麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 18 FEM for the Poisson Problem

FEM for the Poisson Problem in R2 Apri14&16,2003

Formulations Model problem Strong Formulation Find u such that Vu=f in n2 a =0 on I for a polygonal domain N1 SMA-HPO⊙1999M FEM Poisson in R 1

Formulations Model problem Minimization/Weak Formulations Find u= arg min da(w, w)-e(w); ∈X J(w) or find∈ X such that a(u,0)=(),v∈X SMA-HPO⊙1999M FEM Poisson in R 2

Formulations Model problem Minimization/Weak Formulations where X={v∈H(s)|vr=0}=H(9), C(0U。0 Vw. VU dA SPD 0 f vdA bounded SMA-HPO⊙1999M FEM Poisson in R 3

Regularity Model problem In general, ullH1(o)< ClellH-1(0) If f EL (S)and n is convex, ur2()≤Cf(g) N2 important for convergence rate SMA-HPO⊙1999M FEM Poisson in R 4

Finite Element Triangulation Discretization ∪Tn N3 Th∈Th TR: elements k=1 K ai:nodes, interior 1 n boundary SMA-HPO⊙1999M FEM Poisson in R 5

Finite Element Approximation Discretization Space(Linear Elements) Xn={∈x|vn∈P1(T),VTn∈Th} 0 U∈C0() P1(Th):0n=c0+、C+、cyy,c,cmyy∈ SMA-HPO⊙1999M FEM Poisson in R 6

Finite Element Approximation Discretization Basis(Nodal) h=span{1,…,9n} 9;∈Xh,(m;)=6,1≤i,j≤m Support of p ppi nonzero SMA-HPO⊙1999M FEM Poisson in R2 7

Finite Element Approximation Discretization Basis(Nodal) Nodal interpretation: v E Xh 0=∑ vi pila)i 0(a )=∑091(m)=∑016n→=0(m =1 SMA-HPO⊙1999M FEM Poisson in R 8

Finite Element “ Projection Discretization Rayleigh-Ritz or Galerkin Rayleigh-Ritz Wh= arg min ba(o, w)-e(o) D∈Xh J(w) Galerkin: Uh E X, satisfies a(uh,v)=e(v),Vv∈Xh SMA-HPO⊙1999M FEM Poisson in R 9