advertisement

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