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Problem 1
Gronwall’s inequality is used for the analysis of time-dependent problem. There are two versions: the
continuous in time version and the discrete in time version.
(1) Let f, g and h be continuous nonnegative functions defined on an interval a ≤ t ≤ b. Assume that g is
non-decreasing and that for all t ∈ [a, b]
Z t
f (t) + h(t) ≤ g(t) +
f (s)ds
a
Prove that for all t ∈ [a, b]
f (t) + h(t) ≤ et−a g(t)
(2) Let τ > 0 be the time step value. Let tn = nτ for 0 ≤ n ≤ N . Let (f (tn ))n , (g(tn ))n and (h(tn ))
be sequences of nonnegative functions f, g, h evaluated at tn for all n. Assume that g is non-decreasing and
that there exists a positive constant C such that for all 0 ≤ n ≤ N :
f (tn ) + h(tn ) ≤ g(tn ) + Cτ
n−1
X
f (tj )
j=0
Prove that for all 0 ≤ n ≤ N , we have
f (tn ) + h(tn ) ≤ g(tn )eCT
where T = N τ
Problem 2
Let Ω ⊂ IRd , d = 1, 2, 3, let a be a positive constant and let β be a constant vector in IRd . Consider the
problem:
∂u
− a∆u + β · ∇u = f, in (0, T ) × Ω
∂t
u = 0, on (0, T ) × ∂Ω
u(t = 0) = u0 , inΩ
We discretize this problem in space by using the finite element method of order r, and in time by using the
backward Euler method. Let h denote the mesh size and τ denote the time step. Denote the numerical
solution at time tn by unh .
(1) Give a detailed definition of the scheme: in other words define the discrete variational problem.
(2) Show existence and uniqueness of the discrete solution unh
(3) State and prove a stability bound for unh in the L2 norm in space.
(4) State and prove an error estimate for u(tn ) − unh in the L2 norm in space.
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