Matching boundary data exactly in an elliptic problem avoids one of Strang's "variational crimes". (Strang and Fix (1973)). Supporting numerical evidence for this procedure is given by Marshall and Mitchell (1973), who considered the solution of Laplace's equation with Dirichlet boundary data by bilinear elements over squares and measured the errors in the L2 norm. Then Marshall and Mitchell (1978) obtained some surprising results: for certain triangular elements, matching the boundary data exactly produced worse results than the usual procedure of interpolating the boundary data

Barnhill, RE

Brown, JH

Gregory, JA

Mitchell, AR

English

Brunel University Research Archive

                     TR/01/81 EXACT AND APPROXIMATE BOUNDARY DATA INTERPOLATION IN THE FINITE ELEMENT                METHOD    by R . E . B a r n h i l l ,  J . H . B r o w n ,  J  .  A .  G r e g o r y  a n d  A . R . M i t c h e l l  Exact and Approximate Boundary Data Interpolation in the Finite Element Method        R.E. Barnhill* J.H. Brown ** J.A. Gregory *** A.R. Mitchell ** * Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, U.S.A. Science Research Council Senior Visiting Fellow at Institute of Computational Mathematics and Department of Mathematics, Brunel University, during 1979-80.   ** Department of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland. *** Department of Mathematics, Brunel University, Uxbridge, Middlesex, UB8 3PH, England. Table of Contents 1. Introduction 2. Finite Element and Interpolation Remainder Theory 3. Rectangular Elements 4. Triangular Elements 5. Conclusions       References                    ACKNOWLEDGMENTS R. E. Barnhill 's  research was supported by a sabbatical leave from the University of Utah, by the National Science Foundation with Grant MCS 7801966 at the University of Utah and by the Science Research Council with Grant GR/A/78515 at Brunel University. J.  H. Brown's research was supported in part by the same NSF grant.  J.  A. Gregory's was supported by the Science Research Council with Grant GR/A/92238. -1- 1 . IntroductionMatching boundary data exactly in an elliptic problem avoids one of Strang's "variational crimes". (Strang and Fix (1973)).  Supporting numerical evidence for this procedure is given by Marshall  and Mitchell  (1973), who considered the solution of Laplace's equation with Dirichlet boundary data by bilinear elements over squares and measured the errors in the L2 norm. Then Marshall  and Mitchell  (1978) obtained some surprising results:  for certain triangular elements,  matching the boundary data exactly produced worse results than the usual procedure of interpolating the boundary data. This paper provides a partial analysis of these unusual results.  We find that the best approximation properties of the Rayleigh-Ritz- Galerkin approximation in the energy norm are true for both the exact and discretized schemes. We analyze the interpolation remainders for certain rectangular and triangular schemes and we find no direct connection between these and the finite element remainders.  