Problem 1.4: Friedrichs’ inequality

IWR – Universit¨at Heidelberg Prof. Dr. Guido Kanschat Alec Gilbert

Due date:19.10.2018

Homework No. 1

Finite Elements, Winter 2018/19 Problem 1.1: Variational equations in R

Given a symmetric, positive definite matrixA∈R^n×nand a vectorb∈Rⁿand the “energy functional”

E(x) = ¹₂x^TAx−x^Tb, (1.1)

(a) Derive the variational equation of the minimization problem by studying the derivative of the auxiliary functionΦ(t) = E(x+ty)for arbitraryy∈Rⁿ.

(b) Show that a vectorx∈RⁿminimizesE(x), that is,

E(x)≤E(y) ∀y∈Rⁿ, if and only if

Ax=b.

(c) Conclude that the minimizerxexists and is unique.

Problem 1.2: Minimizing sequence

(a) Using the knowledge from Problem 1.1 that a minimiserxof the energy functional in (1.1) exists, show that a sequence {x^(k)}that satisfies

E(x^(k))→ inf

y∈R^d

E(y), (1.2)

necessarily converges tox. The “binomial formula”x^TAx−y^TAy= (x+y)^TA(x−y)and the fact thatAis invertible are useful ingredients to this proof.

(b) Show without assuming the existence of the minimizerx, that a sequence{x^(k)}for which (1.2) holds is necessarily a Cauchy sequence. Can you conclude the existence of a minimizerx?

Problem 1.3: Integration by parts

LetΩbe a domain inR^d. Use the Gauß theorem for smooth vector fieldsϕ: Ω→R^d, namely, Z

Ω

∇·ϕdx= Z

∂Ω

ϕ·nds, to show Green’s first and second formula (for smooth scalar functionsuandv)

− Z

Ω

∆uvdx= Z

Ω

∇u· ∇vdx− Z

∂Ω

∂_nuvds Z

Ω

(u∆v−v∆u) dx= Z

∂Ω

(u∂_nv−v∂_nu) ds.

Here,nis the outward unit normal vector toΩon∂Ω. The differential operators have the meaning:

∇u= (∂1u, . . . , ∂du)^T gradient

∂nu=n· ∇u normal derivative

∇·ϕ=∂₁ϕ₁+· · ·+∂_dϕ_d divergence

∆u=∇·∇u=∂11u+· · ·+∂ddu Laplacian

Problem 1.4: Friedrichs’ inequality

(a) Prove Friedrichs’ inequality

kukL²(Ω)≤cku⁰kL²(Ω), with c=b−a forΩ = (a, b)and functionsu∈C₀¹(Ω).

(b) Generalize the proof for functions inH₀¹(Ω), using that each function inH₀¹(Ω)is the limit of a sequence inC₀¹(Ω).