1. F. Fairag, H Tawfiq and M. Al-Shahrani Department of Math & Stat Department of Mathematics and Statistics, KFUPM. Nov 6, 2013 Preconditioning Technique for Darcys Law in Porous Media King Fahd University of Petroleum & Minerals Dhahran, Saudi Arabia

2. Main Idea Sparse indefinite ill-conditioned We begin with Matrices A and B are generated from specific problem In this talk, we will present a preconditioner for this linear system. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

3. Main Idea the first interval is of small size The preconditioned matrix is nonsymmetric Powell-Silvester [2003]Bramble-Pasiak [1988] The eigenvalues of the preconditioned matrix discrete represent It is self-adjoint in the inner product H and + PCG based on this inner product can be applied P-MINRES can be applied Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

4. Outline 1)Definition ( all matrices A,B,D,N ) 2)self-adjointness (H-symm) 3)Clustering (eigenvalues) 4)Positive definiteness 5)Numerical Examples 1)Definition ( all matrices A,B,D,N ) 2)self-adjointness (H-symm) 3)Clustering (eigenvalues) 4)Positive definiteness 5)Numerical Examples Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

5. Matrices Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

6. We consider the solution of system of linear algebraic equations which is obtained from Raviart-Thomas mixed finite element formulation of Darcys equations. We begin with the mixed formulation of Darcy flow with some suitable boundary conditions u is the velocity p is the pressure K represents the permeability. Main Idea Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

7. Saddle Point Problem We begin with the mixed formulation of Darcy flow We define the two subspaces Velocity and pressure space there exist positive constants Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

8. Saddle Point Problem We begin with the mixed formulation of Darcy flow Multiplying by test functions, integrating the first equation by parts and imposing essential boundary conditions yield the variational problem, Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

9. Saddle Point Problem Department of Mathematics and Statistics, KFUPM. Nov 6, 2013 Two Hilbert spaces Two binlinear forms Consider the following problem we assume that (*) has a unique solution. Thm:Babuska(1971) (Brezzi 1974) Abstract Mixed Formulation

10. Discrete Form We restrict our discussion to the lowest order Raviart-Thomas space On rectangles Department of Mathematics and Statistics, KFUPM. Nov 6, 2013 We start by dividing the domain into triangles or rectangles

11. Saddle Point Problem discrete velocity space discrete pressure space give the problem Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

12. Saddle Point Problem the velocity divergence matrix the pressure mass matrix Department of Mathematics and Statistics, KFUPM. Nov 6, 2013 A+D is the matrix representation of the div-norm N is the matrix representation of the 0-norm

13. Self-adjoint Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

14. Self-Adjoint Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

15.  Meyer and Steidten (2001).  Benzi and Simoncini (2006).  Liesen and Parlett (2006).  Stoll and Wathen (2008). (excellent survey) Extensions to the classical BramblePasciak case: Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

16. Self-Adjoint Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

17. Positive Definiteness Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

18. Positive Definite Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

19. congruence transformation Positive Definite = Sylvester's law of inertia ifpositive ifpositive Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

20. Lemma 1 There exist two positive numbers and such that Lemma 2 There exist two positive numbers and such that For some positive numbers and for all Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

21. Eigenvalues Clustering Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

22. We study the generalized eigenvalue problem We get Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013 We want to compute all eigenvalues

23. Assume that Hence, eigenvalues (multiplicity n). Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

24. If, then we have So that the remaining m eigenvalues satisfy This equation leads to four cases. Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

25. Spectral Properties Theorem 4.3: If is a quasi-uniform mesh, the n+m eigenvalue of the generalized eigenvalue problem, where lie on the intervals, where is the minimum eigenvalue of the Schur complement and c is a constant independent of h. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

26. Example 4.1 is an eigenvalue of multiplicity n the remaining m positive eigenvalues Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

27. Theorem 4.4: If is a quasi-uniform mesh, the n+m eigenvalue of the generalized eigenvalue problem, wherelie on the union of the intervals, where is the minimum eigenvalue of the Schur complement is the area of the smallest element in and c is a constant independent of h. Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

28. Example 4.2 is an eigenvalue of multiplicity n the remaining m positive eigenvalues Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

29. Theorem 4.5: If is a quasi-uniform mesh, the n+m eigenvalue of the generalized eigenvalue problem, where lie on the union of the intervals, Spectral Properties where is the minimum eigenvalue of the Schur complement is the area of the smallest element in and c is a constant independent of h.