A new triangular element that improves the interpolation remainder theory is given. Numerical examples are also included. -2-   2.  Finite Element and Interpolation Remainder Theory    We consider Poisson's equation —Δu = f on polygonal domain D with Dirichlet boundary data u = g on ∂D. The corresponding weak problem is to find a function u such that                        (2.1) )D(10Hgu)D(10HVallfor)v,f()v,u(a∈−∈=where a(.,.) is the usual semi-definite energy  inner product )10H(and {.,.) is the L2 inner product. The exact boundary data matching Rayleigh-Ritz-Galerkin (RRG) approximation UE satisfies          (2.2) )D(10Hi0Svallfor)v,f()V,EU(a ⊂>φ<=∈=∑ φ+= iiEUEGEUWhere  and GE = g on ∂D. Here we use local elements, e.g., the functions Φ i .  can be piecewise bilinear elements with the corresponding "transfinite" function GE(see Sections 3 and 4 for specific examples). The corresponding discrete boundary data matching RRG approximation UD satisfies             (2.3) 0Svallfor)v,f()V,DU(a ∈=∑ φ+= iiDUDGDUWhere  and GD approximates g on ∂D. Let |  .  |  = [a(.,.)]½ be the usual energy semi-norm. Then the following  -3- best approximation properties hold: |u - UE| ≤  |u - ũE| |u - UD| ≤  |u - ũD| (2.4) where     ũE, = GE + Σ  a i  Φ i .  and ũD = GD + Σ  bi  Φ i .  for arbitrary {ai} and {bi} .  Boundary layer effect: Since GE = g on ∂D, then UD can be considered as a RRG approximation to UE, which implies the best approximation property |UE - UD| ≤  |  UE- ũD|  .  (2.5) The possibility    ∑+= toleadsiφiEUDGDu~|UE - UD|  ≤  |GE - GD| (2.6) which is an upper bound on the difference between the exact and discrete RRG approximations, but of course gives no guide as to which of UE and UD is the better approximation to u. For the local schemes considered in this paper, the right hand side of (2.6) involves an integral over the "boundary layer" of elements next to the boundary. Interpolation remainder theory is linked to RRG approximations by the best approximation property. More specifically, ũE and ũD can be chosen to be the interpolants to u from the appropriate approximating sets that contain UE and UD , respectively. For this paper we obtain bounds on |u - ũE| and |u - ũD |  for various local interpolation schemes. In addition, we give numerical results for |u - UE |  and |u - UD |  for some model problems. -4- Theoretically, the result    |u -  ũE|  ≤  |u -  ũD|  (2.7) does not guarantee|u - UE|  ≤  |u- UD|  (2.8) However, our numerical results indicate that (2.7) does imply (2,8).  More surprisingly, even in one case where |u - ũD|  ≤  |u -  ũE|  we find that (2.8) holds. The variational crime of not matching the boundary data exactly provokes a question of norms. The energy semi-norm | . |  = [a(. , .)½  is not a norm on H1 .  By application of the Sobolev Imbedding Theorems, it  can be made into a norm by the definition                                  (2.9).ds2|.|Da(.,.)2||.