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Here's the flow of logic, hopefully in a slightly more readable form: We write the transmission conditions with as-yet-unknown linear operators $\mathcal{S}_1$ and $\mathcal{S}_2$ which operate on the fields $u_1$ and $u_2$ at the interface. Perform a Fourier transform in the direction $y$ which is tangential to the interface. $k$ is the spatial frequency ...


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First, for most sparse matrices you don't actually need $n$ matvecs with unit vectors to determine the diagonal entries. Rather, if you know the sparsity pattern, you can "color" the nodes in your discretization so that nodes of the same color do not couple in the matrix. As a consequence, you should be able to compute the trace of the matrix with something ...


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