Scientific Computing II

Scientific Computing II

Summer term 2018 Priv.-Doz. Dr. Christian Rieger

Christopher Kacwin

Sheet 8

The goal of this exercise sheet is to resolve singular initial conditions for parabolic equations, using a geometrically refined time mesh. We consider the model problem from the lecture

u⁰(t) +Lu(t) =g(t), t∈I = (0,1) u(0) =u0

withL(X, X^∗) an elliptic differential operator,u0 ∈X, andg∈L²(I, H) being analytic as a function in time. We discretize I with the partition T_n,σ ={I_m}ⁿ⁺¹_m=1 with grading factor σ ∈(0,1) andn+ 1 time intervallsI_m given by the nodes t₀ = 0, t_m =σ^n−m+1. The time steps ∆tm = tm−tm−1 satisfy ∆tm =λtm−1, λ= _1−σ^σ . Set γ = max{1, λ}.

Assign to every interval a polynomial approximation orderr_m and assume regularity of the solutionu|_Im ∈H^sm with s_m =α_mr_m,α_m ∈(0,1).

Exercise 1. (sharper estimate away from the initial conditions) Fix an intervalIm,m≥2. Show that there exist constants C, d >0 s.t.

ku−Π^r_I^m

muk²_L2(Im,X)≤Cσ^(n−m+2)

(γd)^2αm

(1−αm)^1−αm (1 +α_m)^1+αm

rm

withC,donly depending onu0,g.

(10 points) Exercise 2. (exponential convergence)

Let u be the exact solution and U ∈ Vr(T_n,σ, X) the computed hp-DGFEM solution.

Show that there exists µ >0 s.t. forrm=bµmc,m= 1, . . . , n+ 1, one obtains ku−Uk²_L2(I,X)≤Cexp(−bN¹²)

with constants C and bindependent of N = dimVr(T_n,σ, X).

You can use the following estimate for the first interval without proof:

ku−Uk²_L2(I1,X)≤cσⁿ, withc >0 independent ofn.

(10 points)

1