1. Local Defect Correction for the Boundary Element Method
Kakuba Godwin
CASA day, Tuesday May 9, 2006

2. Outline Introduction BEM LDC Why LDC for BEM Coarse grid problem
Fine grid problem
The local defect
Defect correction and the Algorithm
Numerical experiments and results
Conclusions and future work

3. Introduction: BEM 2u = 0 in , given u or nu at . Problem
The solution of a function is expressed in terms of
its values and values of its normal derivatives
at the boundary in an integral equation.
(mother equation)
Discretisation: Elements in the boundary

4. LDC in  Coarse grid solution BCs on   Large error Defect BCs dH
Fine grid solution

5. Why LDC for BEM Problems with high local activity Uniform grids
Full matrices

6. Coarse grid problem Solution: Discretisationj
Discretisation
Uniform triangular elements
Constant elements
Solution:

7. Fine grid problem H
H/2 , , Ah l AH c = uh
Solution:
l,0

8. The local defect  AHuH = bH AH c = l
For we have two solutions, coarse and
fine grid solutions
H l 
AHuH = bH
+ defect = c AH l

9. Defect correction and the Algorithm
for i = 1,2,…
The Algorithm
Initialisation
Solve the basic coarse grid problem
Solve the local fine grid problem
Iterations i = 1,2,…
Solve the updated coarse grid problem
Solve the local fine grid problem

10. Numerical experiments and results
Test problem 6 10 4

11. Results  Factor 2 reduction in complexity
Let be the size of the global problem, the size of the local problem,
the number of lH nodes
Composite problem complexity:
Complexity of the BEMLDC algorithm:
 Factor 2 reduction in complexity

12. Conclusions Future work Next
Same accuracy as the uniform grid whose size is equivalent
to the grid size of the local problem
Reduced complexity by a factor 2
Future work
Practical problem
Convergence
Next

13. Global coarse grid error
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14. Updated global coarse grid error
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15. Updated global coarse grid error
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16. THANK YOU.