ,

 :Let us define the following closed linear spaces for functions over   

 



   

 



   

 



   

 



 

    (2)     

(3)

                 

(4)
(5)

where     and     are respectively the first- and second-order Sobolev spaces
for real square integrable functions over  . For the above four spaces, we will again use
abbreviated notations           and     . Moreover, thespaces of constant functions and (at most) linear functions over  are respectively denotedand  .
byLet us consider the usual  interpolation operator   and  one   for functions
well-defined byon  [7, 8, 10] :    for      is a function in    
while    for    

 

 is a function in 

(6)

 such that           
3



(7)

To give error estimates for these interpolation operators, it is natural to evaluate the positive
constants defined by                         
¿







    







(8)





 ¿     



(9)



where        , and              . The existence of these constants easily follows from the standard compactness arguments. We willagain use abbreviated notations         ,       and    
for   .
By a simple scale change, we find that             and
         . These relations and constants are used to evaluate interpolation
errors for functions on  . That is, we can easily have the popular interpolation error estimates [7, 8, 10] :                           

 



(10)

 



(11)

 



(12)

where we have used the fact that       for      and      
for 
    . Moreover, for the partial derivatives of  (  ; right
triangle case), we have


 



½ ¾     





 

   (13)

which are in a sense sharper than (11), cf. [10]. These relations follow from the facts that        for    . It is to be noted that, for   , the results
still hold if   is replaced with    for each of     and the partial derivative for   is done with the directional derivative of  in  direction.
Thus we can give quantitative interpolation estimates if we succeed in evaluating or bounding the constants    ’s explicitly. So we will give upper bounds of these constants as fairly
simple functions of  and  . Notice here that each of such constants can be characterized byminimization of a kind of Rayleigh quotient. Then it is equivalent to finding the minimum
eigenvalue of a certain eigenvalue problem expressed by a weak formulation, which is further
expressed by a partial differential equation with some auxiliary conditions.
For later purposes, let us explain the cases of    and    as examples. From(8),    is characterized by using a kind of Rayleigh quotient :      
 



¼



4

(14)

where all notations and quantities are for  . The infimum in the right-hand side is actuallyanda minimum, and it is the smallest eigenvalue of the eigenvalue problem : Find    that satisfy

Ê

 

            
(15)Here,     denotes the inner products of both    and     , and  is the gradient

operator. The present eigenvalue problem is also expressed in terms of a partial differential
equation, the linear constraint for   and the boundary condition [13, 14]:      

 









(16)

where  denotes the outward normal derivative on edges, and   does the boundary of . The above boundary condition is the homogeneous Neumann one, and the desired minimum eigenvalue is also the second (and positive) one for the same problem without the linear
constraint.
For    , it is characterized in essentially the same fashion as (14) and (15), but the associated space   must be replaced with   . On the other hand, the equations correspondingto (16) become more complicated [13, 14]:        on edges  and  on edge (17)where  denotes an unknown constant to be decided with  and .
The other constants are characterized similarly, but the partial differential equations related
to    and    are of fourth order and are more difficult to deal with than the second
order equations like above, cf. [2, 5]. Since  is a triangle, it is in general difficult to solve
such eigenvalue problems explicitly. However, in certain special cases, we can achieve suchaims as we will see later.

3 Dependence of constants on
This section is devoted to analysis of the effects of the maximum interior angle  on     ’sfor fixed . For    , 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     .
To this end, let us introduce the following simple affine transformation between   : and    









           

(18)This transformation is a bit different from that in [3]. By eigenvalue analysis of matrices
resulting from the above transformation in the Rayleigh quotients like (14), we obtain thefollowing results.
5

Theorem 1. For   , it holds for each      that        

where



  

 

    







 



  








(19)

  



(20)

Remark The function form for   is consistent with the maximum angle condition in[3], since   is bounded on  Æ  for each sufficiently small Æ  . Notice alsothat     for  , as will be shown in the subsequent section. The other’sareuniformly
bounded on  . Moreover, the corresponding result for    by
Natterer [15] is expressed in terms of  ,    and  as







                 (21)This estimation is, however, not consistent with the maximum angle condition. In fact, theright-hand side of the above diverges toas  . When  , our formula for    
is numerically comparable to Natterer’s, even when   in (19) is replaced with  . In
particular, when   , (21) is identical to (19) for   .







