α1 α2 Simplex and Rectangle Elements Multi-index Notation of polynomials of degree Definition: The set P k will be the set of all functions:

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1 Smplex ad Rectagle Elemets Mult-dex Notato = (,..., ), o-egatve tegers = = β = ( β,..., β ) the + β = ( + β,..., + β ) + x = x x x x = x x β β + D = D = D D x x x β β Defto: The set P of polyomals of degree p( x) p( x,..., x ) γ x = = wll be the set of all fuctos: + + Note: P s a vector space of dmeso: dm P = = -Smplex Type () SIMPLEXES Defto: A (o-degererate) -smplex s the covex hull of ( + ) vertex pots v, =,..., where the pots { v } are ot cotaed ay hyperplae H = p+ M s.t. dm M =, so that: = λv : λ, λ = = = Defto: The stadard -smplex S wll be the comvex hull of the vertces: v for = = e for =,..., Note: S = ( λ,..., λ) : λ, λ =

2 Theorem: A -smplex s specfed by gvg + vertex coordates v = v s.t. the matrx: { } { } = = = has o-zero determat. v v... v, v v... v, A = v v... v,... proof: The affe mappg T : S becto. gve by T :( λ,..., λ) = v + λ( v v) s a Defto: The barycetrc coordates λ = λ( x) =,..., of a pot x are the uque soluto to the lear system: Note: λ ( x) P e: = = v λ = x λ = x A λ = Defto: The barycetrc ceter (of gravty/mass) of s the pot x whose barycetrc coordates are all equal to /( + ). -Smplex Type() Theorem: A polyomal p( x) P s uquely determed by ts value o the vertces { v } of ay -smplex. proof: It s requred to show that the system of equatos: γ v = β =,..., =

3 has a uque soluto { } γ for all possble values of { β } λ ( x) satsfy λ ( v ) = δ for, =,..., so that: satsfes the equato. p( x) = βλ( x) = λp( v ) = =. The barycetrc coordates Defto: A -smplex of type() s gve by the trplet (, P, Σ ) where Σ = { p( v )} e: the evaluato fuctoals at the + vertces of. The fuctos at the odes v are gve as f( x) = λ ad we have: = f = =, > λ, > ad -Smplex Type() Defto: The mdpots of the edges of a -smplex wll be deoted by: v = ( v + v ) for < Lemma: λ( v) = ( δ + δ ) Theorem: For all p P we have p( x) = λ (λ ) p( v ) + λλ p( v ) = < proof: We observe that f p, q P ad f pv ( ) = qv ( ) ad pv ( ) = qv ( ) for < the p( x) q( x). Ths follows sce r( λ) = p( x) q( x) P( λ,..., λ ). If λ = for the r( λ ) s a quadratc polyomal wth zeros (for λ =,,). Thus r( λ) s of the form r( λ) = rλλ. But r ( )( ) δ + δl δ + δ l = rl =. So f < < λ λ λλ = < qx ( ) = ( ) pv ( ) + pv ( ) the qv ( ) = pv ( ) ad qv ( ) = pv ( ) for < so p( x) q( x). Defto: A -smplex of type() s gve by the trplet (, P, Σ ) where Σ = { p( v), p( v)} e: the evaluato fuctoals at the ( + )( + )/ vertces ad edge mdpots of.

4 The fuctos at the odes v ad v are gve as: f ( x) = λ (λ ) for =,..., f ( x) = λλ for < The we have: λ =, f λ f, + = λ + λ = = = = λ (λ ) δ λ λ = λδ -Smplex Type() Defto: For let v = ( v + v ) ad let v = ( v + v + v ) for < < Theorem: For all p P we have : λ ( )( ) ( ) λ λ λλ λ p( x) = p( v) + p( v) + 7 λλ λp( v) < < Defto: A -Smplex of type() s gve by the trplet (, P, Σ ) where Σ = { p( v ), p( v ), p( v ):, < l < m} s the set of evaluato fuctoals at the lm ( + )( + )( + )/6 odes v, v ad v. The fuctos at the odes are the gve as: λ ( )( ) ( ) λ λ λλ λ f ( x) = f ( x) = f ( x) = 7λλλ Theorem: Let be a -smplex wth vertces v, =,...,. For, ay polyomal p P s uquely determed by ts values o the set: L( ) = x= λv; λ = ; λ,,...,, = = -Smplex Type( ) Theorem: For ay trple (,, ) wth < < let:

