1 A GENERAL AND SYSTEMATIC THEORY OF DISCONTINUOUS GALERKIN METHODS Ismael Herrera UNAM MEXICO

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

3 I.- ALGEBRAIC THEORY OF BOUNDARY VALUE PROBLEMS

4 NOTATIONS

5 BASIC DEFINITIONS

6

7 NORMAL DIRICHLET BOUNDARY OPERATOR

8 EXISTENCE THEOREM

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

10   PIECEWISE DEFINED FUNCTIONS Σ

11 PIECEWISE DEFINED OPERATORS

12 SMOOTH FUNCTIONS

13

14 EXISTENCE THEOREM for the BVPJ

15 III.- ELLIPTIC EQUATIONS OF ORDER 2m

16 SOBOLEV SPACE OF PIECEWISE DEFINED FUNCTIONS

17 RELATION BETWEEN SOBOLEV SPACES

18 THE BVPJ OF ORDER 2m

19 EXISTENCE OF SOLUTION FOR THE ELLIPTIC BVPJ

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

21 FORMAL ADJOINTS

22 GREEN’S FORMULA FOR THE BVP

23 GREEN’S FORMULA FOR THE BVPJ

24 A GENERAL GREEN-HERRERA FORMULA FOR OPERATORS WITH CONTINUOUS COEFFICIENTS

25 WEAK FORMULATIONS OF THE BVPJ

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

27 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.

28 EXAMPLE SECOND 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.

29 CONJUGATE DECOMPOSITIONS

30 OPTIMAL FUNCTIONS

31 THE STEKLOV-POINCARÉ APPROACH THE TREFFTZ-HERRERA APPROACH THE PETROV-GALERKIN APPROACH

32 ESSENTIAL FEATURE OF FEM-OF METHODS

33 THREE VERSIONS OF FEM-OF Steklov-Poincaré FEM-OF Trefftz-Herrera FEM-OF Petrov-Galerkin FEM-OF

34 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’.

35 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.

36 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.