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

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

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

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

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

Coarse grid problem Solution: Discretisation j Discretisation Uniform triangular elements Constant elements Solution:

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

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

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

Numerical experiments and results Test problem 6 10 4

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

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

Global coarse grid error Back

Updated global coarse grid error Back

Updated global coarse grid error Back

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