A GENERAL EFFECTIVE PROCEDURE FOR COMBINING COLLOCATION AND DOMAIN DECOMPOSITION METHODS

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

2. 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.

3. 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.

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

5. NOTATIONS

6. PIECEWISE DEFINED FUNCTIONS  Σ 

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

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

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

11. WEAK FORMULATIONS OF THE BVPJ

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

13. CONJUGATE DECOMPOSITIONS

14. OPTIMAL FUNCTIONS

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

16. ESSENTIAL FEATURES OFFEM-OF METHODS

17. THREE VERSIONS OF FEM-OF

18. EXAMPLESECOND ORDER ELLIPTIC

19. A POSSIBLE CHOICE OF THE ‘SOUGHT INFORMATION’

20. CONJUGATE DECOMPOSITIONS

21. THE SYMMETRIC POSITIVE CASE

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

23. SECOND ORDER ELLIPTIC EQUATIONS

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

27. COMPARISON WITH ‘OSC’ • Steklov-Poincaré FEM-OF yields the samesolution 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. CONSTRUCTION BY ORTHOGONAL COLLOCATION Linear-Quadratic (One collocation point) Support of an ‘Optimal Function’ Collocation at each

29. THE BILINEAR FORM

30. TH-COLLOCATIONFORELASTOSTATIC PROBLEMS OF ANISOTROPIC MATERIALS AND ITS PARALLELIZATION

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. THE BILINEAR FORM

34. ISOTROPIC MATERIALS

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.