A GENERAL EFFECTIVE PROCEDURE FOR COMBINING COLLOCATION AND DOMAIN DECOMPOSITION METHODS

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A GENERAL EFFECTIVE PROCEDURE FORCOMBINING
COLLOCATION AND DOMAIN DECOMPOSITION METHODS

• Ismael Herrera and Robert Yates
• UNAM and Multisistemas de Computo
• MEXICO

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THE PROBLEM
Combining collocation and DDM presents
difficulties that must be overcome

• The main technical difficulty stems from the fact
  that the standard collocation method (orthogonal
  spline collocation OSC) yields non-symmetric
  matrices, even for formally symmetric
  differential operators.

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SOLUTION OF THE PROBLEM
New collocation methods

• In recent years new collocation methods have been
  introduced which yield symmetric matrices when
  the differential operators are formally
  symmetric . Generically they are known as
  TH-collocation.
• TH-collocation combines orthogonal collocation
  with a special kind of Finite Element Method
  FEM-OF.

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STRUCTURE OF THIS TALK

• This talk is divided into two parts
• Finite Element Method with Optimal Functions
  (FEM-OF).
• TH-collocation

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NOTATIONS
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PIECEWISE DEFINED FUNCTIONS
??
S
?
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THE BOUNDARY VALUE PROBLEM WITH PRESCRIBED JUMPS
(BVPJ)
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GREENS FORMULAS IN DISCONTINUOUS
FUNCTIONS(GREEN-HERRERA FORMULAS,1985)
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A GENERAL GREEN-HERRERA FORMULA FOR OPERATORS
WITH CONTINUOUS COEFFICIENTS
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WEAK FORMULATIONS OF THE BVPJ
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FINITE ELEMENT METHOD with OPTIMAL FUNCTIONS
A target of information is defined. This is
denoted by Su. FEM-OF are procedures for
gathering such information.
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CONJUGATE DECOMPOSITIONS
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OPTIMAL FUNCTIONS
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THE STEKLOV-POINCARÉ APPROACH
THE TREFFTZ-HERRERA APPROACH
THE PETROV-GALERKIN APPROACH
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ESSENTIAL FEATURES OFFEM-OF METHODS
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THREE VERSIONS OF FEM-OF
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EXAMPLESECOND ORDER ELLIPTIC
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A POSSIBLE CHOICE OF THE SOUGHT INFORMATION
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CONJUGATE DECOMPOSITIONS
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THE SYMMETRIC POSITIVE CASE
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TH-COLLOCATION

• This is obtained by locally applying orthogonal
  collocation to construct the approximate optimal
  functions.

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SECOND ORDER ELLIPTIC EQUATIONS
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CONSTRUCTION OF THE OPTIMAL FUNCTIONS

• An optimal function is uniquely defined when its
  trace is given on S.
• Piecewise polynomials, up to a certain degree,
  are chosen for the traces on the internal
  boundary S.
• Then the well-posed local problems are solved by
  orthogonal collocation.

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CONSTRUCTION BY ORTHOGONAL COLLOCATION Cubic-Cubic
Four Collocation Points
Support of an Optimal Function
Collocation at each
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COMPARISON WITH OSC

• Steklov-Poincaré FEM-OF yields the same solution
  as OSC. However, now the system-matrix is
  positive definite for differential systems that
  are symmetric and positive.
• Trefftz-Herrera FEM-OF yields the same order of
  accuracy as OSC, although its solution is not
  necessarily the same. The system-matrix is
  positive definite for differential systems that
  are symmetric and positive.

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CONSTRUCTION BY ORTHOGONAL COLLOCATION Linear-Quad
ratic (One collocation point)
Support of an Optimal Function
Collocation at each
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THE BILINEAR FORM
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TH-COLLOCATION FORELASTOSTATIC PROBLEMS OF
ANISOTROPIC MATERIALS AND ITS PARALLELIZATION
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CONSTRUCTION OF THE OPTIMAL FUNCTIONS

• The displacement fields are chosen to be
  piecewise polynomials, up to a certain degree, on
  the internal boundary, S.
• Then the well-posed local problems are solved by
  orthogonal collocation.

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THE BILINEAR FORM
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ISOTROPIC MATERIALS
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CONCLUSIONS

• For any linear differential equation or system of
  such equations, TH-collocation supplies a new and
  more effective manner of using orthogonal
  collocation in combination with DDM. It has
  attractive features such as
• 1. Better structured matrices,
• 2. The approximating polynomials on the internal
  boundary and in the element interiors can be
  chosen independently,
• 3. The number of collocation points can be
  reduced.