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(adsbygoogle = window.adsbygoogle || []).push({});

$$ \nabla^2 \alpha = \alpha \left(K_{ij}K^{ij}+4\pi\left(\rho+S\right)\right), $$

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.

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# I Non-diagonal metric for testing elliptic PDE solver

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