Gregory, JA

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             TR/56       1975 ERROR BOUNDS FOR LINEAR        INTERPOLATION ON TRIANGLES, by J. A. Gregory        This  paper  is  to  appear  in  the proceedings of  the conference on the Mathematics of Finite Elements and Applications held at Brunel University, April, 1975.  Error  Bounds for  Linear  Interpolat ion on Triangles  John A. Gregory Department of Mathematics, Brunel University  1.  Introduction. The existence of Sobolev norm bounds for interpolation remainders def ined on t r iangles  is  wel l  known [7,8] ,  as  is  the  re la t ionship of  such bounds to finite element analysis.   A constructive method for the derivation of such bounds is  proposed in a recent paper by Barnhill  and the author [3].  This method is used here to derive error bounds for l inear interpolation o n  a  t r i a n g l e  T  t o  f u n c t i o n s  d e f i n e d  i n  t h e  S o b o l e v  s p a c e  ( T ) .  I n  2Hparticular,  the constants  which are  involved in the bounds are estimated.  Knowledge of such constants can lead to computable finite element error bounds for second order elliptic problems defined on convex domains [5]. L e t  T  b e  t h e  t r i a n g l e  w i t h  v e r t i c e s  a t   ( 1 , 0 ) ,  ( 0 , 1 ) ,  a n d  ( 0 , 0 )  .  The Sobolev space (T)  is  the  space of  a l l  real  valued funct ions which,  2Htoge ther  wi th  a l l  the i r  genera l ized  der iva t ives  of  o rder  ≤  2  ,  a re  in  L 2  (T) .  A norm and semi-norm on (T) are respectively defined by 2H (1.1)   ,2/1)T(22Lj,iu2ji0)T(2Hu ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧≤+≤= ∑  (1.2)   ,2/1)T(22Lj,iu2ji)T(2Hu ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=+= ∑   Let linear interpolation remainders be defined by  (1.3) ⎪⎪⎪⎩⎪⎪⎪⎨⎧+−=∂∂=+−=∂∂=−−−−−=(0,0).u(0,1)ut)(s,0,1ut)R[u](s,tt)[u](s,0,1R(0,0),u(1,0)ut)(s,1,0ut)[u](s,Rst)[u](s,1,0R0),(0,ut)s(1tu(0,1)su(1,0)t)u(s,t)R[u](s,  -2-  T h e n  t h e  b o u n d s  o f  i n t e r e s t  f o r  )T(Hu 2∈  a r e  t h o s e  o n  t h e  L 2 ( T )  n o r ms  of (1.3).  The bounds which are given in this paper are derived by means of a piecewise defined Taylor expansion of .  This expansion is )T(Cu 2∈defined in Section 2 and the bounds are given in Section 3. The use of the func t ion  space  (T)  can  be  jus t i f ied  by  the  use  of  ' smooth '  o r  'mol l i f ied '  2Cfunctions from (T).  Finally,  in Section 4,  bounds are derived for an 2Harbitrary triangle. In particular,  i t  is shown that the bounds hold under the maximal angle condition that no angle of the triangle should approach .  This condition is derived by Synge [11] for functions in (T) and 2Cπis also the subject of recent studies [1,2,9].   2. A Taylor expansion in (T) . 2CLet  (a,  b) ∈  T  and let   A1 ,  A2 ,  and A3 be subsets of T defined by (2.1)   ⎪⎩⎪⎨⎧−>∈=−>∈=∪−=a},1yandTy)y)/(x,{(x,3Ab},1xandTy)y)/(x,{(x,2Awhere3,A2AT1Asee Figure 2.1.   -3-  Then  can be written in the piecewise defined form )T(Cu 2∈ (2.2) ,T)y,x(),y,x(u)y,x(3Ax)y,x(u)y,x(2Ax)y,x(u)y,x(1Ax)y,x(u ∈++≡   Where  (2.3)  ⎩⎨⎧ ∈= .otherwise0,iA)y,x(if1)y,x(iAX   For  each region A,  i  = l ,2 ,3 ,  s ingle  var iable  expansions of   are  )T(Cu 2∈made along parallels to the coordinate axes, see Figures 2.2 and 2.3.  