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Numerical Solvers for BVPs

This document provides a comprehensive overview of advanced numerical solvers for Boundary Value Problems (BVPs). It introduces key concepts such as the relaxation method, conjugate gradient method, and multigrid techniques. The discussion covers the discretization of regular and irregular grids, the efficiency of solving linear systems, and the significance of preconditioning. Key insights into the iterative process, smoothing operators, and interpolation are highlighted. Concluding remarks focus on the application of these methodologies for improved computational efficiency in solving elliptic partial differential equations.

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Numerical Solvers for BVPs

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  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

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