Problem 2.3: Trilinearform

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

Due date:26.10.2018

Homework No. 2

Finite Elements, Winter 2018/19 Problem 2.1:

f_n(x) = |x|³

|x²|+_n¹.

(a) Show thatf_nis continuously differentiable.

(b) Show thatf_n → |x|inH¹(−1,1)asn→ ∞, where we equipH¹(−1,1)with the norm

kvk_H1(−1,1) =

kvk²_L2(−1,1)+kv⁰k²_L2(−1,1)

¹₂

= Z 1

−1

|v(x)|²+|v⁰(x)|² dx

1 2

.

Hint:Use de l’Hˆopital’s rule for quotients of sequences which diverge to infinity.

Problem 2.2:

x∈Ω

|f(x)| and kfk2= Z

Ω

|f(x)|²dx ¹₂

are not equivalent.

Hint:Find a sequence which is bounded in one norm and tends to zero with respect to the other.

Problem 2.3: Trilinearform

ForΩ⊂R²consider the term

c(w;u, v) = (w· ∇u, v), c(·;·,·) :W×V ×V →R, withW =H₀¹(Ω;R²), and V =H₀¹(Ω).

Here(·,·)denotes theL²(Ω)inner product, and we equip the spacesV andW with the norms (2.3) and (2.4), respectively.

Note: The semicolon indicates that we will later on usew as data, such thatc(w;., .)is a bilinearform which fits into our existing framework.

(a) Show thatc(·;·,·)is linear in each variable (altogether trilinear) and continuous.

Hint:Show that|c(w;u, v)|is bounded. Use the Sobolev embedding theorem.

(b) Show that the following identity holds:

(w· ∇u, u) =−1

2 ∇·w, u² .

Hint:The notation

∇·ϕ := ∂ ϕ₁

∂x1

+∂ ϕ₂

∂x2

denotes the divergence of a sufficiently regular vector fieldϕ(x) = ϕ1(x), ϕ2(x)T

.

(c) Deduce an analogous formula forw∈H¹(Ω;R²)andu∈H¹(Ω)without zero boundary conditions.

Problem 2.4: Stationary convection-diffusion equation

Consider the stationary convection-diffusion equation

−∆u+w· ∇u = f, inΩ, u = 0, on∂Ω

for a bounded domainΩ⊂R²and a given functionw: Ω→R²with∇ ·w = 0. Consider also the spaceH₀¹(Ω), which we equip with the Sobolev norm (2.3).

(a) Formulate the problem weakly for functionsu∈H₀¹(Ω).

(b) Show that there is a unique solution of the weak formulation by using the theorem of Lax-Milgram. Make suitable assumptions on the datawandf. Why don’t we use Riesz’ representation theorem directly to show existence?

(c) What happens, if we do not assume∇ ·w= 0? Can we still guarantee existence?

Definition of norms

In general, letΩ⊂ R^d for somed ∈ Nbe bounded, and denotex ∈ Ωbyx = (x1, x2, . . . , xd). We define the following norms,

kvk_L2(Ω) :=

Z

Ω

|v(x)|²dx ¹₂

, forv∈L²(Ω), (2.1)

kvk_H1(Ω) := kvk²_L2(Ω)+

d

X

j=1

∂ v

∂x_j

2

L²(Ω)

!¹₂

forv∈H¹(Ω), (2.2)

kvk_H1 0(Ω) :=

d

X

j=1

∂ v

∂xj

2

L²(Ω)

!¹₂

, forv∈H₀¹(Ω). (2.3)

For vector-valued functionsv: Ω→R^s(denotedv= (v₁, v₂, . . . , v_s)) we define the norm onH₀¹(Ω;R^s)by

kvk_H1

0(Ω;R^s) :=

s

X

i=1

kvik²_H1 0(Ω)

!¹₂

, forv∈H₀¹(Ω;R^s). (2.4)