Error: Energy Norm Theory Definition Of interest: for a(a)(exact solution wn(a)(finite element approximation) =ela)=(u-un(a)(discretization error) find bound for elll in terms of h, w SMA-HPO⊙1999M Poisson in R: Theory &Practice 15
Error: Energy Norm Theory Orthogonality Since a(u,)=e(U),V∈X then a(u, v)=e(o), VvE Xn(Xh C X) but a(uh,v)=e(),v∈Xh so a(u-Wh,v)=0, Vve Xh(bilinearity SMA-HPO⊙1999M Poisson in R: Theory &Practice16
Error: Energy Norm Theory General Bound For any wh=wh+h∈Xh, 0h∈X alu- Wn, u-wn)=a((u- un)-Uh, (u-an)-Wh -hl‖l alu- uh, u-uh)- 2a(a-Wh, Un)+a(oh, Un) 0: orthogonality >0ifh≠0 inf‖l-h‖l uh∈Xh SMA-HPO⊙1999M Poisson in R: Theory Practice 17
Error: Energy Norm Theory General Bound In words: even if you knew a you could not find a wn in Xh more accurate than uh in the energy norm SMA-HPO⊙1999M Poisson in R: Theory &Practice 18
Error: Energy Norm Theory General Bound Geometry X ⊥ X a(u-h,u)=0, Vu∈Xh >uh=lh u: the projection of(closest point to u on Xh in the a norm SMA-HPO⊙1999M Poisson in R: Theory &Practice 19