A GENERAL AND SYSTEMATIC THEORY OF DISCONTINUOUS GALERKIN METHODS

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A GENERAL AND SYSTEMATIC THEORY OF
DISCONTINUOUS GALERKIN METHODS

• Ismael Herrera
• UNAM MEXICO

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THEORY OF PARTIAL DIFFERENTIAL EQUATIONS IN
DISCONTINUOUS FNCTIONS
A SYSTEMATIC FORMULATION OF DISCONTINUOUS
GALERKIN METHODS MUST BE BASED ON THE
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I.- ALGEBRAIC THEORY OF BOUNDARY VALUE PROBLEMS
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NOTATIONS
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BASIC DEFINITIONS
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NORMAL DIRICHLET BOUNDARY OPERATOR
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EXISTENCE THEOREM
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II.- BOUNDARY VALUE PROBLEMS FORMULATED IN
DISCONTINUOUS FUNCTION SPACES
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PIECEWISE DEFINED FUNCTIONS
??
S
?
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PIECEWISE DEFINED OPERATORS
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• SMOOTH FUNCTIONS

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EXISTENCE THEOREM for the BVPJ
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III.- ELLIPTIC EQUATIONSOF ORDER 2m
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SOBOLEV SPACE OF PIECEWISE DEFINED FUNCTIONS
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RELATION BETWEEN SOBOLEV SPACES
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THE BVPJ OF ORDER 2m
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EXISTENCE OF SOLUTION FOR THE ELLIPTIC BVPJ
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IV.- GREENS FORMULAS IN DISCONTINUOUS FIELDS
GREEN-HERRERA FORMULAS (1985)
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FORMAL ADJOINTS
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GREENS FORMULA FOR THE BVP
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GREENS FORMULA FOR THE BVPJ
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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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V.- APPLICATION TO DEVELOP FINITE ELEMENT
METHODS WITH OPTIMAL FUNCTIONS (FEM-OF)
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GENERAL STRATEGY

• A target of information is defined. This is
  denoted by Su
• Procedures for gathering such information are
  constructed from which the numerical methods
  stem.

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EXAMPLESECOND ORDER ELLIPTIC

• A possible choice is to take the sought
  information as the average of the function
  across the internal boundary.
• There are many other choices.

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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 FEATURE OFFEM-OF METHODS
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THREE VERSIONS OF FEM-OF

• Steklov-Poincaré FEM-OF
• Trefftz-Herrera FEM-OF
• Petrov-Galerkin FEM-OF

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FEM-OF HAS BEEN APPLIED TO DERIVE NEW AND MORE
EFFICIENT ORTHOGONAL COLLOCATION METHODS
TH-COLLOCATION

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

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CONCLUSION

• The theory of discontinuous Galerkin methods,
  here presented, supplies a systematic and general
  framework for them that includes a Green formula
  for differential operators in discontinuous
  functions and two weak formulations. For any
  given problem, they permit exploring
  systematically the different variational
  formulations that can be applied. Also, designing
  the numerical scheme according to the objectives
  that have been set.

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MAIN APPLICATIONS OF THIS THEORY OF dG METHODS,
thus far.

• Trefftz Methods. Contribution to their
  foundations and improvement.
• Introduction of FEM-OF methods.
• Development of new, more efficient and general
  collocation methods.
• Unifying formulations of DDM and preconditioners.