Finite Difference Discretization of Elliptic Equations: 1D Problem Lectures 2 and 3
Poisson Equation in 1D Model Problem Boundary Value Problem(BVP) Wra(ac)= f(a) N1 x∈(0,1),w(0)=(1)=0,f∈C0N2 Describes many simple physical phenomena(.9 /.M o Deformation of an elastic bar N4 Deformation of a string under tension N5 e Temperature distribution in a bar N6 SMA-HPC⊙2003MT Finite Differences 1
Poisson Equation in 1D Model Problem Solution Properties The solution u(a) always exists ●u(a) is always“ smoother” than the data f(ac) ff(x)≥0 for all a, then u(x)≥0 for all a ello≤(1/8)flo N7 Given f(a) the solution u(e) is unique N8 SMA-HPC⊙2003MT Finite Differences 2
Numerical Finite Differences Solution Discretization Subdivide interval(0, 1) into n+ 1 equal subintervals 7+1 0 0T1 n犯n+1 j=y △ ≈wn三(a for0≤j≤m+1 SMA-HPC⊙2003MT Finite Differences 3
Numerical Finite Differences Solution Approximation For example 0(ax)1 △a(0(j+1/2)-0(-12) 10j+1 0 0 △c △c 0+1-20; a for△ a small SMA-HPC⊙2003MT Finite Differences 4