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

vL2(K) 6 c(vL2(K) + vL2(K)

).

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

vL2(K) 6 c(h1/2vL2(K) + h1/2vL2(K)

).

Exercise 2 40%

Let R2 be a polygonal domain, C0()2, C0() and f L2(). Assume that for somepositive constants 0, 1, 1, there holds 0 < 0 6 (x) 6 1 and |(x)| 6 1 for all x . Theoutward pointing unit normal to is denoted by (defined a.e.) and we consider a decompositionof the domain boundary := in two disjoint pieces = 1 2. In addition, we suppose thatdiv() = 0 and > 0 on 2. We propose to analyze a stabilized numerical method for thefollowing convection-diffusion problem: Seek u : R such that

div(u) + u = f, in

together with the boundary conditions

u = 0 on 1 and

u = 0 on2.

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) > 0v2L2() |a(w, v)| 6MwL2()vL2()

and deduce the existence and uniqueness of a weak solution.

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

Vh :={vh C0() | vh|T Pk(T ), T Th

}X

and define the approximate bilinear form on Vh Vh

ah(vh, wh) := a(vh, wh) + h

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) > hvh2L2() with h := 0 + h. Deduce the existence anduniqueness of uh Vh.

4. Show that for all vh Vh

h(vh uh)2L2() 6 ah(vh, vh uh) F (vh uh),

deduce that

(vh uh)L2() 61

hsup

whVh

|ah(vh, wh) a(vh, wh)|whL2()

+M

h(vh u)L2()

and therefore

(uuh)L2() 6 infvhVh

{1

hsup

whVh

|ah(vh, wh) a(vh, wh)|whL2()

+

(1 +

M

h

)(vh u)L2()

}.

(Strangs lemma).

5. Prove that for all vh Vh, there holds

supwhVh

|ah(vh, wh) a(vh, wh)|whL2()

6 hvhL2().

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

(IThu u)L2() 6 C1hk|u|Hk+1()

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

(IThu)L2() 6 uL2() + C2hk|u|Hk+1().

7. Show the existence of a constant C3 such that

(u uh)L2() 6 C3{(

1 +M

h

)|u|Hk+1()hk +

huL2()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.

40%40%20%

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