Hi, I'd like to thank the developers for an excellent tool. I have a short question, which is presented more legibly in the attached document.
Denote the unit square by Ω and consider the problem
max( ΔV / 2, g - V) = 0 on Ω
V = 0 the boundary of Ω,
where g is a known function. Problems or this form are known as
elliptic obstacle
problems <http://en.wikipedia.org/wiki/Obstacle_problem> or, to
probabilists, as stopping problems
<http://en.wikipedia.org/wiki/Optimal_stopping>, since, under reasonable
conditions
V(x) = sup_{stopping times τ} E_{B_0=x}[g(B_τ)],
where $B$ is a two-dimensional Brownian motion.
This is (informally) equivalent to minimizing the Dirichlet energy
functional
J=\int_Ω|\nabla V|^2dx
over sufficiently regular V with V>= g.
1. Can FEniCS solve this problem? This would be incredibly useful for
solving optimal stopping and control problems. [1] and [2] discuss solving
the elliptic obstacle problems using the FEM, but I am not experienced
enough to know whether they are relevant here.
2. Are there free boundary problems which FEniCS can solve?
3. Are there other tools well suited to solving problems of this form?
Many thanks,
Ben
[1]:
http://www.win.tue.nl/casa/meetings/seminar/previous/_abstract080423_files/ObstacleProblem.pdf
[2]: http://www.math.tifr.res.in/~publ/ln/tifr49.pdf pp 95-101
fenics_obstacle_question.pdf
Description: Adobe PDF document
_______________________________________________ fenics mailing list [email protected] http://fenicsproject.org/mailman/listinfo/fenics
