IWR – Universit¨at Heidelberg Prof. Dr. Guido Kanschat Alec Gilbert
Due date:14.12.2018
Homework No. 9
Finite Elements, Winter 2018/19
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).
Problem 9.1: A priori error analysis for general smooth, symmetric coefficients
(a) Show thata(·,·)defines an inner product onV, and hence thatk·kadefined above is in fact a norm.
(b) Argue that for allf ∈V∗there exist a unique solutionu∈V to the variational problem a(u, v) = f(v) for allv∈V ,
and that the solution satisfies
kuka = kfk−a , where the dual norm is given by
kfk−a = kfkV∗ := sup
kvka=1
|a(f, v)|.
(c) 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. The discrete problem is to finduh∈Vhsuch that
a(uh, vh) = f(vh) for allvh∈Vh.
Prove that the FE solutionuh∈Vhsatisfies thebest approximation propertywith respect to the energy norm:
ku−uhka = inf
v∈Vh
ku−vhka .
Note that in comparison to (2.24), the constant here is 1.
(d) Prove that the energy norm is equivalent to the usual Sobolev normk·k1,Ω:
C1kvk1,Ω ≤ kvka ≤ C2kvk1,Ω for allv∈V .
(e) Assuming additionally thatu∈Hk+1(Ω), prove the following bound on the error of the FE solution ku−uhka ≤ C3hk|u|k+1,Ω.
(f) Discuss the differences to Poisson’s equation. Discuss what happens if the coefficients are no longer symmetric.
Problem 9.2: A posteriori error analysis for general smooth symmetric coefficients
(a) Write the residual fora(w, v) = f(v)in weak (Definition 2.3.8) and strong (Definition 2.3.11) form. Compare to the Laplacian.
(b) Prove an isometry as in Lemma 2.3.9, you will need to define suitable norms.
