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  "body": "\n## Summary\n\nThe discrete Laplace operator is the analogue of the [[Laplacian|0E0B6RA8K0]] for functions on a grid or on the vertices of a graph.\nAt each point it adds up the differences between the values at the neighbours and the value at the point.\nOn the grid $h\\Z^d$ it gives the standard finite difference approximation of $\\sum_i \\partial^2 u / \\partial x_i^2$.\nOn a finite graph it is, up to sign, the [[Laplacian matrix|TN6M6DVZRP]], and it links random walks, electrical networks and discrete harmonic functions.\n\n## Explanations\n\n### L1 human\n\nAsk at each point how much its value differs from the values around it.\nIf the point sits in a dip, its neighbours are higher, and the discrete Laplacian there is positive.\nIf it sits on a bump, the discrete Laplacian is negative.\nOn the integers, $u(m) = m^2$ gives $(m+1)^2 - 2m^2 + (m-1)^2 = 2$ at every point, just as the second derivative of $x^2$ is $2$.\n",
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