Proof. We will use the coordinate transformation (18) between  and . By simple calculations, we have for       under the present transformation :









 


      



 






where  and  are assumed to be sufficiently smooth. Then we can easily derive



 



 





















 

 










Moreover, the Jacobian of the present transformation is evaluated as        . From these estimates, we have                   !
where     , for example, denotes    for  . The results for      are now easy toobtain by using the above and the definitions of the constants     ’s.
Similarly, we obtain


 


 

              



6





        







           Let us consider the following real symmetric matrix related to the quadratic form in the righthand side of the above expression :





 
 














We can see that this has three eigenvalues so that we have the estimates      







 


   




 







  and     ,   

As ! , we have now



 





          

Applying ! and " to the definitions of the constants, we have the results for  ".

4 Dependence of constants on 
Up to now, we have given some basic results for dependence of error constants on  and  .
In this section, we will consider the dependence of such constants on  when   . With
this regard, we owe much the following results to the analysis by Babuška and Aziz [3]. In below is an important consequence derived in [3] andparticular, the estimation  
also in [13, 18], and so we here call  the Babuška-Aziz constant.

 

Theorem 2. For    and   ,     are continuous positive-valued
functions of   (here we consider also for   ). In addition, except for   ,they are monotonically increasing in . Thus,            
On the other hand, it holds for   and    that          

(22)

(23)

Proof. We just give sketches since the arguments employed here are standard. It is convenient
to consider over the common domain by applying a simple coordinate transformation in [3]to . For the continuity, we first show the uniform boundedness over compact intervals,
which assures the existence of    # and    # for each   . Then we canprove the continuity by adopting the weakly lower semi-continuity of   -norm and the Rellich
compactness theorem. The monotonicity and (23) can be concluded completely as in [3].

7Consequently, we can give  and  interpolation error estimates in terms of   and . That is, from the preceding considerations, we have       ,        and         , so that (10) through (12) become







                                   

(24)
(25)
(26)

These may be rough but are still correct upper bounds. As was already noted, such error
bounds are available for triangles of general configuration by applying appropriate congruent
transformations [3, 7, 8, 10].
Thus we can obtain a quantitative error bound for  interpolation of    and those for  interpolation of     , provided that numerical values or concrete
upper bounds of  ,    and  are known. Rough upper bounds of these constants
 (seecan be given even by manual calculation [5, 15]. For example, we found that  
Acknowledgements of this paper). To obtain accurate upper (and lower) bounds, however,
we need numerical computations with verification. Quite fortunately, we can get exact valuesfor  and  (Babuška-Aziz constant) as will be shown in the subsequent section. An upperbound for  was first given by Natterer [15]. By numerical computations without verification,  [2, 11, 18]. The relation between  and  was fully
it is now known that 
discussed in [18] and [13], and in certain cases  is more essential than  itself as wealready noted, cf. [10]. We should also mention that   was verified numerically in [13, 14]with estimate     . Thus 0.493 or so is a nice upper bound to   for most ofpractical purposes. In fact, 0.5 is recommended in [18] for use as an upper bound for   .









5 Determination of

 and



First let us determine  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  .
Theorem 3. With regard to  , i.e.,    for    and   , it holds that   .

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 .







Ê





be a unit square domain :         . Let  be an arbitrary eigenpair of (15) or (16) for      , anddefine the (symmetric) extension  of  to by reflection with respect to the line     :1Æ . Let  Ê        if     

We can find that   

  

 

  is an eigenpair of the eigenvalue problem for :           8

 if 







where denotes the inner products of  



 

 

and  , and  is defined by     Conversely, any eigenpair of  with  restricted to satisfies (15), if  is symmetric withrespect to the line    . Notice here the orthogonal decomposition of  in    aswell as in   :     subspace of symmetric functions in   subspace of antisymmetric functions in Consequently, for the present purposes, it suffices to deal with  in  .

