# I Non-diagonal metric for testing elliptic PDE solver

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1. Jun 22, 2017

### Pablo Brubeck

Hello, I am working with numerical relativity and spectral methods. Recently I finished a general elliptic PDE solver using spectral methods, so now I want to do Physics with it. I am interested in solving the lapse equation, which fits into this category of PDEs
$$\nabla^2 \alpha = \alpha \left(K_{ij}K^{ij}+4\pi\left(\rho+S\right)\right),$$
where $\nabla^2$ is the Laplace-Beltrami operator. Could you please provide a non-diagonal spatial metric $\gamma_{ij}$ and a corresponding intrinsic curvature $K_{ij}$ (and maybe non-zero matter sources $\rho, S$) for which I might obtain an interesting solution for the lapse? Or could you recommend a book exercise? Also I would like a way to verify my numerical solution.

I am also willing to solve for those parameters via similar elliptic equations.

Thank you.

2. Jun 27, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.