Numerical Solvers for BVPs

1. Numerical Solvers for BVPs By Dong Xu State Key Lab of CAD&CG, ZJU

2. Overview • Introduction • Numerical Solvers • Relaxation Method • Conjugate Gradient • Multigrid Method • Conclusions

3. Introduction • What is Boundary Value Problems? Typical BVPs

4. Discretization • Regular Grid • Irregular Grid

5. Linear System (Matrix)

6. Relaxation Methods 0<w<2

7. Conjugate Gradient • Steepest Descent Method • Search in the direction of the gradient of given point (local approximation). • The local gradient doesn’t point to the elliptic center. • Conjugate Gradient Method • Search in the direction pointing to the elliptic center. • Iterate at most n steps. (n – the order of the matrix) • Only need Ap & ATp (matrix multiplies vector), especially efficient for sparse matrix. • Preconditioning

8. Multigrid Methods • Multigrid Methods – NOT a single algorithm, BUT a general framework. • Solve elliptic PDEs (BVPs) discretized on N grid points in O(n) operations. • Multigrid means using fine-to-coarse hierarchy to speed up the convergence of a traditional relaxation method. • Another approach is discretizing the same underlying PDE in multiple resolution. (FMG method)

9. Equations • Equation • Discretization • Correction • Residual/Defect • Linear relation between correction and residual • Only knows residual how to get correction? • Approximation • Jacobi iteration: diagonal part • Gauss-Seidel iteration: lower triangle • Get new approximation

10. A New Way • “Coarsify” rather than “Simplify” • Take H = 2h • New residual equation Approximation • Restriction operator • Prolongation operator • Get new approximation

11. Coarse-grid Correction Scheme • Compute the defect on the fine grid. • Restrict the defect. • Solve exactly on the coarse grid for the correction. • Interpolate the correction to the fine grid. • Compute the next approximation.

12. Two-Grid Iteration • Pre-smoothing: Compute by applying steps of a relaxation method to . • Coarse-grid correction: As above, using to give . • Post-smoothing: Compute by applying steps of the relaxation method to . Key Insight: Relaxation methods are good smoothing operators. (High freq. attenuates faster than low freq.)

13. Operators • Smoothing Operator S • Gauss-Seidel, NOT SOR. • Restriction Operator R • Prolongation Operator P Straight injection, half weighting, full weighting. Relationship Bilinear interpolation

14. Multi-Grid • Cycle – One iteration of a multigrid method, from finest grid to coarser grids and back to finest grid again. • , the number of two-grid iterations at each intermediate stage (resolution/level). • V-cycle – • W-cycle – (named by shape)

15. Multigrid Demo

16. Full Grid Algorithm • First approximation • Arbitrary, on the finest grid. (Simple Multigrid, uh = 0) • Interpolating from a coarse-grid solution. • Nested Iteration • Get coarse-grid solution from even coarser grids. • At the coarsest grid, start with the exact solution. • Need f at all levels, while simple multigrid only needs f at the finest level. • Produce solutions at all level, while simple multigrid at the finest level.

17. Full Grid Demo

18. Conclusions • One Grid • Two Grid • Multi-Grid • Full Grid

19. Reference • Numerical Recipe in C

20. Thank you