2Æ . As is well known, a complete system of functions for    is given by the totality of(orthogonal) eigenfunctions of (c) with  replaced with the whole    :  



$ 



  $     

Since we are interested in symmetric eigenfunctions only, we should make a complete system,of symmetric functions in    from the above : for $  $     %     







 



 

These are orthogonal in   , and also orthogonal with respect to the bilinear form (and in    ). More important to note is that all %  ’s for $  except for % belongto  and are eigenfunctions of  . Thus the desired eigenvalueisassociated to % ,
  .
and hence   

Next we determine    . See also [9] for a slightly different approach.
Theorem 4. The minimum eigenvalue  associated to    is equal to the minimum
positive solution of the transcendental equation for :The concrete value of !





(27)

 can be obtained numerically with verification. For example, we find , and hence     is bounded as    

(28)



Remark Numerical computation without verification gives     " . The
present transcendental equation can be commonly seen in vibration analysis of strings withspecial boundary conditions [16].
Proof. The use of reflection and trigonometric functions is common to the proof of the preceding theorem.

9  Ê   

1Æ . Let be the same as before. Let   be an arbitrary eigenpair of (17) for, and define the symmetric extension  of  to by reflection. Then is an eigenpairof the eigenvalue problem for :

where   is defined by        

 





 

             &Conversely, any eigenpair of  with  restricted to satisfies the weak form of (17), if  issymmetric with respect to the line     . Notice here the orthogonal decomposition of  in    as well as in   :     where  and  are respectively the symmetric and antisymmetric subspaces of   . Consequently, for the present purposes, it suffices to deal with  in  .



2Æ . We use the complete system of symmetric functions %  ’s for $  $     in    defined in the proof of the preceding theorem. From & , the condition for a symmetric     to belong to  is expressed by

!




   !   for  ! % with






$

 !

where ! ’s are real coefficients, and we can show the series    ! is absolutely
convergent under the conditions imposed on the coefficients. Eliminating ! by the above
equation,   is expressed by

 




! %

  







! %

' 

Clearly, % ’s for $  are eigenfunctions of  with the homogeneous Neumannboundary condition, and the minimum of the associated eigenvalues is   .
3Æ . Taking notice of (f),  is expressed by the direct sum  (  (where ( = closure of linear combinations of %    $   and ( = closureof linear combinations of % $   . Here, ( and ( are orthogonal to each other inboth   and    , and moreover, from the observation in 2Æ , all the eigenfunctions in (are known. Consequently, our aim will be attained if we obtain the minimum of eigenvalues
associated with eigenfunctions in (  . If it is smaller than   , the obtained one is nothing but
the desired eigenvalue  .

 

10

4Æ . Let us now solve the eigenvalue problem (d) in (  by expressing 

 





! ) with

$ !(  as*

where )    %      $ $      $
Thus, using the theory of Fourier series with the inequality in * taken into account, must be of the form, for an unknown single-variable function *  * + ,   * 

* 



*   

* 

Æ.
(

Substituting the above into  , we have* +  * +



+





* + +  

Notice in this derivation that  in  can be taken from whole   so that (17) is available, since( is orthogonal to ( and  both in   and    . Solving this eigenvalue problem, we
obtain (27). Clearly, the minimum positive solution of (27) lies in the interval     , andis the unique solution there. It is surely smaller than   , and is exactly the desired eigenvalue   .
 . Moreover, an eigenfunction associated to  is 
 
 
5Æ . To obtain    numerically with verification, we can use various methods. Here
we just use a method based on modification of the equation +  +   for +   : Let us
find the minimum positive zero of' + #+ +     $  + +  
+  $  $The series appearing above is an alternating one, and the absolute value of each term for, monotonically for sufficiently large $. Moreover, ' + isfixed + converges to 0 as $monotonically decreasing for  +. Thus, as is well known in elementary calculus, we
can compute upper and lower bounds for the minimum zero + by utilizing appropriate partial
sums: ' + # partial sum up to the term of $  . It is to be noted here that, at least in
principle, all the computations can be performed in the finite-digit binary arithmetic without
computer errors, provided that + is a rational number. For example, by taking    ", we can+  since '   '    (even ) andbound +  as  !
'  !  '  !   (odd ).



