Polynomial Preserving Gradient Recovery in Finite Element Methods

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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

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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

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Polynomial Preserving Recovery
Recovery operator Gh Sh,k ? Sh,k Sh,k
. Nodal values of Ghuh are defined by 1) At a
vertex ?pk1(0, 0 zi) 2) At an edge node
between two vertices zi1 and zi2 ??pk1(x1, y1
zi1) (1-?)?pk1(x2, y2 zi2), 0lt?lt1 3) At an
interior node on the triangle formed by zij's
Here pk1(. 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 .
The Procedure
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Linear Element
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Quadratic Element
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Cubic Element
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Q1 Element
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Q2 and Q2 Element
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p27
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P23a-c
Mesh geometry(a-c)
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p23d-e
Mesh geometry(d-e)
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p23f-g
Mesh geometry(f-g)
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Polynomial Preserving Recovery
Examples on Uniform Mesh I

• 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.

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p18
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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,
• where zij are nodes involved.
• If zij distribute symmetrically around zi, then
  coefficients
• cj(zi) distribute anti-symmetrically.

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p19
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p20
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p21
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p22
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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?k2 (?i), then
• If zi is a grid symmetry point and u ?W?k2 (?i)
  with k2r,
• then
• The ZZ patch recovery does not have this
  property.

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Polynomial Preserving Recovery
Key Observation

• Ghu(z) difference quotient on translation
  invariant mesh,
• Example Linear element, regular pattern, vertex
  O
• Translations are in the directions of

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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 ?W?k2 (?), 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,

Superconvergence Property I
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Polynomial Preserving Recovery

• Th triangulation for ?.
• Condition (?) Th T1,h ? T2,h with
• every two adjacent triangles inside T1,h form an
  O(h1?)
• (?gt0) parallelogram
• 2) ?2,h O(h?), ? gt 0 2,h ???T2,h
  .
• Observation Usually, a mesh produced by an
  automatic mesh generator satisfies Condition (?).
  

Irregular Grids
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Polynomial Preserving Recovery

• 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
  coeffi-
• cients of the operator A are smooth. Then

Superconvergence Property II
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Polynomial Preserving Recovery
Comparison with ZZ

• 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

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P24_1
Mesh distortion
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Polynomial Preserving Recovery
Numerical Tests

• 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, 12, u 0
  on ??.

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p24_2
Linear element (Chevron) case 1
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p24_3
Linear element (Chevron) case 2
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p25_1
Quadratic element case 1
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p25_2
Quadratic element case 2
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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