I am implementing finite difference scheme for the simple 2D Laplace
equation. I am facing problem in imposing boundary conditions in 9-point
stencil. What boundary conditions should I give at the vertex i.e. at
the corner where two boundaries intersect.
e.g.- For square plate what should be boundary condition at four vertices?
What should be the boundary condition at vertices in Dirichlet problem?either homogeneous or non-homogeneous?
Please help in this regard.
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B.C. of Laplace's Equation
The Dirichlet Boundary Conditions adapted to 0≤ x ≤ 1 and 0≤ y ≤ 1 are:
Phi(0,y)=a(y)
Phi(1,y)=b(y)
Phi(x,0)=c(x)
Phi(x,1)=d(x)
without initial conditions (your case).
But which Scheme have you applied ?
Mohammed Lamine
The 9-point stencil
Mohammed, prerakchitnis's question concerns a 9-point stencil and not a 5-point one. He apparently wishes to have a higher-order approximation, though I can't readily see why---what aspect in which application in particular drives this need in what anticipated way.
prerakchitnis, check out Prof. Sanjeev Sehra[^]'s notes here[^], which, incidentally, was the second result for a Google search on "Laplace equation stencil" (the first one being the Wiki [^]). And, also appearing on the very first page of this same Google search are: (i) solution to an assignment [^] in a module taught by Prof. Sawyer [^], then at CMS [^], at the University of Pune, and (ii) this IJNME paper [^]. Advisable: First, work it all out in 1D.
--Ajit
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Never reveal your password
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--Ajit
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Points
Hello Ajit,
Do you mean that the 2nd derivatives are approximated with 5 symmetric points (centred formula) => 9 points for the two directions ?
You have also to know that the corners are located with their cartesian coodinates.
Mohammed Lamine.
9 point stencil
Thanks for the reply. But the problem is still unanswered.
For the 5 point stencil it is ok to go by approach suggested by Mohammed Lamine.
Ajit, I have gone through the resources provided by you.
I belive nine point srencil approach can not be used for 'at least for near edge nodes' in dirichlet BC problem.