These expansions are such that their arguments are contained in T for all (a, b) ∈  T and the following piecewise defined Taylor expansion of the function u(x,y) about the point (a, b) results:  (2.4) )b,a(u)by()b,a(u)ax()b,a(u)y,x(u 1,00,1 −+−+=           [ y~dx~d)y~,x~(uxaybx~d)b,x~(u)x~x(xa)y,x(1Ax 1,10,2 ∫∫+−∫+               ]y~d)y~,a(u)y~,y( 2,0yb∫+             [ x~ddy)y,a(ux~d'dx)x~1,'x(u)y,x(2Ax **1,1x~1bxa0,2x~axa−∫∫+−∫∫+      ]y~d)y~,a(u)y~y(x~dy~d)y~,x~(u 2,0yb1,1y x~1xa −∫+∫∫+ −              + [ y~dx~d)y~,x~(ux~d)b,x~(u)x~x()y,x(3Ax 1,1xy~1yb0,2xa −∫∫+−∫                  ]y~d'dy)'y,y~1(udydx)b,x(u 2,0y~byb**1,1y~1ayb −∫∫+∫∫+ −   (A detailed derivation of the generalized form of this expansion for  )T(Cu n∈  R e m a r k  1 .  T h e  f a c t  t h a t  ( a , b )  c a n  b e  a n  a r b i t r a r y  p o i n t  o f  T  i s  s i g n i f i c a n t  s i n c e  ( a , b )  =  ( s , t )  i s  p e r m i s s i b l e ,  w h e r e  ( s , t )  i s  t h e  p o i n t  o f  eva lua t ion  of  the  in te rpola t ion  remainders  (1 .3) .  This  choice  of  (a ,b)  enables bounds to be derived in Sobolev function spaces. This is observed -5-  by Birkhoff, Schultz, and Varga 6 , who use the Kernel Theory of Sard 10    to derive error bounds for Hermite interpolation on rectangles.  However, the theory of Sard is based on Taylor expansions which correspond to the case of (x,y) ∈  A1  in (2.4), and such expansions have a rectangular domain of i n f l u e n c e ,  s e e  F i g u r e  2 . 2 .  T h i s  p r e c l u d e s  t h e  c h o i c e  o f  ( a , b )  =  ( s , t )  i n  the application of Sard's  theory on triangles (although this theory can be applied with (a,b) = (0,0) ,   [2,4]).  Remark 2.  The remainder terms which involve u2 , 0  in the Taylor expansion (2.4)  are  constants  in  the var iable  y  and,  dual ly ,  the  terms which i n v o l v e  u 0 , 2  a r e  c o n s t a n t s  i n  t h e  v a r i a b l e  x .  T h i s  p r o p e r t y ,  w h i c h  i s  a  result  of  the development of the Taylor expansion along parallels to the coordinate axes, has an important consequence in the following section and leads to the maximal angle condition of Section 4.   3. Error bounds for linear interpolation on T . The application of the l inear interpolation remainders defined by (1.3) to the Taylor expansion (2.4),  with (a,b) = (s, t)  ,  gives the following representations:  (3.1) x~ddy)y,s(usx~d'dx)x~1,'x(us)t,s](u[R **1,1x~1t1s0,2x~s1s−∫∫−−∫∫−=       y~dx~d)y~,x~(1,1u0y~11ttx~dy~d)y~,x~(1,1u0x~11ss −∫∫−−∫∫−    y~d)dy'y',y~(10,2uy~t1ttdy*dxt),*(x1,1uy~1s1tt −∫∫−−∫∫−       y~dx~d)y~,x~(1,1u0s0tt)s(1x~dt),x~(2,0ux~0ss)(1 ∫∫−−−∫−+             ,y~d)y~(s,0,2uy~0tt)(1 ∫−+-6-  (3.2) x~d dy ) y , s ( ux~1t1sx~d'dx)x~1,'x(ux ~s 1 s )t , s ]( u [ 0 , 1 R * * 1 ,  1 0,2 −∫ ∫ − − ∫ ∫ − =            x~dt),x~(2,0ux~0sx~dy~d)y~,x~(1,1u0 x~11s ∫−−∫∫−         y~dx~d)y~,x~(1,1u0s0t ∫∫+ (  R 0 , 1  [u](s , t )  has  a  dual  representat ion to  (3 .2)) .  Equat ions (3 .1)  and (3.2) contain terms which involve only second derivatives since R and R1 , 0  annihilate the l inear terms in (2.4).  Also,  a consequence of the form of the Taylor expansion (2.4) and the form of the interpolation remainders (1.3) is t h a t  ( 3 . 