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A GENERAL AND SYSTEMATIC THEORY OF DISCONTINUOUS GALERKIN METHODS

A GENERAL AND SYSTEMATIC THEORY OF DISCONTINUOUS GALERKIN METHODS. Ismael Herrera UNAM MEXICO. A SYSTEMATIC FORMULATION OF DISCONTINUOUS GALERKIN METHODS MUST BE BASED ON THE. THEORY OF PARTIAL DIFFERENTIAL EQUATIONS IN DISCONTINUOUS FNCTIONS.

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A GENERAL AND SYSTEMATIC THEORY OF DISCONTINUOUS GALERKIN METHODS

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  1. A GENERAL AND SYSTEMATIC THEORY OFDISCONTINUOUS GALERKIN METHODS Ismael Herrera UNAM MEXICO

  2. A SYSTEMATIC FORMULATION OF DISCONTINUOUS GALERKIN METHODS MUST BE BASED ON THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS IN DISCONTINUOUS FNCTIONS

  3. I.- ALGEBRAIC THEORY OF BOUNDARY VALUE PROBLEMS

  4. NOTATIONS

  5. BASIC DEFINITIONS

  6. NORMAL DIRICHLET BOUNDARY OPERATOR

  7. EXISTENCE THEOREM

  8. II.- BOUNDARY VALUE PROBLEMS FORMULATED IN DISCONTINUOUS FUNCTION SPACES

  9. PIECEWISE DEFINED FUNCTIONS  Σ 

  10. PIECEWISE DEFINED OPERATORS

  11. SMOOTH FUNCTIONS

  12. EXISTENCE THEOREM for the BVPJ

  13. III.- ELLIPTIC EQUATIONSOF ORDER 2m

  14. SOBOLEV SPACE OF PIECEWISE DEFINED FUNCTIONS

  15. RELATION BETWEEN SOBOLEV SPACES

  16. THE BVPJ OF ORDER 2m

  17. EXISTENCE OF SOLUTIONFOR THE ELLIPTIC BVPJ

  18. IV.- GREEN´S FORMULAS IN DISCONTINUOUS FIELDS“GREEN-HERRERA FORMULAS (1985)”

  19. FORMAL ADJOINTS

  20. GREEN’S FORMULA FOR THE BVP

  21. GREEN’S FORMULA FOR THE BVPJ

  22. A GENERAL GREEN-HERRERA FORMULA FOR OPERATORS WITH CONTINUOUS COEFFICIENTS

  23. WEAK FORMULATIONS OF THE BVPJ

  24. V.- APPLICATION TO DEVELOP FINITE ELEMENT METHODS WITH OPTIMAL FUNCTIONS (FEM-OF)

  25. GENERAL STRATEGY • A target of information is defined. This is denoted by “S*u” • Procedures for gathering such information are constructed from which the numerical methods stem.

  26. 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.

  27. CONJUGATE DECOMPOSITIONS

  28. OPTIMAL FUNCTIONS

  29. THE STEKLOV-POINCARÉ APPROACH THE TREFFTZ-HERRERA APPROACH THE PETROV-GALERKIN APPROACH

  30. ESSENTIAL FEATURE OFFEM-OF METHODS

  31. THREE VERSIONS OF FEM-OF • Steklov-Poincaré FEM-OF • Trefftz-Herrera FEM-OF • Petrov-Galerkin FEM-OF

  32. 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’.

  33. 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.

  34. 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.

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