麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 15 Discretization of the Poisson

Discretization of the Poisson Problem in rl Formulation Apri2,2003

Dirichlet Model Problems Strong Form Domain: Q =(0, 1) Find u such that (0)=(1)=0 for given f SMA-HPO⊙1999M Poisson in Rl. Formulation 1

Dirichlet Model Problems Minimization Statement Define X≡Hb(2) Find u= arg min J(w) U∈X where au dr a ac 2 0 SMA-HPO⊙1999M Poisson in Rl. Formulation 2

Dirichlet Model Problems Weak Formulation Find u∈ X such that 6J(u)=0,V0∈X E U ai fdm,u∈X 0 SMA-HPO⊙1999M Poisson in Rl. Formulation 3

Dirichlet Model Problems Notation Define a(00,0 UmUr d 0 e(o v d Minimization: u=arg min a(w, w)e(w) ∈X Weak u∈X:a(u,)=e(v),Vv∈X SMA-HPO⊙1999M Poisson in Rl. Formulation 4

Dirichlet Model Problems Generalization For any e(v)∈H1(92) find w∈Hb(S) such that w= arg min a(w, w)-e(w);or E∈Hb(2) au, v=e(o), VUE Ho(); for example, e(o)=Sco, =v(eo) is admissible SMA-HPO⊙1999M Poisson in Rl. Formulation 5

Dirichlet Model Problems regularity fe∈H1(s), ulra()≤ c eh-1() fe∈D2(92),e()=/fda lulr()≤Cofz2(g) Ni SMA-HPO⊙1999M Poisson in Rl. Formulation 6

Neumann” Model Problems Strong Form Domain: Q=(0, 1) Find u such that in 2, (0)=0, (1)=g for given f, g SMA-HPO⊙1999M Poisson in Rl. Formulation 7

Neumann” Model Problems Minimization Statement Define X={v∈H(32)|0(0)=0} Find u=arg min J(o) U∈X where (c)=1/2 wU- aC f w dac-g w(1 0 0 SMA-HPO⊙1999M Poisson in Rl. Formulation 8

Neumann” Model Problems Weak Statement Find u∈ X such that 6J(u)=0,V0∈X 1 Wr vr dac=/f vda +g u(1) Vu∈X 0 0 SMA-HPO⊙1999M Poisson in RI. Formulation 9