Solution Methods Iterative Techniques Lecture 6
Motivation Consider a standard second order finite difference discretization of V2u=f on a regular grid, in 1, 2, and 3 dimensions Au= f SMA-HPC⊙2003MT Solution Methods: Iterative Techniques 1
1D Finite Differences Motivation △m=b +1 6 1z=13 n points m× n matrix bandwidth b=1 Cost of Gaussian elimination O(bin)=O(n SMA-HPC⊙2003MT Solution Methods: Iterative Techniques 2
2D Finite Differences Motivation 2,y+1 了,+1, -1 0510152025 nz=105 7× points 72×m2 matrix bandwidth b=n Cost of Gaussian elimination O(b2n)=O(n4) SMA-HPC⊙2003MT Solution Methods: Iterative Techniques 3
3D Finite Differences Motivation ,k+1 2+1,,k k 了a,y+1,k 元-1,了,k i,,k-1 020406080100120 nz=725 7×× n points 3×m3 matrix bandwidth b Cost of Gaussian elimination 0(b2n)=0(n) SMA-HPC⊙2003MT Solution Methods: Iterative Techniques 4