|| ∫∂+=For this norm,            .ds2EUuD)EU~u,EUu(a2EUu −∂+−=− ∫     = a(u-UE, u-UE)             .ds2DUuD)DU~u,DUu(a2EUu −∂+−=− ∫                .ds2DGgD)DU~u,DUu(a −∂+−= ∫  The best approximation properties stil l  hold in this new norm. (2.10) -5- 3. Rectangular ElementsWe consider piecewise bilinear interpolation over a union of squares. Let Bu be the usual bilinear interpolant over [0,1] × [0,1] and Ju be the semi-discretized interpolant over [0,1] × [0,1] with the x-axis being part of the boundary of the region, that is,  Ju = (1 -y) u(x,0) + (1 -x)y u(0,l) + xy u(l, l)      = Bu + (1 - y) [u(x,0) - (1 -x) u(0,0) -x u(l ,0)] (3.1) So the identifications with the general ũE and ũD  are ũE = Ju and ũD = Bu in this Section. From (3.1) we obtain                  (3.3)β2yBu][u2yJu][u(3.2)α2xBu][u2xJu][u+⎥⎦⎤⎢⎣⎡∂−∂=⎥⎦⎤⎢⎣⎡∂−∂+⎥⎦⎤⎢⎣⎡∂−∂=⎥⎦⎤⎢⎣⎡∂−∂  For suitable functions α  and β .  We show that                                   (3.4)0dydx)(1010<β+α∫∫and hence that |u -  Ju|  ≤  |u -  Bu| (3.5)  Marshall (1975) used a truncated Taylor expansion to estimate the functions α  and β .  In order to compute α  and β  exactly, we first used a Sard kernal analysis similar to that of Barnhill  and Gregory (1976). However, a simpler analysis can be used in some cases. If we show that (3.4) is true whenever u(x,y) is a monomial xm yn ,  m, n ≥  0, then the continuity of the energy norm suffices to prove (3.4) for all  -6- relevant u, from which the desired result (3.5) follows. Notice that B(xm yn ) -  J(xm yn ) = xm yn for 0 ≤  m, n ≤  1 and B(xm yn) =J(xm yn) ≢  xmyn for m > 1 and n > 0. Hence  B(x m  y n )  and  J (x m  y n  )  d i sag ree  on ly  fo r  m  >  1  and  n  =  0 .  Ca re fu l  u se  o f  t he  de f in i t i ons  o f  α  and  β ,  ( 3 .2 )  and  (3 .3 )  r e spec t ive ly ,  app l i ed  to  u (x ,y )  =  x m  ,  even tua l ly  y i e lds  the  fo l lowing :              )1m2()2m(2)1m(3210101m22)1m(321010++−−=β−−−=α∫∫∫∫  so that               )1m2()1m2()2m()3m32m2(2)1m(32)(1010 +−+++−−=β+α∫∫          (3.7)which is negative for m ≥  2. As mentioned earlier,  numerical verification of this theoretical result was given by Marshall and Mitchell (1973)and by Marshall (1975). (See also Mitchell and Wait (1977).) It  should be remembered of course that these authors measured errors in the L2 norm and not in the energy semi-norm used in the proofs in this paper. - 7 -  4 .  T r i a n g u l a r  E l e m e n t sM a r s h a l l  a n d  M i t c h e l l ' s  ( 1 9 7 8 )  r e s u l t s ,  t h a t  m a t c h i n g  t h e  b o u n d a r y  d a t a  e x a c t l y  i n  a  t r i a n g u l a t e d  r e g i o n  p r o d u c e s  w o r s e  r e s u l t s  t h a n  i n t e r p o l a t i n g  t h e  b o u n d a r y  d a t a ,  w e r e  q u i t e  s u r p r i s i n g .  T h e y  c o n s i d e r e d  t w o  C o  t r i a n g u l a r  s c h e m e s :  t h e  " s i d e  v e r t e x "  m e t h o d  a n d  t h e  " N i e l s o n "  m e t h o d .  ( T h e  s i d e - v e r t e x  m e t h o d  i s  a l s o  c a l l e d  t h e  " r a d i a l "  m e t h o d  a n d  t h e  N i e l s o n  s c h e m e  i s  a n  i n s t a n c e  o f  a  p o l y n o m i a l  b l e n d e d  t r i a n g u l a r  i n t e r p o l a n t .  S e e  B a r n h i l l  ( 1 9 7 7 ) ,  p .  1 0 1  f .  f o r  a d d i t i o n a l  i n f o r m a t i o n  a b o u t  s u c h  t r i a n g u l a r  i n t e r p o l a n t s . )  H e r e  w e  g i v e  a n  a n a l y s i s  o f  t h e  N i e l s o n  s c h e m e .  