Numerical Methods for PDEs Homework 4MATH 610 - 600 Spring 2018A. Bonito

Homework 4

Exercise 1 40%

Let K be the reference triangle in R2 and K be a triangle in Rd, d > 1. Let h = diam(K) and

F : K → K be an affine tranformation.Show that there exists a constant c such that for any v ∈ H1(K) there holds

‖v‖L2(∂K) 6 c(‖v‖L2(K) + ‖∇v‖L2(K)

).

Deduce that there exists a constant c independent of h such that for any v ∈ H1(K),

‖v‖L2(∂K) 6 c(h−1/2‖v‖L2(K) + h1/2‖∇v‖L2(K)

).

Exercise 2 40%

Let Ω ⊂ R2 be a polygonal domain, β ∈ C0(Ω)2, µ ∈ C0(Ω) and f ∈ L2(Ω). Assume that for somepositive constants µ0, µ1, β1, there holds 0 < µ0 6 µ(x) 6 µ1 and |β(x)| 6 β1 for all x ∈ Ω. Theoutward pointing unit normal to Ω is denoted by ν (defined a.e.) and we consider a decompositionof the domain boundary Γ := ∂Ω in two disjoint pieces Γ = Γ1 ∪ Γ2. In addition, we suppose thatdiv(β) = 0 and β · ν > 0 on Γ2. We propose to analyze a stabilized numerical method for thefollowing convection-diffusion problem: Seek u : Ω→ R such that

−div(µ∇u) + β · ∇u = f, in Ω

together with the boundary conditions

u = 0 on Γ1 and µ∂

∂νu = 0 onΓ2.

1. Derive an adequate weak formulation of the above problem: Seek u ∈ X such that

a(u, v) = F (v), ∀v ∈ X.

2. Show that for all v, w ∈ X, there exist a constant M > 0 such that

a(v, v) > µ0‖∇v‖2L2(Ω) |a(w, v)| 6M‖∇w‖L2(Ω)‖∇v‖L2(Ω)

and deduce the existence and uniqueness of a weak solution.

3. Given a shape-regular, quasi-uniform triangulation Th of Ω (h denotes the largest outer circlediameter), we set for k > 1

Vh :=vh ∈ C0(Ω) | vh|T ∈ Pk(T ), ∀T ∈ Th

∩X

and define the approximate bilinear form on Vh × Vh

ah(vh, wh) := a(vh, wh) + αh

∫Ω

∇wh · ∇vh, ∀wh, vh ∈ Vh.

Consider the finite element approximation uh ∈ Vh of u ∈ X defined as satisfying

ah(uh, vh) = F (vh) ∀vh ∈ Vh.

Show that ah(vh, vh) > µh‖∇vh‖2L2(Ω) with µh := µ0 + αh. Deduce the existence anduniqueness of uh ∈ Vh.

4. Show that for all vh ∈ Vh

µh‖∇(vh − uh)‖2L2(Ω) 6 ah(vh, vh − uh)− F (vh − uh),

deduce that

‖∇(vh − uh)‖L2(Ω) 61

µhsup

wh∈Vh

|ah(vh, wh)− a(vh, wh)|‖∇wh‖L2(Ω)

+M

µh‖∇(vh − u)‖L2(Ω)

and therefore

‖∇(u−uh)‖L2(Ω) 6 infvh∈Vh

1

µhsup

wh∈Vh

|ah(vh, wh)− a(vh, wh)|‖∇wh‖L2(Ω)

+

(1 +

M

µh

)‖∇(vh − u)‖L2(Ω)

.

(Strang’s lemma).

5. Prove that for all vh ∈ Vh, there holds

supwh∈Vh

|ah(vh, wh)− a(vh, wh)|‖∇wh‖L2(Ω)

6 αh‖∇vh‖L2(Ω).

6. Recall that there exists a constant C such that (interpolation estimates)

‖∇(IThu− u)‖L2(Ω) 6 C1hk|u|Hk+1(Ω)

provided u ∈ Hk+1(Ω) (which is assumed from now on). (OPTIONAL) Deduce that thereexist a constant C2 such that

‖∇(IThu)‖L2(Ω) 6 ‖∇u‖L2(Ω) + C2hk|u|Hk+1(Ω).

7. Show the existence of a constant C3 such that

‖∇(u− uh)‖L2(Ω) 6 C3

(1 +

M

µh

)|u|Hk+1(Ω)h

k +α

µh‖∇u‖L2(Ω)h

.

Exercise 3 20%

Let K = [0, 1]2 be the unit square and denote by qi, i = 1, ..., 4, its vertices and by ai, i = 1, .., 4,the midpoints of its sides. Set P = Q1 := p(x, y) = (ax+ b)(cy + d) : a, b, c, d ∈ R be the spaceof polynomial of degree at most 1 in each direction.

1. For N := N1, N2, N2, N4, where Ni(p) = p(qi), i = 1, ..., 4, show that the finite elementtriplet (K,P,N ) is unisolvent.

2. For N :=N1, N2, N3, N4

, where Ni(p) = p(ai), i = 1, ..., 4, show that the finite element

triplet (K,P, N ) is not unisolvent.