5 φ ( p) = pv ( ) + pv ( ) pv ( ) l llm l=,, l, m=,, l m = = < < s The ay polyomal the space P { p P : φ ( p) for all } uquely determed by ts values at the odes v ad v. Also P P. proof: + = = + = dm P dm P ( ) degrees of freedom p P we have: p = λ(λ )(λ ) λ( λλ ) pv ( ) + <, 7 λλ (λ ) + λλ λ pv ( ),. The for { }. Cosequetly, for all Now let p P I =,, wth < < we have pv ( l ) = pv ( ) + Dpv ( )( vl v ) + Av ( l v ) where A= D ps costat The: pv ( l ) = pv ( ) + Av ( l v ) as ( vl v ) = l I l I l I Smlarly: pv ( llm) = 6 pv ( ) + Av ( llm v ) Hermte -Smplex Type() lm, I lm, I l m l m Theorem: Ay polyomal p( x) P s uquely determed by ts values at the vertces v, the odes v ad the frst dervatves Dp( v ) at the vertces. Moveover: p = ( λ + λ 7 λ λ λ ) pv ( ) + λλ (λ + λ ) Dpv ( )( v v) <, + 7 λλ λ pv ( ) < < proof: Let qx ( ) deote the above expresso. The clearly: 5

6 qv ( ) = ( + ) pv ( ) = pv ( ) 7 qv ( ) = + pv ( ) + ( )( ) 7 ( ) ( ) 7 + Dpv v v + pv = pv 7 v + v qv ( ) = q( ) = pv ( ) + pv ( ) + Dpv ( )( v v ) + Dpv ( )( v v ) However, We observe that o the segmet ( λ) v + λv we have that for ay p P () that: p( λ) = p( v ) H ( λ) + Dp( v )( v v ) H ( λ) + p( v ) H ( λ) Dp( v )( v v ) H ( λ) ad that : H ( λ) = H ( λ) ad H ( λ) = H ( λ) where: H ( λ) = λ λ + ad H ( λ) = λ( λ) 7 So that p(/ ) = pv ( ) + pv ( ) + Dpv ( )( v v) Dpv ( )( v v) ad we have p( v ) = q( v ) Now Dq( v) = Dp( v)( v v) Dλ ad Dλ ( v v) = δ λ( v) for whch follows from λ = Bx, x = Aλ ad λ = b ad Dλ ( v x) = δ λ ( x) so that : Dq( v )( v v ) = Dp( v )( v v ) RECTANGLES -Rectagles Type() Defto: Q wll deote the set of all polyomals varables of maxmum degree each separate varable: Q = p( x) = γx = γx x x : for =,..., Note: dm Q = ( + ) ad P Q P. 6

7 x Theorem: The Lagrage polyomal ( ) x l x = = = = l( ) = δ for, has the property that Defto: Let l ( x) = l ( x ) l ( x ) l ( x ) β β Theorem: If xβ = (,..., ) the l ( x ) = δ β β Theorem: If p Q the p l ( x) p( x ) = Defto: A -rectagle s a set of the form: [ ] = A face F of s a set of the form: = a, b where a < b =,...,. { } { } F = a a, b or b a, b = = A edge E of s a set of the form: E = [ a, b] { c } where c = a or c = b. = A vertex ( ) v of s a pot of the form v= v,..., v wth v = a or v = b. Note: Ay rectagle s the mage of a vertble dagoal affe mappg ψ :[,] of the form: ψ ( x) = b+ Ax where A s a dagoal matrx Defto: A -rectagle of type() s the trplet (, Q, Σ ) where s a -rectagle Σ ( ): ( ), [,] = pv v = ψ x x are the evaluato fuctoals at the pots ad { β β β β } x β β β = (,..., ). Examples: = Type() 7

8 Lettg x, x deote the depedet varables o the ut square S = [,] [,] we defe the coordates x = x ad x = x. We also umber the vertces v = (,), v = (,), v = (,) ad v = (,). The base fuctos p ( x, x) =,..., are gve as: p = x x p = x x p = x x p = x x whch are equvalet to p = xx ad crcular dex rotato for p, p ad p Type() Let v5 = ( v+ v), v 6 = ( v + v), v7 = ( v+ v), v8 = ( v + v). The we ca wrte: p = x(x ) x(x ), p =..., p =..., p =... p5 = x( x ) x(x ), p5 =...,... p = x x x x Type() Let v5 = ( v+ v ),... v = ( v+ v ),... v = ( v5 + v ) = (v + v + v + v),... The p = x(x )(x ) x(x )(x ),... p5 = x(x )( x ) x(x )(x ),... p = x(x )( x ) x(x )(x ),... 8 p = x ( x )( x ) x ( x )( x ) Type( ) 8 Let Q = p Q : p( v) + p( v) p( v) = = = 5 The: 8

9 a) Ay p Q s determed by the values at v,..., v b) P Q 8 c) p = x x (x + x ),... p = x x ( x ),... 5 Type( ) : ( ),..., = = = where ψ ( p) = pv ( ) + pv ( ) + pv ( ) + pv ( ) + pv ( ) 6 pv ( ) pv ( ) pv ( ) 6 pv ( ) Let Q { p Q ψ p } 5 6 permuted crcularly over the sets { } The v,..., v,{ v,..., v },{ v,..., v },{ v,..., v } a) Ay p Q s determed by the values at v,..., v b) P Q c) p = xx( + x( x ) + x( x )),... p5 = x( x )( x ) x,... p = x( x )(x ) x,...