Fuxiang 2009-07-09 12:11:52
I am trying to learn the grid method to solve the dirichlet problem
(my main reference is Jost’s PDE). I can understand most of it but
still have a few questions. I Hope somobody can help me out.
1, What if the discrete region $U_h$ is not discretely connected? Does
the method still works? My first thought is Yes.
2, In Jost’s book (and any book I read), the points in U_h must also
be in U. Is it OK to pick some points outside of U? That is, can we
let U_h include some points that are out of U but within a distance of
less than h to the boundary of U? My first thought is also Yes.
Question 1 and 2 are related. If the answer for question 1 is yes,
then U_h does not have to be “connected”; if the answer for question
2 is yes, then U_h is more poosible to be “connected” but it is
possible that it is not “simply connected”.
3, what is the best book for error analysis for this method? The books
I read do not focus on error analysis.
Spellucci 2009-07-10 09:25:13
email@example.com (Sean) writes:
this depends on whether you have the necessary boundary values for
noninterior grid points.
this depnds on the properties of the tru solution. if you can assume that
the (true) solution can be continued outside the given domain then the answer
is indeed “yes” and the technique is known as “method of fictitious points”
this is often useful for approximating von Neumann boundary data
but there are cases there this is not possible
a simply connected domain is not necessary in this connetion, but of course
a path connected one. and if U_h is not path comnnetced
in the discrete snese this will be an exception (net too crude)