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OVERVIEW

OVERVIEW. MULTIGRID. Why multigrid? Basic multigrid principles Full multigrid (FMG) Nonlinear multigrid (FAS) Adaptive multigrid. Why Multigrid?. Error Components.

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OVERVIEW

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  1. OVERVIEW

  2. MULTIGRID • Why multigrid? • Basic multigrid principles • Full multigrid (FMG) • Nonlinear multigrid (FAS) • Adaptive multigrid

  3. Why Multigrid?

  4. Error Components

  5. The main idea of multigrid (MG) is that a boundary value problem is discretized on a sequence of grids with increasing mesh sizes rather than on a single grid.

  6. Consider the nonlinear boundary system of equations presented in the form OR

  7. , ,

  8. Adaptivity Why? 1- accurate numerical solution of partial differential equations 2- reduce memory requirements and computing time. Where? 1- Locally non-smooth solutions (boundary layers, interior layers, shocks, turbulence) 2- non-smooth domains. 3- Singularities/discontinuities in PDE How? 1- Grid resolution (h, r – refinement) 2- Order of discretization (p-refinement)

  9. For new applications or hard problems, one(or more) MG components need to be tuned to retain MG Convergence.

  10. In the presented thesis, we address some main multigrid difficulties and propose some new developments in the MG field that could have an impact in extending MG applicability.

  11. First Contribution: MG for nonlinear bifurcation problems Bratu Equation is an elliptic nonlinear partial differential equation which comes from a simplification of the solid fuel ignition model in thermal combustion theory We propose Krylov subspace methods as smoothers

  12. Comparisons with Hackbusch work [40] Results presented in [40] demonstrate that MG-based methods that use Gauss–Seidel or Jacobi smoothers are excellent in obtaining the first solution of Bratu problem.However, they even fail to converge to the second solution when h is smaller than 1/8. Our Results show that MG-based methods can converge at higher grid levels ( up to h=1/256)

  13. Solutions at Small Parameter Values Comparison with Washio and Oosterlee [143] Results in [143], FAS method converges after 91 W(2,2) cycles to the second solution starting by very near initial guess on a grid with mesh size h = 1/128 for C = 0.2 , however, it diverges for C = 0.1. Also, the method is very sensitive to initial guess and that they cannot apply FMG since coarser grids cannot provide good approximation to the solution. Our proposed smoothers could solve at small value of C for different grid sizes. Moreover, FMG technique is employed with extrapolation to provide good initial guess on the finest grid starting at very coarse grid h = 1/2.

  14. Second Contribution: Assessment of Truncation Error Estimation within FE and MG methods. Tau-Extrapolation, Tau-Stopping criterion, Tau-Indicator Local Truncation Error Relative Local Truncation Error Correct Computations Incorrect Corrected

  15. Prolongation of truncation error estimator We define weighted interpolation

  16. Applications of Truncation Error Stopping Criterion for MG Iterations Tau Extrapolation

  17. Indicator for Adaptive FEMG -indicator is compared with the well known robust energy norm indicator

  18. Third Contribution: Data handling for 3D MG to solve Anisotropic PDE. Anisotropy due to PDE coefficients Anisotropy due to stretched grids

  19. Plane(x,y,Nz) z Plane (x,y,z) y Plane (x,y,1) x MG needs block smoothers(Plane smoother)

  20. Apply Kronecker tensor product Given Ax, Ay and Az, then Also, the same concept can be applied to the 1D transfer operators

  21. Future Work • Extension of the proposed smoother within MG to other hard PDEs. • Studying smoothing properties of Krylov methods to provide efficient smoothers aiming to extend the applicability of MG. • Extension of the proposed data handling technique to3D-MG unstructured tetrahedron meshes with FE method on general 3D domains. • Extension of the truncation error estimator technique to FE method on general 3D unstructured meshes. • Comparison and integration between geometric and algebraic multigrid methods.

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