麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 2 Finite Difference Discretization

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

Numerical Finite Differences Solution Equations Waa=f suggests j+l 2i;+uj f(∞)1≤j≤7 0 7+1 0 A i=f SMA-HPC⊙2003MT Finite Differences 5

Numerical Finite Differences Solution Equations 2-10 f(a1) 12-1 f(a2) A △ -12-1 f(an-1 0…0-12 f(on) (Symmetric) A∈IRm f∈R SMA-HPC⊙2003MT Finite Differences 6

Numerical Finite Differences Solution Solution Is A non-singular For any 0 01,02 0 2410△a2( (2+(02-01-1)2+ 7=2 Hence vT A v>0, for any v#0(A is SPD)N9 Ai=f i exists and is unique N10 SMA-HPC⊙2003MT Finite Differences 7

Numerical Finite Differences Solution Example um2=(3x+x2)e,x∈(0,1) with u(0)=u(1)=0 ake=5,△a=1/6 SMA-HPC⊙2003MT Finite Differences 8

Numerical Finite Differences Solution Example SMA-HPC⊙2003MT Finite Differences 9