Using c Coefficient to Encode First-order PDE Eigenvalue Problem
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I want to apply pdeeig to solve an N = 2 PDE eigenvalue problem. For complex-valued u = [ u(1) , u(2) ]:
−( ∂x − i ∂y ) u(2) + a u(1) = λ u(1)
−( ∂x + i ∂y ) u(1) + b u(2) = λ u(2).
Here, a and b are scalar functions of x and y, but I know how to encode them in the form a u using the a coefficient matrix. My question is whether it's possible to encode the first terms in the form − ∇ · ( c ⊗ ∇u ) using the c coefficient tensor.
On that latter page, there is a formula for the components of this vector, but it is slightly ambiguous. Does the notation mean that if c is a function of x and y, then it will be differentiated via chain rule?
If I interpret the formula in this way, it seems like I can use the following trick:
c(1,2,1,2) = − i x − y
c(1,2,2,1) = i x + y
c(2,1,1,2) = i x − y
c(2,1,2,1) = − i x + y,
with all other components of c zero. Since the mixed partials cancel after chain rule expansion, I'm left with exactly the system above. In the function implementation, of course, x and y are the centroids of the triangles in the mesh, and c is represented by a 16-row matrix, as specified.
I would like to know whether this kind of trick is within the specification of the c coefficient, or whether I have interpreted the documentation incorrectly. Moreover, even if this trick works, I'd like to know if perhaps it might lead to numerical issues.
I have made an attempt to encode my problem using this method, with no good results so far. But I can't come up with a quick way to tell if the issues are due to this technical point or more fundamental issues with my physical setup. I'm hoping the community can help me either implicate or eliminate this technicality as an underlying cause.
2 Comments
Deepak Ramaswamy
on 24 Feb 2014
Edited: Deepak Ramaswamy
on 24 Feb 2014
Edwin, let me comment on the first part of the question. I believe, your C coefficient is correct. I ran it through a PDE coefficients extractor I have and I get identical results:
Before commenting on the numerical validity of the approach, I'd like to understand your problem better. Could you let me know the source of your eigenvalue problem? What do these equations represent?
Regards, Deepak
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