N i e l s o n  S c h e m eL e t  N u  b e  t h e  s e m i - d i s c r e t i z e d  N i e l s o n  i n t e r p o l a n t  o n  t h e  s t a n d a r d  t r i a n g l e  T  w i t h  v e r t i c e s  ( 1 , 0 ) ,  ( 0 , 1 )  a n d  ( 0 , 0 ) .  T h i s  i n t e r p o l a n t  p i c k s  u p  t h e  f u n c t i o n  v a l u e s  a t  ( 0 , 1 )  a n d  a l o n g  t h e  l i n e  s e g m e n t  y  =  0 ,  0  ≤  x  ≤  1 ,  a n d  i s  g i v e n  b y  N u  =  x u ( l , 0 )  +  y u ( 0 , l )  +  ( 1  - y )  u ( x , 0 )  -  x u ( l  - y , 0 )  .  ( 4 . 1 )  I t s  c o m p l e t e  d i s c r e t i z a t i o n  i s  t h e  C o u r a n t  l i n e a r  i n t e r p o l a n t  L u  =  x u ( l , 0 )  +  y u ( 0 , l )  +  ( 1  - x - y )  u ( 0 , 0 )  .  ( 4 . 2 )  ( S o  ũ E  =  N u  a n d  ũD  =  L u  h e r e ) .  B y  r e a s o n i n g  a s  i n  S e c t i o n  3 ,  w e  c o n s i d e r  t h e  e q u a t i o n       ∫ ∫ β+α+−=− dydx)(T2Luu2Nuu             ( 4 . 3 )f o r  t h o s e  m o n o m i a l s  f o r  w h i c h  N u  a n d  L u  d i s a g r e e ,  n a m e l y  u  =  x m  ,  m  ≥  2. -8 -    We even tua l ly  ob ta in   )4.4()1m2)(2m()2m2()1m2(3)12m32m2()1m(2)1m(2dydx)(Tand)2m()2m2()1m2(3)6m9m2(2)1m(Tand)2m()2m2()2m2()3m2(2)1m(T∫ ∫∫ ∫∫ ∫−+++−++−−=β+α+++++−−=β++++−−=α which  i s  pos i t ive  fo r  a l l  m ≥  2 .    Th i s  seems  to  ind ica te  tha t  the  exac t  match ing  t echn ique  i s  worse  in  th i s  case ,  a  resu l t  conf i rmed  numer ica l ly  by  Marsha l l  and  Mi tche l l  us ing  the  L 2  norm.  However ,  we  ob ta in  the  surpr i s ing  numer ica l  r esu l t  (Tab le  4 .1 )  tha t ,  in  the  energy  norm,   | u  -  U E |  <  | u  -  U D |  (4 .5 )   fo r  the  p rob lems  computed .  The  Barnh i l l ,  B i rkhof f ,  Gordon  (BBG)  t r i angu la r  e lement  was  a l so  cons idered .  The  in te rpo la t ion  remainder  theory  p roduced  no  def in i t e  conc lus ions .  The  numer ica l  r esu l t s  in  Tab le  4 .1  ind ica te  tha t  exac t  match ing  i s  usua l ly  be t t e r  fo r  th i s  scheme.  -9 -  Tab le  4 .1    RRG Errors  in  Semi-Norm )10H(-Δu  =  0   on   [0 ,1 ]  ×  [0 ,1 ]  =  R u  =  g   on   ∂R  Prob lem 1 :  u  =  s in  2x  e - 2 y    P rob lem 2 :  u  =  In  r  -  2       where  r 2  =  (x -0 .437) 2  +  (y  +  0 .3 ) 2h   Couran t  L inear  Nie l son  BBG Couran t  L inear  Nie l son  BBG 21   0 .4544  0 .3965  0 .4070  0 .53861  0 .51558  0 .51962  41   0 .2364  0 .2098  0 .2124  0 .30157  0 .27193  0 .27575  81   0 .  1195  0 .1115  0 .  1121  0 .15732  0 .  14858  0 .14956  116  0 .05989  0 .05776  0 .05791  0 .07  966  0 .07737  0 .07761    P rob lem 3 :  u  =  e x  cos  y     P rob lem 4 :  u  =  s in  4x  e  - 4 y h   Couran t  L inear   Nie l son   BBG  Couran t  L inear   Nie l son   BBG 21  0 .4055  0 .3324  0 .3427  1 .2088  1 .1938  1 .2149  41  0 .1815  0 .1567  0 .1595  0 .6649  0 .6337  0 .6416  81  0 .0714  0 .0625  0 .0634  0 .3053  0 .2873  0 .2897  161  0 .0266  0 .0235  0 .0237  0 .1244  0 .1175  0 .1182  -10-  Improved  Nie l sonWe now c rea