1 / 17

Convergence properties of the Local Defect Correction method for Time-Dependent PDEs

Explore the convergence behavior of Local Defect Correction (LDC) method for solving time-dependent PDEs. The iterative procedure combines coarse and fine grid solutions to achieve accuracy. Discover the impact of safety regions on convergence through numerical experiments.

plumb
Download Presentation

Convergence properties of the Local Defect Correction method for Time-Dependent PDEs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Convergence properties of the Local Defect Correction methodfor Time-Dependent PDEs Remo Minero Eindhoven University of Technology 16th November 2005

  2. Outline • Introduce Local Defect Correction (LDC) • Iterative procedure • Investigate the convergence behavior of LDC • Numerical experiments Convergence of LDC for Time-Dependent PDEs

  3. H What is LDC? • An adaptive method to solve PDEs with highly localized properties • A coarse grid solution and a fine grid solution are iteratively combined Uniform structured grids h Convergence of LDC for Time-Dependent PDEs

  4. t tn-1 tn t tn-1 tn t tn-1 tn One time step with LDC • Integrate on the coarse grid • Provide boundary conditions locally • Integrate on the local fine grid • Until convergence • Compute a defect at forward time • Solve a modified coarse grid problem • Provide new boundary conditions locally • Integrate on the fine grid with updated boundary conditions t tn-1 tn Δt δt Convergence of LDC for Time-Dependent PDEs

  5. Boundary conditions Coarse grid solution at tn Fine grid solution at tn Defect LDC iteration Convergence of LDC for Time-Dependent PDEs

  6. The defect • PDE • Coarse grid discretization • Fine grid solution is more accurate • Defect • Correction Convergence of LDC for Time-Dependent PDEs

  7. The safety region Points where the defect is actuallycomputed No safety region With safety region Convergence of LDC for Time-Dependent PDEs

  8. The iteration matrix Theorem: if the LDC iteration converges on the interface ΓH, then the entire LDC iteration converges. • Motivation: fix BC for fine grid problem • (interface)iteration error: • Iteration matrix: • Convergence if: Convergence of LDC for Time-Dependent PDEs

  9. Measuring ||Miter||∞ experimentally • Consider • Discretization • centered differences + Euler backward • Perform one time step with LDC • Measure interface iteration errors Convergence of LDC for Time-Dependent PDEs

  10. What do we expect to see? • Δt  0, ||Miter||∞  0 • Very little to correct • Δt  +∞, stationary case limit (0=2u+f) (*) M.J.H. Anthonissen, R.M.M. Mattheij, and J.H.M. ten Thije Boonkkamp, Numerische Matematik, 2003 • In general ||Miter||∞ <1 Convergence of LDC for Time-Dependent PDEs

  11. 1D numerical experiments Local region = (0,0.5) h = H/5 δt = Δt/5 x 0.5 1 0 Convergence of LDC for Time-Dependent PDEs

  12. 1D results: no safety region Convergence of LDC for Time-Dependent PDEs

  13. 1D results: with safety region Convergence of LDC for Time-Dependent PDEs

  14. 2D numerical experiments y 1 Local region = (0,0.5)x(0,0.5) h = H/2 δt = Δt/2 0.5 x 0 0.5 1 Convergence of LDC for Time-Dependent PDEs

  15. 2D results: no safety region Convergence of LDC for Time-Dependent PDEs

  16. 2D results: with safety region Convergence of LDC for Time-Dependent PDEs

  17. Conclusions • LDC: an adaptive method for solving PDEs • Coarse and fine grid solution iteratively combined • We study iteration on the interface only • Numerical experiments show • LDC has good convergence properties • Faster convergence if we use a safety region Convergence of LDC for Time-Dependent PDEs

More Related