homework 4 - math.tamu.edubonito/teaching/mth610/hw4.pdf · numerical methods for pdes homework 4...
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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.