Polynomial Preserving Gradient Recovery
in Finite Element Methods
Zhimin Zhang
Department of Mathematics
Wayne State University
Detroit, MI 48202
http://www.math.wayne.edu/~zzhang
Collaborator: Ahmed A. Naga
Research is partially supported by the NSF grants:
DMS-0074301 and DMS-0311807
Polynomial Preserving Recovery
Motivation
The ZZ patch recovery is not perfect!
1. Difficulty on the boundary, especially curved boundary.
2. Not polynomial preserving.
3. Superconvergence cannot be guaranteed in general.
EVERY AVERAGING WORKS! C. Carstensen, 2002
Polynomial Preserving Recovery
The Procedure
Recovery operator Gh: Sh,k  Sh,k × Sh,k .
Nodal values of Ghuh are defined by
1) At a vertex: pk+1(0, 0; zi);
2) At an edge node between two vertices zi1 and zi2:
pk+1(x1, y1; zi1) + (1-)pk+1(x2, y2; zi2), 0<<1;
3) At an interior node on the triangle formed by zij's:
 j pk 1 x j , y j ; zi ,  j  1,  j  0.
3
j 1
3
j
j 1
Here pk+1(.; zi) is the polynomial from a least-squares fitting
of uh at some nodal points surrounding zi .
Ghuh is defined on the whole domain by interpolation
using the original basis functions of Sh,k .
Linear Element
Quadratic Element
Cubic Element
Q1 Element
Q2 and Q2’ Element
p27
P23a-c
Mesh geometry(a-c)
p23d-e
Mesh geometry(d-e)
p23f-g
Mesh geometry(f-g)
Polynomial Preserving Recovery
Examples on Uniform Mesh I
Vertex value Ghu(zi) for linear element.
I.1. Regular pattern.
1  2(u1  u4 )  u2  u3  u6  u5 

.
6h  2(u2  u5 )  u3  u4  u1  u6 
I.2. Chevron pattern.
6(u6  u4 )

1 

.
12h   u1  4u2  u3  u4  2u5  u6  6u7 
• Regular pattern, same as ZZ and simple averaging.
• Chevron pattern, all three are different.
p18
Polynomial Preserving Recovery
Examples on Uniform Mesh II
Quadratic element on regular pattern.
II.1. At a vertex;
II.2. At a horizontal edge center;
II.3. At a vertical edge center;
II.4. At a diagonal edge center.
In general,

1 

Gh u ( zi )   c j ( zi )u ( zij ),  c j ( zi )  0,
j
h j
where zij are nodes involved.
• If zij distribute symmetrically around zi, then coefficients
cj(zi) distribute anti-symmetrically.
p19
p20
p21
p22
Polynomial Preserving Recovery
Polynomial preserving Property
i, a union of elements that covers all nodes needed for
the recovery of Ghuh(zi).
Theorem 1. Let u Wk+2 (i), then
u  Gh u
L ( i )
 Ch k 1 | u |W
k 2
( i

)
.
If zi is a grid symmetry point and u Wk+2 (i) with k=2r,
then
u  Ghu ( zi )  Ch k  2 | u |W ( ) .
k 3

i
• The ZZ patch recovery does not have this property.
Polynomial Preserving Recovery
Key Observation
Ghu(z): difference quotient on translation invariant mesh,
 (i )
Ghu ( z )    C ,hu ( z  hli ).
 M i
Example: Linear element, regular pattern, vertex O:
 hxu (O)0 ( x, y )
1
 u2 2 ( x, y  h)  u33 ( x  h, y  h)  2u11 ( x  h, y )
6h
 2u44 ( x  h, y )  u66 ( x  h, y  h)  u55 ( x, y  h).
Translations are in the directions of
l1  (1,0), l2  (0,1), l3  (1,1).
Polynomial Preserving Recovery
Superconvergence Property I
Theorem 2. Let the finite element space Sh,k be translation invariant in directions required by the recovery operator Gh on D, let u Wk+2 (), and let A(u-uh,v)=0
for vS0h,k(). Assume that Theorem 5.5.2 in Wahlbin's
book is applicable. Then on any interior region 0,
there is a constant C independent of h and u such that for
some s  0 and q1,
1 r k 1
u  Ghuh L (  )  C (ln ) h u W (  )  C u  uh W ( D ) ,
h
2
2
w v
w
A( w, v)    aij
  bi
v  cwv.
xi x j i1 xi
i , j 1

0
k 2

s
q
Polynomial Preserving Recovery
Irregular Grids
Th: triangulation for .
Condition (): Th = T1,h  T2,h with
1) every two adjacent triangles inside T1,h form an O(h1+)
(>0) parallelogram;
2) |2,h| = O(h),  > 0;  2,h = T2,h .
•
Observation: Usually, a mesh produced by an
automatic mesh generator satisfies Condition ().
Polynomial Preserving Recovery
Superconvergence Property II
Theorem 3. Let u W3() be the solution of
A(u, v) = (f, v), v  H1(),
let uhSh,1 be the finite element approximation, and let
Th satisfies Condition (). Assume that f and all coefficients of the operator A are smooth. Then
1 
u  Ghuh 0 ,  Ch1  u 3, , ,   min( , , ).
2 2
Polynomial Preserving Recovery
Comparison with ZZ
1. Linear element on Chevron pattern:
O(h2) compare with O(h) for ZZ.
2. Quadratic element on regular patter at edge centers:
O(h4) compare with O(h2) for ZZ.
3. Mesh distortion at a vertex for ZZ:
 2h 2
2
4 3
(
533

454


26

) x u
4
2
120(11  50  44)

 (829  1409 2  218 4 ) 2x  yu  (275  1483 2  514 4 ) x  2yu

 (11  34 2  18 4 ) 3yu .
P24_1
Mesh distortion
Polynomial Preserving Recovery
Numerical Tests
Case 1. The Poisson equation with zero boundary condition on the unit square with the exact solution
u(x, y) = x (1 - x) y (1 - y).
Case 2. The exact solution is u(x, y) = sinx siny.
-  u = 22 sinx siny in  = [0, 1]2, u = 0 on .
p24_2
Linear element (Chevron) case 1
p24_3
Linear element (Chevron) case 2
p25_1
Quadratic element case 1
p25_2
Quadratic element case 2
Polynomial Preserving Recovery
ZZ Patch Recovery in Industry
Purpose: smoothing and adaptive remeshing.
• ANSYS
• MCS/NASTRAN-Marc
• Pro/MECHANICA (product of Parametric Technology)
• I-DEAS (product of SDRC, part of EDS)
• COMET-AR(NASA): COmputational MEchanics Testbed
With Adaptive Refinement