An evolving surfaceΓ(t)is given directly by the mappingX(·,t):Γ0 →R3: Γ(t

Universit¨at T ¨ubingen T ¨ubingen, den 12.06.2018 Mathematisches Institut

Dr. Bal´azs Kov´acs

7. Exercise sheet for Numerik f ¨ur Differentialgleichungen auf Oberfl¨achen

Exercise 16. An evolving surfaceΓ(t)is given directly by the mappingX(·,t):Γ⁰ →_R³: Γ(t) =

x ∈_R³ _with x₁= p₁+_max{_0,p₁}_,x₂ = ^g(p,t)p₂ q

p²₂+p²₃

,x₃ = ^g(p,t)p₃ q

p²₂+p²₃

p∈_Γ⁰

,

where the initial surfaceΓ⁰is the unit sphere, and the function g(p,t) =e^−2t

q

p²₂+p²₃+ (1−e^−2t)

(1−p²₁)(p²₁+0.05) +p²₁ q

1−p²₁

.

Write a short code which visualises the surface evolution using the direct mapping. As an initial triangulation use a mesh from a previous programming exercise.

Hint.The code is indeed short. Apart from the information given above, nothing else is needed.

Exercise 17. Visualise the evolving surface Γ(t)from Exercise 12, using the computed velocity, and the pseudo-code from part (c).

Please, download the codehttps://na.uni-tuebingen.de/ex/surfPDE_ss18/func_v.m, and bring your laptop along!

Exercise 18. (a) Derive the fully discrete scheme ak-step BDF method for the numerical solution of the heat equation on an evolving surface. Recall that the corresponding matrix vector formulation is

d dt

M(t)u(t)+A(t)u(t) =b(t), with given initial data.

(b) Write a pseudo-code.

Discussed on the tutorials on 26.06.2018.