6 Asymptotic behaviors of constants as ·¼

Moreover, we can analyze the asymptotic behaviors of the constants    ’s as ,cf. [12]. In particular, the right limit values     ’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,      is equal to the first positive zero of the Bessel function , - .
11

Table 1: Right limits of    :  i01

ODE’s foreigenvalue problems                         (Constraints and/or

Numerical valuesfor   

boundary conditions







( : unknown constant)        
3          (reduces to case :   )
4         2   )       

 ’s

0.26098     

0.32454  0.41583       

0.32454       

0.10790

For the analysis, we use various techniques including compactness arguments. We will
publish the detailed analyses and results elsewhere, since they become rather lengthy. Instead,
we list up the related ODE’s with the constraint and/or boundary conditions in Table 1.

7 Nonconforming  triangle
We have mainly considered the conforming  triangle, which can naturally construct subspaces of   space over the entire domain. But there also exists a non-conforming counterpart,which is also based on  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 theerrors induced by the interelement discontinuity of the approximate functions. Still we canobtain some results for the interpolation errors. The estimates shown below are based on thepreceding results for the usual  and  interpolations. They may be fairly rough, but can beused for some purposes. To give sharper estimates, we must introduce and analyze some new
constants.
We define here the non-conforming  interpolation operator  as follows: for    ,   is a function in  such that      %&'% &    





(29)

where  denotes the infinitesimal line element on edges.
Then we have the following results for      in terms of the constants introduced
for the original  and  interpolations :       
                      
12

(30)
(31)



To show (30), we use (29) and the Gauss formula to derive          for    . Then we can easily obtain (30) by noting the definition of    . To derive



(31), we should evaluate          and         . The
former can be evaluated by using    and    , while the latter can be done by (30).













8 Numerical results
We performed numerical computations to see the actual dependence of various constants on
 and . Here, we just show the results for   ,   and   by the  FEM with theuniform triangulation of the domain . In such calculations,is subdivided into a numberof small congruent triangles  with   . The penalty method in [18] was also
adopted to calculate   approximately. The resulting approximate problems are matrix
eigenvalue ones, and can be solved numerically if the linear constraint conditions imposed on
eigenfunctions are appropriately dealt with.
Figure 2 illustrates the graphs of approximate    ’s    versus   . The
exact value    at    is also included as a horizontal line. At   , the approximate
values coincide well with the exact one, and, for general , the monotonically increasing
behaviors of these functions are also well represented. The present numerical results suggestthat   is also monotonically increasing, but we have not succeeded in proving such a, the numerical results agree well with the exact limits
conjecture. Moreover, when given in Table 1 based on the asymptotic analysis.





Æ
Æ





Æ
Æ
Æ



Æ
 

Æ

Æ

Æ

Æ


¾ 
¿ 
½

Æ
Æ



















Figure 2: Numerical results for   ,   and  

13





As a simple example of application of our results, let us consider a kind of a posteriori   by  FEM. By Schultz [17] and many others, weestimate for approximation ofhave the following a priori error estimate for the approximation  to :  # )   (32)   
where  is a positive constant such that      for all  in the triangulation, and    in the triangulation. Since the function ) above is monotonically increasing,  

it has the inverse function. Thus we have the following a posteriori estimate for  :)  

  

(33)

Table 2 gives an application of (33) based on numerical results by the   FEM. Here, theemployed meshes are uniform ones composed of small triangles similar to the entire domain that are necessary to use (32) and (33) are also shown. The values of parameters  and in the table. We can observe that this simple method can actually bound  from both aboveand below. It is straightforward to apply it to give a posteriori estimates to general    .
By slight modification, it can be also used to bound     and    .

