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

2. Convergence of LDC for Time-Dependent PDEs
Outline
Introduce Local Defect Correction (LDC)
Iterative procedure
Investigate the convergence behavior of LDC
Numerical experiments
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

3. Convergence of LDC for Time-Dependent PDEs
What is LDC? H
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
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

4. Convergence of LDC for Time-Dependent PDEs
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 Δt tn t tn
tn-1 δt t tn
tn-1 t tn
tn-1
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

5. Convergence of LDC for Time-Dependent PDEs
LDC iteration
Coarse grid
solution at tn
Fine grid
Boundary conditions
Defect
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

6. Convergence of LDC for Time-Dependent PDEs
The defect
PDE
Coarse grid discretization
Fine grid solution is more accurate
Defect
Correction
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

7. Convergence of LDC for Time-Dependent PDEs
The safety region
Points where the defect is
actually computed
No safety region
With safety region
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

8. Convergence of LDC for Time-Dependent PDEs
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:
Sept 28th, 2005
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
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

10. Convergence of LDC for Time-Dependent PDEs
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
1D Poisson eq.
2D Poisson eq.(*)
No safety region
||Miter||∞ = 0, H
||Miter||∞ = C·H, (H0)
With safety region
||Miter||∞ = 0 , H
||Miter||∞ = C·H2 , (H0)
Sept 28th, 2005
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
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

12. 1D results: no safety region
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

13. 1D results: with safety region
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

14. 2D numerical experimentsy 1
Local region = (0,0.5)x(0,0.5)
h = H/2
δt = Δt/2
0.5 x
0.5 1
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

15. 2D results: no safety regionH
||Miter||∞
ratio
1/64
9.1·10-3
1/128
4.8·10-3
0.53
1/256
2.4·10-3
0.50
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

16. 2D results: with safety region
||Miter||∞
ratio
1/64
8.3·10-5
1/128
2.2·10-5
0.26
1/256
5.3·10-6
0.25
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs

17. Convergence of LDC for Time-Dependent PDEs
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
Sept 28th, 2005
Convergence of LDC for Time-Dependent PDEs