2 )  c o n t a i n s  n o  t e r m  i n  u 0 , 2 .  T h i s  i s  b e c a u s e  R [ g ( y ) ] ( s , t )  i s  a  cons tan t  in  the  va r iab le  s  fo r  a l l  un ivar ia te  func t ions  g (y) ,  and hence     R1 , 0  [g(y)] (s, t)  = 0 (see Remark 2,  Section 2).  A generalization of this property forms the basis of the Zero Kernel Theorem in Barnhill and Gregory [2] The L2(T) norm of (3.1) and (3.2) is  now taken with respect to (s, t)  and the triangle inequality for norms is applied to the sum of terms of the right hand sides. For u  (T), the norms of the terms which result can be 2H∈bounded appropiately. For example, for one such term (cf. (3.2)) the following inequalities hold:  (3.3) 22**1,1x~1t1s )T(L||x~ddy)y,s(u|| −∫∫    22**1,11*y1t0**1,1*y1ss1t )T(L||dyx~d)y,s(udyx~d)y,s(u|| −−− ∫∫−∫∫=  { } 2/1*2*1,1s1t*2*s1t dy|)y,s(u|dy)sy1(|| −− ∫−−∫≤     222/1*2*1,1t0*2*t0 )T(L||dy)y,s(udy)y( ⎭⎬⎫⎩⎨⎧ ∫∫+    -7-  ⎢⎣⎡ ∫+∫−−∫∫= −− *2*1,1t03*2*1,1s1t3s1010 dy|)y,s(u|3tdy|)y,s(u|3)ts1(   2/1*2*1,1t0*2*1,1s1t2/32/3dy|)y,su(|dy|)y,s(u|3t)ts1(⎭⎬⎫⎩⎨⎧ ∫∫−−+ − dt ds dsdy|)y,s(u|)32/6/1()s1( *2*1,1s10410−∫π+−∫≤  221,1 )T(L||u||)32/6/1( π+≤  The details of the calculations which involve the other terms are omitted for brevity, and the final bounds obtained are (3.4) ,)T(L||u||17.0)T(L||u||39.0)T(L||u||17.0)T(L||]u[R|| 22,021,120,22 ++≤   (3.5) )T(L||u||93.0)T(L||u||65.0)T(L||]u[R|| 21,120,220,1 +≤  , (3.6) )T(L||u||65.0)T(L||u||93.0)T(L||]u[R|| 22,021,121,0 +≤  .  4. Error bounds for linear interpolation on a general triangle. Consider the linear transformation defined by (4.1)      )]y,x(),y,x([u),(uˆ ηξ=ηξ  , where (4.2)    ⎪⎩⎪⎨⎧≡η=η+≡ξ=ξ,by)y,x(ayx)y,x(and  . Equations (4.2) define a transformation of the .0b1,0a1 ≥≥≥≥t r i a n g l e  T  w i t h  v e r t i c e s  a t  ( 1 , 0 ) ,  ( 0 , 0 ) ,  a n d  ( 0 , 1 )  i n  t h e  ( x , y )  p l a n e ,  t o  t h e  t r i a n g l e  ˆ  w i t h  v e r t i c e s  a t  ( 1 , 0 ) ,  ( 0 , 0 ) ,  a n d  ( a , b )  r e s p e c t i v e l y  T-8-  i n  t h e   p l a n e .  F u r t h e r ,  l e t  t h e  a n g l e s  a t  t h e  v e r t i c e s  o f  Tˆ  b e  ),( ηξdenoted  by  α ,  β ,  and  γ respec t ive ly ,  where  γ≤β≤α ,  see  F igure  4 .1 .  Thus ,  t he  l eng th  o f  t he  g r ea t e s t  s i de  o f   Tˆ  i s  1 .  ( R e s u l t s  fo r  a  t r i a n g l e  of  greatest  s ide h  fol low immediately from the analysis  in  this  sect ion.)    L e t  R ,  ,  a n d  d e n o t e  t h e  l i n e a r  i n t e r p o l a t i o n  r e m a i n d e r  Rˆ ,Rˆ 0,1 ,Rˆ 1,0opera tors  on  .  Then ,  f rom (3 .4) - (3 .6)  and  the  t ransformat ion  (4 .1) ,  the  Tˆfollowing results can be obtained:  (4.3) )T(L|||u|R||b)Tˆ(L|||uˆ|Rˆ||22=            )Tˆ(L||uˆ||)a17.0a39.017.0(20,22++≤   , )Tˆ(L||uˆ||b17.0)Tˆ(L||uˆ||)ba34.0b39.0(22,0221,1 +++(4.4) )T(L||]u[R||b)Tˆ(L||]u[Rˆ||20,120,1 =     )Tˆ(L||uˆ||b93.0)Tˆ(L||uˆ||)a93.065.0(21,120,2 ++≤ , (4.5) )T(L||]u[Rb1]u[Rba||b)Tˆ(L||]u[Rˆ||21,00,121,0 +−=     )Tˆ(L||uˆ||)a1(58.1ba20,2+≤         ||uˆ||b65.0||uˆ||)a3.193.0( +++  -9-  From (4.4) and (4.5),  the following semi-norm result is obtained: (4.6)   2/1221,020,11 )Tˆ(L||]uˆ[Rˆ||ˆ||]uˆ[Rˆ||)Tˆ(H|]uˆ[Rˆ| ⎭⎬⎫⎩⎨⎧ += )T(L2          , )Tˆ(H|uˆ|)βtancc( 221 +≤where  C 1  and  C 2  a re  cons tan ts  independent  of  Tˆ  and  uˆ ,  and  tan  β  =  a /b  .  