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Old   February 25, 2004, 05:13
Default Spectral Method Boundary condition
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Hi all, I am working with spectral method to resolve 2D partial differential equation in 2D. I have two boundary condition for each of the 4 boundary. I dont know how to impose the boundary conditions as in spectral method(using collocation) number of equations are same as number of collocation points.... if i take just one of the boundary condition then i have equal number of equations ans unknowns and if i intend to incorporate both the boundary condition then i have more equations for the same number of unknowns. I tried using multiple collocation point on the boundary but .. then i have zero diagonal elements in the final matrix.. at three of the neighbouring node(of each corner). can someone help me here?? Thanx in advance,
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Old   February 25, 2004, 09:48
Default Re: Spectral Method Boundary condition
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Basically you need to remove the equations near the boundary and replace them by the boundary condtions; i.e you have something like (in 1D)

u = a_1.P_1 + a_2.P_2 + ... + a_N.P_N

where P_j(x) are your basis functions. If you now use N-k (where k is the number of boundary conditions that need to be applied - 4 in your case) collocation points then you can prescribe your boundary conditions in the remaining k equations.

( If P_N is a Tchybshev polynomial for example you would use the zeroes of the next polymial (P_N+1) as yor collocation points and throw away the zeroes nearest the wall )

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