Homework #5. (Due Oct. 15)
Math. 610:601
(Last modified on 2015-10-12 10:54)
Problem 1. Let Ω = (0, 1)2 and consider the boundary value problem:
−∆u = f,
in Ω,
un + 2u = g1 ,
on Γ1 ,
un = 0,
on ∂Ω \ Γ1 .
Here Γ1 = {(x, 0), x ∈ (0, 1)} and un denotes the normal derivative on ∂Ω.
(a) Derive a variational formulation for the above problem (in H 1 (Ω)).
(b) Show that the resulting bilinear form is coercive on H 1 (Ω).
Problem 2. Let 0 = x0 < x1 < . . . < xN = 1 denote a partition of [0, 1]
and set Vh to be the set of continuous functions which are piecewise linear
with respect to the intervals [xi−1 , xi ], i = 1, 2, . . . , N . Let u be smooth and
Ih u denote the interpolant of u in Vh , i.e.,
Ih u(xi ) = u(xi ),
i = 0, 1, . . . , N.
Set eh = u − Ih u. Show that
keh kL2 (0,1) ≤ hke′h kL2 (0,1) .
Here
N
h = max{xi − xi−1 }.
i=1
1