2D Resolvent Analysis on NS equation (how to do multiplication with complex conjugate)

I am trying to solve this resolvent formulation

I have (iw-L) ready from matrix transformation from https://github.com/FreeFem/FreeFem-sources/blob/develop/examples/hpddm/navier-stokes-2d-SLEPc-complex.edp. but I don’t know how to get J’ J to solve the minimum eigenvalues, where J’ is the complex conjugate. Can someone help me with that.


varf vJ([u1, u2, p], [v1, v2, q]) = int2d(Th)(
(UgradV(uc, u) + UgradV(u, uc))’ * [v1, v2]
+ nut * (grad(u1)’ * grad(v1) +
grad(u2)’ * grad(v2))
- p * div(v)
- div(u) * q - omega * u1 * v1 - omega * u2 * v2)
+ on(1, 3, 4, 5, u1 = 0, u2 = 0);
matrix Loc = vJ(Wh, Wh);
J = Loc;

varf vM([u1, u2, p], [v1, v2, q]) = int2d(Th)(
(u1 * v1 + u2 * v2) * wu1) ;

matrix Loc = vM(Wh, Wh);
Mat M(J, Loc, clean = true);

int nev = 1;
complex[int] val(nev); // array to store eigenvalues
Wh[int] def(vec)(nev); // array to store eigenvectors
complex s = 0;
string params = "-eps_tol 1.0e-11 -eps_nev " + nev + " " +
"-eps_type krylovschur -st_type sinvert " +
“-eps_target " + real(s) + “+” + imag(s) + “i”+
" -eps_smallest_magnitude”;

int k = EPSSolve(J, M, vectors = vec, values = val, sparams = params);


What you are showing in your image does not match the code snippet. What is M in the picture? It looks like you want singular values, not eigenvalues. You should use SVDSolve() to solve what is shown in your picture.

J = (iw-L). I don’t want to solve the inverse of the Laplace operator. To use SVDSolve, I have to get (iw-L)^-1.

No, but it looks like you know what you are doing, so OK.

I am not sure you are able to do this. Is not finding the interior eigenvalues, in general, problematic?
I you find a solution please let me know I am interested in something like this.

See if this thread has what you need.

It’s a very simple problem. You must know the work of professor Denis Sipp because you are doing resolvent analysis in turbulent flow? He had an article around 2015 and explained how to transform the svd problem to a generized eigenvalue problem, which is also mentioned in a PRF article of professor Beverley Mckeon around 2018. The svd is never called directly, you should also add the weighting matrix for the response and forcing fields.