1. 1 A GENERAL EFFECTIVE PROCEDURE FOR COMBINING COLLOCATION AND DOMAIN DECOMPOSITION METHODS Ismael Herrera* and Robert Yates** *UNAM and **Multisistemas de Computo MEXICO

2. 2 THE PROBLEM 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. Combining collocation and DDM presents difficulties that must be overcome

3. 3 SOLUTION OF THE PROBLEM 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. New collocation methods

4. 4 STRUCTURE OF THIS TALK This talk is divided into two parts: 1.Finite Element Method with Optimal Functions (FEM-OF). 2.TH-collocation

5. 5 NOTATIONS

6. 6   PIECEWISE DEFINED FUNCTIONS Σ

7. 7 THE BOUNDARY VALUE PROBLEM WITH PRESCRIBED JUMPS (BVPJ)

8. 8 GREEN´S FORMULAS IN DISCONTINUOUS FUNCTIONS (GREEN-HERRERA FORMULAS,1985)

9. 9

10. 10 A GENERAL GREEN-HERRERA FORMULA FOR OPERATORS WITH CONTINUOUS COEFFICIENTS

11. 11 WEAK FORMULATIONS OF THE BVPJ

12. 12 FINITE ELEMENT METHOD with OPTIMAL FUNCTIONS A target of information is defined. This is denoted by “S*u”. FEM-OF are procedures for gathering such information.

13. 13 CONJUGATE DECOMPOSITIONS

14. 14 OPTIMAL FUNCTIONS

15. 15 THE STEKLOV-POINCARÉ APPROACH THE TREFFTZ-HERRERA APPROACH THE PETROV-GALERKIN APPROACH

16. 16 ESSENTIAL FEATURES OF FEM-OF METHODS

17. 17 THREE VERSIONS OF FEM-OF

18. 18 EXAMPLE SECOND ORDER ELLIPTIC

19. 19 A POSSIBLE CHOICE OF THE ‘SOUGHT INFORMATION’

20. 20 CONJUGATE DECOMPOSITIONS

21. 21 THE SYMMETRIC POSITIVE CASE

22. 22 TH-COLLOCATION This is obtained by locally applying orthogonal collocation to construct the approximate optimal functions.

23. 23 SECOND ORDER ELLIPTIC EQUATIONS

24. 24

25. 25 CONSTRUCTION OF THE OPTIMAL FUNCTIONS An optimal function is uniquely defined when its ‘trace’ is given on Σ. Piecewise polynomials, up to a certain degree, are chosen for the traces on the internal boundary Σ. Then the well-posed local problems are solved by orthogonal collocation.

26. 26 Support of an ‘Optimal Function’ CONSTRUCTION BY ORTHOGONAL COLLOCATION Cubic-Cubic: Four Collocation Points Collocation at each

27. 27 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.

28. 28 Support of an ‘Optimal Function’ CONSTRUCTION BY ORTHOGONAL COLLOCATION Linear-Quadratic (One collocation point) Collocation at each

29. 29 THE BILINEAR FORM

30. 30 TH-COLLOCATION FOR ELASTOSTATIC PROBLEMS OF ANISOTROPIC MATERIALS AND ITS PARALLELIZATION

31. 31

32. 32 CONSTRUCTION OF THE OPTIMAL FUNCTIONS The displacement fields are chosen to be piecewise polynomials, up to a certain degree, on the internal boundary, Σ. Then the well-posed local problems are solved by orthogonal collocation.

33. 33 THE BILINEAR FORM

34. 34 ISOTROPIC MATERIALS

35. 35

36. 36 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.