Mathematical Problems in Industry Assignment
Q1. In lectures we defined the Sobolev space H10(V ) with the norm
||f||H10 = (Z∫Vf,if,idV)1/2.
However, since H10(V) ⊂ H1(V), it will also have the norm associated with the Sobolev space H1(V),
||f||H1 = (∫V(f2 + f,if,i)dV)1/2.
We want to show that these are equivalent norms. That is, there are constants k, K > 0 such that for any f ∈ H10(V),
k||f||H10 ≤ ||f||H1 ≤ K||f||H10. (1)
(a) Show that if (1) is true, then there also exists k', K' > 0 such that
k'||f||H1 ≤ ||f||H10 ≤ K'||f||H1. (2)
So equivalence is symmetric, as we would expect.
(b) We will show this equivalence for the one-dimensional case where V is the interval (a, b), though it is true for n = 2, 3 dimensions also. The left hand inequality of (1) is obvious (with k = 1). Let f be any bounded and continuous function with bounded and continuous first derivatives, and which is zero on the boundary of V, that is at x = a and x = b. Then for any x ∈ (a, b),
f(x) = a∫xf'(x) dx,
where f' = df/dx. Why?
(c) Apply the Cauchy-Schwarz inequality to the right hand side of this to obtain
|f(x)| ≤ (a∫x|f'|2dx)1/2 (a∫x1 dx)1/2.
(d) Show that
|f(x)|2 ≤ ||f'||2L_2|x - a|
where ||.||L2 is the L2 norm defined in lectures.
(e) By integrating both sides with respect to x over the entire interval (a, b), conclude that
||f||2L_2 ≤ C||f'||2L_2
for some C > 0 which does not depend on f. What does C depend upon?
(f) Use the result (e) to prove the right hand inequality of (1).
(g) We have now proved (1) for the special case of f as described in (b). How do you think we would prove it for any f ∈ H10(a, b)? (Describe in words.)
2. We want to prove the result that there exists M > 0 such that for any f ∈ H1(V),
∫Vf2dV ≤ M(∫∂Vf2dS + ∫Vf,if,i dV). (3)
Intuitively, this result is saying that the size of f on V is "controlled" by the size of f on the boundary and the size of the first partial derivatives of f on V, where size is measured by the L2 norm. Again we will only consider the case where V is the interval (a, b), though the result is true for n = 2, 3 dimensions also. In the one-dimensional case the boundary integral over ∂V on the right hand side takes the simple form f(a)2 + f(b)2.
(a) Let f be any bounded and continuous function with bounded and continuous first derivatives. Using a similar argument as in Question 1(b), show that or any x ∈ V,
|f(x)| ≤ |f(a) + a∫x f'dx | ≤ |f(a)| + |a∫x f'dx|,
and so
|f(x)|2 ≤ 2(|f(a)|2 + |a∫xf'dx|2)
≤ 2|f(a)|2 + ||f'||2L_2|x - a|)
(c) Integrate both sides with respect to x over the entire interval (a, b) to conclude that
||f||L_2 ≤ C(f(a)2 + ||f'||L_2)
where C does not depend on f. What does C depend upon?
(d) Deduce from (3) that there exists M' > 0 such that for any f ∈ H1(V),
||f||2H^1 ≤ M' (∫∂Vf2dS + ∫Vf,if,i dV). (4)
3. We next want to prove the result that there exists M > 0 such that for any f ∈ H1(V), the boundary integral
∫∂Vf2 dS ≤ M||f||2H^1.
Intuitively, this is saying that the size of f on the boundary is controlled by the H1 norm of f on V. Again we will only consider the case where V is the interval (a, b), though the result is true for n = 2, 3 dimensions also. In this one-dimensional case the boundary integral on the left hand side takes the simple form f(a)2 + f(b)2.
(a) Let f be any bounded and continuous function with bounded and continuous first derivatives. Using a similar argument as in Question 2(a), show that or any x ∈ V,
|f(a)| ≤ |f(x)| + |a∫xf'dx|,
and
|f(a)|2 ≤ 2(|f(x)|2 + |a∫xf'dx|2)
≤ 2|f(x)|2 + ||f'||2L_2|x - a|)
(c) Integrate both sides with respect to x over the entire interval (a, b) to conclude that
|f(a)|2 ≤ C||f||2H^1
where C does not depend on f. What does C depend upon?
(d) Prove a similar result for |f(b)|2.
Q4. Consider the weak formulation of the steady state Dirichlet problem in the form: Find v ∈ H10(V) such that
a(v, φ) = f(φ) ∀φ ∈ H10(V),
where the bilinear form a(., .) and the linear functional f(.) are defined by
a(v, φ) = ∫Vk(x)v,iφ,i dV and f(φ) = - ∫Vgφ dV - ∫Vk(x)U˜,iφ,i dV.
Here k(x) is the non-constant conductivity, g(x) is a source term and U˜ is a H1(V) extension of the boundary value U(x) to all of V.
(a) Suppose that there are constants m, M > 0 such that for any x ∈ V, m ≤ k(x) ≤ M. Show that the energy norm (a(., .))1/2 derived from the bilinear form a(., .) is equivalent to the H10 norm. (Hint: Use the property of integrals that for any functions f(x) ≥ h(x) then
∫Vf dV ≥ ∫Vh dV.)
(b) Assume that the source term g ∈ L2(V). Show that the linear functional f(.) is bounded on H10(V). (Hint: Use the result (1) from Question 1 above to bound the first integral in f(.).)
We have therefore shown that the assumptions of the Lax-Milgram Theorem are true for this problem.
Q5. Consider the weak formulation of the cooling problem described in Question 3(b) of Assignment 1: Find u ∈ H1(V) such that
a(u, φ) = f(φ) ∀φ ∈ H1(V),
where the bilinear form a(., .) and linear functional f(.) are defined by
a(u, φ) = ∫Vk(x)u,iφ,i dV + ∫∂V huφ dS and f(φ) = -∫Vgφ dV + ∫∂Vhu0φ dS.
Here k(x) is the non-constant conductivity, g(x) is a source term, h(x) is the heat transfer coefficient on the boundary and u0(x) is the temperature of the external environment.
(a) With the same conditions on k as in Question 4(a) above, and suppose that there are constants hmin, hmax > 0 such that hmax ≥ h(x) ≥ hmin, show that the energy norm (a(., .))1/2 derived from the bilinear form a(., .) is equivalent to the H1 norm. (Hint: Use the general results (4) above (5) to handle the surface integral term in the bilinear form.)
(b) Assume that the source term g ∈ L2(V ). Show that the linear functional f(.) is bounded on H1(V).
We have therefore shown that the assumptions of the Lax-Milgram Theorem are also true for this problem.
Attachment:- Assignment Files.rar