麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 16 Discretization of the Poisson Problem in RI: Theory and Implementation

Discretization of the Poisson Problem in RI: Theory and Implementation Apr|7&9,2003

Goals Theory A priori A priori error estimates N1 bound various“ measures” of u exact]-un [approximate] in terms of C(n, problem parameters h [mesh diameter, and u SMA-HPO⊙1999M Poisson in IR: Theory Practice 1

Goals Theory A priori uL2xn=f,(0)=(1)=0 a(u,)=e(u) V∈X (00。0)三 Wr Ur da, e(o)=f 0 a 0 X={0∈H(92)|(0)=0(1)=0} SMA-HPO⊙1999M Poisson in IR: Theory Practice 2

Goals Theory A priori Up a(uh,)=e(u),V0∈Xh (00。0)三 Wr Vr da, e(o)=f 0 a 0 Xh={v∈Xvl2∈m1(T),VTh∈Th} SMA-HPO⊙1999M Poisson in IR: Theory Practice 3

Goals Theory A posteriori A posteriori error estimates N2 bound various“ measures” of a [exact]-uh [approximate in terms of C(Q, problem parameters h[mesh diameter], and uh SMA-HPO⊙1999M Poisson in IR: Theory Practice 4

Projection Theory Definition Given Hilbert spaces Y andZCY (Iy,)y=(y,)y,0∈z ∈Y defines the projection of y onto Z, Ily I:Y→z SMA-HPO⊙1999M Poisson in IR: Theory Practice 5

Projection Theory Property The projection Ily minimizes‖y-21|y,Vz∈Z. Why? ly-(Ily +u)y=((y-Ily)-v,(y-Ily)-vy anyz∈z Iy-IIyly-2(y-IIy, uy +la, VUEZ 0:v∈z SMA-HPO⊙1999M Poisson in IR: Theory Practice 6

Projection Theory Geometr Geometry of projection y y Orthogonality:(y-IIy, UY=0, VUEZ SMA-HPO⊙1999M Poisson in IR: Theory Practice 7

The Interpolant Theory Definition Recall Xh={∈X|vln2∈P1(Tn),WTh∈h} U∈Xh2 0 1=1 SMA-HPO⊙1999M Poisson in IR: Theory Practice 8

The Interpolant Theory Definition Given w E X, the interpolant Lh w satisfies Tn∈Xh; and th(a2)=w(c2),a=0,…,,m+1 hw(a w(i) pila m+1 SMA-HPO⊙1999M Poisson in IR: Theory Practice 9