te  an  " improved"  Nie l son  e lement  N*  u  =  P 1  u  +  P 2  u  -  P 1  P  2  u  (4 .6 )  where  P 1  u  =  yu(x , l  -x )  +  xu( l  -y ,y )  P 2  u  =  u (x ,0 )  +  u (0 ,y )  -  u (0 ,0 )  +  4xy{u(21 ,21 )  -  u (21 ,0 )  -  u (0 ,  21 )  +  u (0 ,0 )}  P  1  P  2  u  =  y  {  u  (  x  ,  0  )  +  u  (  0  ,  1  -  x  )  -  u (0 ,0 )  +  4x  (1  -x )Cu(21 ,21 )  -  u (21 ,0 )  -  u (0 ,  21 )  +  u (0 ,0 ) ]}  +  x{u( l  -y ,0 )  +  u (0 ,y )  -  u (0 ,0 )  +  4y( l -y ) [u (21 ,21 ) -u (21 ,0 )  -  u (0 ,  21 )  +  u (0 ,0 ) ]}  N i s  exac t  fo r  the  func t ion  xy  and  N i s  no t  We d i sc re t i ze  N*u  l inea r ly  a long  x  =  0  and  quadra t i ca l ly  a long  y  =  1  -  x  to  ob ta in  J*  u  =  (1  -y )  u (x ,0 )  -  xu( l  -y ,0 )  +  2xy(x  +  y -  1 )  u (0 ,0 )  +  y ( l -2x)  u (0 , l )  +  x{y(2x-  1 )  +  (1  -  2y)  (1  -  y )  }  u ( l  , 0 )  +  4xy  (1  -x -y)  u (21 ,0 )  +  4xyu(21 ,21 )  (4 ,7 )  The  fu l ly  d i sc re t i zed  N*  i s  B*  u  =  (1  -x -y)  u (0 ,0 )  +  y ( l  -  2x)  u (0 , l )  +  x ( l  -2y)  u ( l ,0 )  +  4xyu(21 ,21 )  (4 .8 )  - 1 1 -   *  *  F i g u r e  4 .  1  S t e n c i l s  f o r  J *  u  a n d  B *  u  T h e  m o n o m i a l s  f o r  w h i c h  J *  u  a n d  B *  u  d i s a g r e e  a r e  x  m  ,  m  ≥  2 .  I n  t h e  c a s e  m  =  2 ,  w e  o b t a i n  (α  +β  )  d x  d y  <  0 ,  w h i c h  a g r e e s  ∫ ∫w i t h  t h e  r e s u l t s  o b t a i n e d  f r o m  t h e  t r u n c a t e d  T a y l o r  e x p a n s i o n  a n a l y s i s .  S u p p o r t i n g  n u m e r i c a l  e v i d e n c e  i s  g i v e n  i n  T a b l e  4 . 2 ,  H o w e v e r ,  f o r  l a r g e  m ,  α  =  -  0 ( 1 )  a n d   β  =  0 ( m )  s o  t h a t  ∫ ∫ ∫ ∫t h e  a n a l y s i s  b e c o m e s  i n c o n c l u s i v e .  -12-  Tab le  4 ,2  RRG Er rors  in Semi-Norm 10HJ*u is  the  exac t  b i l inea r  cor responding  to  (4 .7 )  and  B*ui t s  d i sc re t i za t ion  in  (4 .8 ) .  The  same  four  mode l  p rob lems  a re  used .   P rob lem 1   P rob lem 2   h  B*u  J*u  B*u  J*u  21  0 .31372  0 .20391  0 .36198  0 .27907  41  0 .15616  0 .12665  0 .19811  0 .16065  81  0 .078001  0 .070180  0 .09938  0 .08793  116 0 .038991  0 .036953  0 .04972  0 .04649   P rob lem 3   P rob lem 4    h   B*u    J*  u          B*u    J*  u  21  0 .31187  0 .21452  0 .73338  0 .50324  41  0 .15566  0 .13115  0 .38091  0 .28547  81  0 .07780  0 .07146  0 .19098  0 .16192  161  0 .03889  0 .03726  0 .09552  0 .08752  - 1 3 -  5 .  Conc lu s ions( i )  E x a c t  ma t c h i n g  u s u a l l y  p r o d u c e s  b e t t e r  n u me r i c a l  r e s u l t s  i n  t h e  e n e r g y  s e mi - n o r m.  I f  t h i s  s e mi - n o r m i s  ma d e  i n t o  a  n o r m,  t h e  e x a c t  ma t c h i n g  t e c h n i q u e  w o u l d  b e  e v e n  b e t t e r .  ( i i )  E x a c t  ma t c h i n g  n e e d  n o t  b e  b e t t e r ,  i n  t h e  L 2  n o r m.  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Exact and approximate boundary data interpolation in the finite element method

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