30. Theorem 4.6: If is a quasi-uniform mesh, the n+m eigenvalue of the generalized eigenvalue problem, where lie on the union of the intervals, Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

31. Example 4.4 Then the eigenvalue of the preconditioned matrix lie on the union of the following intervals Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

32. We study the problem on with uniform mesh The observed eigenvalues of the preconditioned matrix system are listed in the following table Our goal is to verify that the bounds (obtained) agree with the computed bounds. Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

33. The last four columns confirm that the bounds in Theorem (4.3) and Theorem (4.5) are tight. left bound Right bound 0.3124 0.0124100 1.00951.01011.00951.0101 0.3124 0.2125 1.16191.17631.16191.765 Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

34. The last four columns confirm that the bounds in Theorem (4.3) and Theorem (4.5) are tight. left bound Right bound 0.3124 22 1.90471.99871.90472.0000 0.0774 2 2 1.90391.99971.90392.0000 0.3124 99 1.07141.12341.07141.1250 0.0774 99 1.07091.12481.07091.1280 Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

35. In this example, we consider the same problem in Example (1) and we compute the eigenvalues of the original matrix and the preconditioned matrix. The Figure shows the eigenvalues clustering for the original matrix in (TOP) and the preconditioned matrix using preconditioner from case2 with (BOTTOM) Spectral Properties Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

36. Numerical Experiments Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

38. We solve the resulting linear system in three different ways. 1) without a preconditioner. 2-3) using GMRES method with two different set of parameters The Table shows GMRES iterations for these different ways. Without preconditioner 4683136 Preconditioner with 334 333 GMRES Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

39.  In this example, we apply H-CG with two different sets of parameters. These two sets ensures that the matrices H and are symmetric positive definite. These choices of parameters make the preconditioned matrix self-adjoint and positive definite with respect to the inner product H·, The first set of parameters is (γ1, γ2, α) = (0.001, 0.5, 0.002).  Theorem (4.5) implies that the eigenvalues of the preconditioned matrix lie on the union of the intervals, [1.0009, 1.0010) ∪ [1000]. The second set of parameters is (γ1, γ2, α) = (0.0001, 0.0001, 0.9999).  Theorem (4.3) implies that the eigenvalues of the preconditioned matrix lie on the union of the intervals, [0.9525, 1.0001) ∪ [10000].  We carried out computations with various mesh sizes.  For our numerical computations we have chosen TOL = 10−3 and the initial guess x0 to be the zero vector.  Iteration counts are reported in Table (6.4). This table shows that the number of iterations is insensitive to the linear system size. PCG based on H inner product Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

40. MeshSysSizeset1set2 8x820845 10x1032045 12x1245645 16x1680045 20x20124045 32x32313645 PCG based on H inner product Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

41. The following experiment report on P-H_MINRES convergence with different parameters. We apply P-H-MINRES (Algorithm 2) to the system with a stopping tolerance of 10^-6 on the residual norm with respect to H. If the number of iterations is more than 1,000, we use the symbol *. Example 6.5 We consider the same problem in Example 6.1 and we solve it using both the classical MINRES on the unpreconditioned system and the H- MINRES on the preconditioned system with a stopping tolerance of 10^-6. We choose the parameters These values are chosen such that H is symmetric positive definite. The number of iterations are reported in the following Table. P-MINRES based on H inner prod

42. h Unpreconditioned system Preconditioned system 633 893 1113 1453 1813 2753 *3 P-MINRES based on H inner prod Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

43. Example 6.6: We test, in this example, the performance of the preconditioner in the presence of discontinuous coefficients. Let with Iteration counts are in the Table. *35533333444 *35533333444 *35533344456 *37533334444 *37533444559 *3*533444446 Department of Mathematics and Statistics, KFUPM. Nov 6, 2013

44.  We present a preconditioner so that PCG based on H-inner product can be applied.  Several numerical tests confirm that the computational bounds agree with the theoretical bounds.  Numerical experiments illustrate good convergence properties.  The k-th error can be written as  We study the eigenvalues of the matrix instead of Lemma 1 [Pestana and Wathen 2013].  We found the parameters experimentally (with no relations). Conclusions and Remarks

45. Questions or comments