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

Convergence properties of the Local Defect Correction method for Time-Dependent PDEs. Remo Minero Eindhoven University of Technology 16th November 2005. Outline. Introduce Local Defect Correction (LDC) Iterative procedure Investigate the convergence behavior of LDC Numerical experiments.

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

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  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

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