1. 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

2. 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 Motivation

3. Polynomial Preserving Recovery Recovery operator G h : S h,k  S h,k × S h,k. Nodal values of G h u h are defined by 1) At a vertex:  p k+1 (0, 0; z i ); 2) At an edge node between two vertices z i 1 and z i 2 :  p k+1 (x 1, y 1 ; z i 1 ) + (1-  )  p k+1 (x 2, y 2 ; z i 2 ), 0<  <1 ; 3) At an interior node on the triangle formed by z i j ' s: Here p k+1 (.; z i ) is the polynomial from a least-squares fitting of u h at some nodal points surrounding z i. G h u h is defined on the whole domain by interpolation using the original basis functions of S h,k. The Procedure

4. Linear Element

5. Quadratic Element

6. Cubic Element

7. Q 1 Element

8. Q 2 and Q 2 ’ Element

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10. P23a-c Mesh geometry(a-c)

11. p23d-e Mesh geometry(d-e)

12. p23f-g Mesh geometry(f-g)

13. Polynomial Preserving Recovery Vertex value G h u(z i ) 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 I

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15. 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 z ij are nodes involved. If z ij distribute symmetrically around z i, then coefficients c j ( z i ) distribute anti-symmetrically. Examples on Uniform Mesh II

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20. Polynomial Preserving Recovery  i, a union of elements that covers all nodes needed for the recovery of G h u h (z i ). Theorem 1. Let u  W  k+2 (  i ), then If z i is a grid symmetry point and u  W  k+2 (  i ) with k = 2r, then The ZZ patch recovery does not have this property. Polynomial preserving Property

21. Polynomial Preserving Recovery G h u(z): difference quotient on translation invariant mesh, Example: Linear element, regular pattern, vertex O: Translations are in the directions of Key Observation

22. Polynomial Preserving Recovery Theorem 2. Let the finite element space S h,k be transla- tion invariant in directions required by the recovery opera- tor G h on  D, let u  W  k+2 (  ), and let A ( u-u h,v )=0 for v  S 0 h,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, Superconvergence Property I

23. Polynomial Preserving Recovery T h : triangulation for . Condition (  ) : T h = T 1,h  T 2,h with 1)every two adjacent triangles inside T 1,h form an O ( h 1+  ) (  >0) parallelogram; 2) |  2,h | = O(h  ),  > 0; 2,h =   T 2,h. Observation: Usually, a mesh produced by an automatic mesh generator satisfies Condition (  ). Irregular Grids

24. Polynomial Preserving Recovery Theorem 3. Let u  W  3 (  ) be the solution of A(u, v) = (f, v),  v  H 1 (  ), let u h  S h,1 be the finite element approximation, and let T h satisfies Condition (  ). Assume that f and all coeffi- cients of the operator A are smooth. Then Superconvergence Property II

25. Polynomial Preserving Recovery 1.Linear element on Chevron pattern: O(h 2 ) compare with O(h) for ZZ. 2. Quadratic element on regular patter at edge centers: O(h 4 ) compare with O(h 2 ) for ZZ. 3. Mesh distortion at a vertex for ZZ: Comparison with ZZ

26. Mesh distortion P24_1

27. 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) = sin  x sin  y. -  u = 2  2 sin  x sin  y in  = [0, 1] 2, u = 0 on . Numerical Tests

28. p24_2 Linear element (Chevron) case 1

29. p24_3 Linear element (Chevron) case 2

30. p25_1 Quadratic element case 1

31. p25_2 Quadratic element case 2

32. 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 Industry