Polynomial Preserving Gradient Recovery in Finite Element

MethodsZhimin 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

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

MotivationMotivation

Polynomial Preserving Recovery

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:

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 interpolationusing the original basis functions of Sh,k .

The ProcedureThe Procedure

.0,1,;,3

1

3

11

j

jj

jijjkj j

zyxp

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

Vertex value Ghu(zi) for linear element. I.1. Regular pattern.

I.2. Chevron pattern.

• Regular pattern, same as ZZ and simple averaging.

• Chevron pattern, all three are different.

Examples on Uniform Mesh IExamples on Uniform Mesh I

.)(2

)(2

61

614352

563241

uuuuuu

uuuuuu

h

.624

)(6

121

7654321

46

uuuuuuu

uu

h

p18

Polynomial Preserving Recovery

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,

where zij are nodes involved. • If zij distribute symmetrically around zi, then

coefficients cj(zi) distribute anti-symmetrically.

Examples on Uniform Mesh IIExamples on Uniform Mesh II

,0)(,)()(1

)(

jij

jijijih zczuzc

hzuG

p19

p20

p21

p22

Polynomial Preserving Recovery

i, a union of elements that covers all nodes needed for

the recovery of Ghuh(zi).

Theorem 1. Let u Wk+2 (i), then

If zi is a grid symmetry point and u Wk+2 (i) with

k=2r,then

• The ZZ patch recovery does not have this property.

Polynomial preserving PropertyPolynomial preserving Property

.||)(

1

)( 2i

ki W

k

Lh uChuGu

.||)()(

23

ikW

k

ih uChzuGu

Polynomial Preserving Recovery

Ghu(z): difference quotient on translation invariant mesh,

Example: Linear element, regular pattern, vertex O:

Translations are in the directions of

Key ObservationKey Observation

M i

i

i

hh hlzuCzuG

).()( )(

,

.),(),(),(2

),(2),(),(6

1

),()(

556644

113322

0

hyxuhyhxuyhxu

yhxuhyhxuhyxuh

yxOuh

x

).1,1(),1,0(),0,1( 321 lll

Polynomial Preserving Recovery

Theorem 2. Let the finite element space Sh,k be transla-

tion invariant in directions required by the recovery opera-

tor Gh on D, let u Wk+2 (), and let A(u-uh,v)=0

for vS0h,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 q1,

Superconvergence Property ISuperconvergence Property I

2

1,

2

1

)()(

1

)(

.),(

,)1

(ln 20

ji i i

i

ji

ij

DWhW

kr

Lhh

cwvvx

wb

x

v

x

wavwA

uuCuhh

CuGu sq

k

Polynomial Preserving Recovery

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 ().

Irregular GridsIrregular Grids

Polynomial Preserving Recovery

Theorem 3. Let u W3() be the solution of

A(u, v) = (f, v), v H1(),

let uhSh,1 be the finite element approximation, and let

Th satisfies Condition (). Assume that f and all coeffi-

cients of the operator A are smooth. Then

Superconvergence Property IISuperconvergence Property II

).2

,2

1,min(,

,,3

1

,0

uChuGu hh

Polynomial Preserving Recovery

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:

Comparison with ZZComparison with ZZ

.)183411(

)5141483275()2181409829(

)26454533()445011(120

342

242242

34224

22

u

uu

uh

y

yxyx

x

Mesh distortion

P24_1

Polynomial Preserving Recovery

Case 1. The Poisson equation with zero boundary condi-

tion 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) = sinx siny.

- u = 22 sinx siny in = [0, 1]2, u = 0 on .

Numerical TestsNumerical Tests

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

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

ZZ Patch Recovery in IndustryZZ Patch Recovery in Industry