Finite Difference Discretization of Elliptic Equations: FD Formulas and Multidimensional Problems Lecture 4
Finite Difference Problem Definition Formulas Given l+r+ 1 distinct points (2-1,3-+1,……,0,…,r), find the weights o such that dmy S!, dam IS of optimal order of accuracy N1 TWo approaches Lagrange interpolation o Undetermined coefficients SMA-HPC⊙2003MT Finite Differences 1
Finite Difference Lagrange interpolation Formulas Lagrange polynomials c- - Li(a) 1)(m-+1)…( - 1)…(x;-21-1)(c-m13+1)…(a Lagrange interpolant 6(a)=∑L(a) SMA-HPC⊙2003MT Finite Differences 2
Finite Difference Lagrange interpolation Formulas Approximate do d dL da =:0 doc lc=co j=-I doc x=ro Therefore dmL Te drm 0 SMA-HPC⊙2003MT Finite Differences 3
Finite Difference Lagrange interpolation Formulas Example sett=7=1,(a-1,j,m+1 Second order Lagrange interpolant 0(c) (m-21)(m-2+1) a 0-1+ )(x-31+1) (2-1j-1)(mj-mj+1 0+ 1-0j+1 (m-1-1)(m-y) +1 SMA-HPC⊙2003MT Finite Differences 4