IWR – Universit¨at Heidelberg Prof. Dr. Guido Kanschat Alec Gilbert
Due date:7.12.2018
Homework No. 8
Finite Elements, Winter 2018/19 Problem 8.1: Higher-order variational problem
LetΩ ⊂Rd be a bounded, convex, polyhedral domain. LetV =H02(Ω)and forf ∈ V∗(the dual space ofV) consider the following variational problem: Findu∈V such that
a(u, v) = f(v) for allv∈V , (8.1)
where the bilinear forma(·,·) :H02(Ω)×H02(Ω)→Ris given by
a(u, v) :=
Z
Ω
Xd
i,j=1
∂2
∂xi∂xj
u(x) ∂2
∂xj∂xi
v(x) dx.
(a) Show that anyu∈V also satisfies
∂ u
∂xj
∈H01(Ω) forj= 1,2, . . . , d . Recall thatV =H02(Ω)is the completion ofC00∞(Ω)with respect to the normk·k2,Ω. (b) Show that
a(u, u) = |u|22,Ω for allu∈V . (c) Prove that there exists a constant0< C0<∞such that for allu∈V
kuk2,Ω ≤ C0|u|2,Ω.
(d) Show that for eachf ∈V∗there exists a unique solutionu∈V to (8.1).
(e) For a family of shape-regular meshes{Th}h>0ofΩ, consider the conforming finite element spacesVh ⊂V, where for each cellT ∈Ththe restriction ofVhtoT containsPk(T)fork≥2. The discrete problem is to finduh∈Vhsuch that
a(uh, vh) =f(vh) for allvh∈Vh. Assuming additionally thatu∈Hk+1(Ω), prove the following energy error estimate
ku−uhk2,Ω ≤ C1hk−1|u|k+1,Ω.
(f) Prove the following error estimate in theH1norm
ku−uhk1,Ω ≤ C2hk|u|k+1,Ω, and state the extra regularity assumptions that are required in order for this to hold.
Hint:Consider an appropriate dual problem and use the Aubin–Nitsche trick.
(g) Discuss the construction of such a finite element spaceVh. What conditions must the functions inVhsatisfy in order to be conforming?
