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'||²_{L_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||²_{L_2} ≤ C||f'||²_{L_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'||²_{L_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||²_H^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||²_H^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'||²_{L_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||²_H^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 h_min, h_max > 0 such that h_max ≥ h(x) ≥ h_min, 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