Moreover, since the semi-norms in (4.6) are invariant under translation and ro ta t ion ,  (4 .6 )  i s  t rue  fo r  any  t r i ang le  o f  g rea tes t  s ide  1  and  wi th  ang les   ≤β≤α γ  From (4 .6)  there  fo l lows  the  condi t ion  tha t  t an  ,  ∞<≤β Kthat  is  ,  where 0>β≥β ο γ≤β≤α .  This  is  equivalent  to  the maximal  angle  condi t ion that  π<γ≤γ o .  Final ly ,  i t  should be noted that  had (3.6)  conta ined  a  t e rm in  ,  then  (4 .5)  would  have  conta ined  the  addi t iona l    0,2uf r o m   T h i s  t e r m  w o u l d  g i v e  t h e  m o r e  u s u a l l y  q u o t e d  .||uˆ||b/1 )Tˆ(2L0,2minimum angle condition that .0o >α≥α-10-  References 1. Babu s( ka, I.  and Aziz, A.K., On the Angle Condition in the Finite Element Method, Technical Note BN-8O8, IFDAM, University of Maryland, 1974.  2. Barnhill, R.E. and Gregory, J.A., Sard Kernel Theorems on Triangular Domains with Applications to Finite Element Error Bounds, TR/11, Department of Mathematics, Brunel University, 1972, revised 1974, Numer. Math, (to appear).  3. Barnhill ,  R.E. and Gregory, J.A.,  Interpolation Remainder Theory from Taylor Expansions on Triangles, TR/16, Department of Mathematics, Brunel University, 1972, revised 1975.  4. Barnh i l l ,  R .E .  and  Whi teman ,  J .R . ,  Er ro r  Ana lys i s  o f  F in i t e  Element Methods with Triangles for Elliptic Boundary Value Problems, The Mathematics of Finite Elements and Applications, Whiteman, J.R. (ed), Academic Press, London, 1973, pp 83-112.  5. Barnhill, R.E. and Wilcox, C.H., Computable Finite Element Error Bounds in Terms of Boundary Data, in preparation.  6. Birkhoff,  G.,  Schultz, M.H. and Varga, R.S.,  Piecewise Hermite Interpolation in One and Two Variables with Applications to Par t i a l  Di f fe ren t i a l  Equa t ions ,  Numer .  Math .  11 ,  1968 ,  pp 232-256.  7. Bramble,  J .H.  and Zlamal ,  M.,  Triangular  Elements  in  the Fini te  Element Method, Math. Comp. 24, 1970, pp 809-820.  8. Ciar le t ,  P .G.  and Raviar t ,  P .A. ,  General  Lagrange and Hermite  Interpolat ion in  Rn  with Applicat ions to  Fini te  Element  Method, Arch. Rat. Mech. Anal. 46, 1972, pp 177-199.  9. Jamet ,  P . ,  Es t imat ions  d 'Er reur  pour  des  Elements  F in is  Droi t s  Presque Degeneres, Centre d'Etudes de Limeil, 1974.  10. Sard, A.,  Linear Approximation, Math. Survey 9, American Math. Soc . ,  Providence ,  Rhode  Is land ,  1963.   11. Synge,  J .L. ,  The Hypercircle  in  Mathematical  Physics ,  Cambridge University Press, London, 1957.     

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Estimations d'Erreur pour des Elements Finis Droits Presque Degeneres, Centre d'Etudes de Limeil,

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Piecewise Hermite Interpolation in One and Two Variables with Applications to Partial Differential Equations,

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The Hypercircle in Mathematical Physics,

Error bounds for linear interpolation on triangles

Error bounds for linear interpolation on triangles

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