Table 2: A posteriori estimates for .

bounds for" 
 !"     (3! " "
 ("(  (   "(4
 !
 "(     (8
 " (!  (   !
16     !    !
32  " " !!     "(64  ("(
 !(  !        ((      
2  .

.   above   "bounds for 

149 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 offinite element solutions based on the   and/or  approximate functions. To obtain moreclear picture for the dependence of the interpolation error constants, we should also perform
various analyses including numerical analysis with verifications, asymptotic analysis etc. We
will continue such study, and more detailed results will be reported in due course.
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  was obtained through fruitful discussion with Prof. A. Kaneko of
Ochanomizu University, who presented an estimation method in his textbook : Introduction
to Partial Differential Equations (in Japanese), University of Tokyo Press, 1998. Although he
did not give any concrete values for upper bounds of   there, it is not difficult to apply his
method to obtain such values. Some of the exact computations in this paper were performed
with the aid of Mathematica R 5. This work was partially supported by Japan Society for
the Promotion of Science Grant-in-Aid for Scientific Research (C) (2) 16540096, and also by
Yoshida Scholarship Foundation for the second author.



References[1] M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis,
John Wiley & Sons, 2000.
[2] P. Arbenz, “Computable finite element error bounds for Poisson’s equation”, IMA J. Numer. Anal., 2, 475-479 (1982).
[3] I. Babuška and A.K. Aziz, “On the angle condition in the finite element method,” SIAM
Journal on Numerical Analysis, 13, 214-226 (1976).
[4] I. Babuška and T. Strouboulis, The Finite Element Method and Its Reliability, Clarendon
Press, 2001.
[5] R.E. Bahnhill, J.H. Brown and A.R. Mitchell, “A comparison of finite element error
bounds for Poisson’s equation”, IMA J. Numer. Anal., 1, 95-103 (1981).
[6] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Birkäuser, 2003.
[7] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 2ndedn., Springer, 2002.
15

[8] P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, 2002.
[9] F. Kikuchi and X. Liu, “Determination of the Babuška-Aziz constant for the linear triangular finite element”, Japan Journal of Industrial and Applied Mathematics, 23(1), 75-82(2006).
[10] P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial
Differential Equations, Springer, 2003.
[11] R. Lehmann, “Computable error bounds in finite-element method”, IMA J. Numer. Anal.,
6, 265-271 (1986).
[12] J.L. Lions, Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Springer, 1973.
[13] M.T. Nakao and N. Yamamoto, “A guaranteed bound of the optimal constant in the error estimates for linear triangular element,” Computing [Supplementum], 15, 163-173(2001).
[14] M.T. Nakao and N. Yamamoto, “A guaranteed bound of the optimal constant in theerror estimates for linear triangular elements, Part II: Details,” Perspectives on Enclosure Methods (eds. U. Kulisch et al.), the Proceedings Volume for Invited Lectures of
SCAN2000, Springer-Verlag, Vienna, p. 265-276, 2001.
[15] F. Natterer, “Berechenbare Fehlerschranken für die Methode der finite Elemente,” International Series of Numerical Mathematics, 28, p. 109-121, Birkhäuser, 1975.
[16] J.W.S. Rayleigh, The Theory of Sound, Vol. 1, Dover, 1945.
[17] M.H. Schultz, Spline Analysis, Prentice-Hall, 1973.
[18] G.L. Siganevich, “On the optimal estimation of error of the linear interpolation on atriangle of functions from (  (in Russian),” Doklady Akademii Nauk SSSR, 300(4),811-814 (1988).
[19] R. Temam, Numerical Analysis, D. Reidel Publishing Company, 1973.
[20] E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th edn., Cambridge
University Press, 199616

Preprint Series, Graduate School of Mathematical Sciences, The University of Tokyo
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