IWR – Universit¨at Heidelberg Prof. Dr. Guido Kanschat Alec Gilbert
Due date:21.12.2018
Homework No. 10
Finite Elements, Winter 2018/19
Problem 10.1: A posteriori error analysis for general smooth, symmetric coefficients cont.
LetΩ ⊂ Rd be a bounded, convex, polyhedral domain and letV = H01(Ω). Letai,j ∈ C∞(Ω) fori, j = 1,2, . . . , dbe symmetric coefficients (ai,j=aj,i) such that for allx∈Ωthe coefficient matrixA(x) = [ai,j(x)]satisfies
ξTA(x)ξ ≥ C0|ξ|2 for allξ∈Rd. Consider the bilinear forma:V ×V →Rgiven by
a(u, v) =
d
X
i,j=1
Z
Ω
ai,j(x)∂iu(x)∂jv(x) dx,
and define the correspondingenergy normonV by
kukV = kuka := p a(u, u).
Forf ∈V∗, the variational equation corresponding to the above bilinear form is to findu∈V such that
a(u, v) =f(v) for allv∈V. (10.1)
Further, let{Vh}h>0be a family of conforming finite element spaces each defined on a shape regular meshTh, and letPk(T)⊂ Vh|T for all cellsT ∈Th.
(a) Let alsoThbe locally quasi-uniform. Formulate and then prove an estimate of the form (2.89) from Lemma 2.3.12 in the notes.
Hint:Recall from the previous class that forv, w∈V the strong form of the residual is
a(Rv, w) = X
T∈Th
Z
T
rT(v)wdx− X
F∈Fih
Z
F
2{{n·A∇v}}wds , (10.2)
whererT(v) := f +∇ ·(A∇v), and we proved that forv∈V the weak residual satisfies ku−vka = kRvk−a = sup
kwka=1
|a(Rv, w)|. (10.3)
(b) As in Definition 2.3.15, define the residual based error estimatorηa,hfor (10.1).
(c) Formulate and then prove an estimate of the form (2.95) from the notes for (10.1) and the energy norm.
(d) Formulate and then prove an estimate of the form (2.96) from the notes for (10.1) and the energy norm.
