Difference and functional equations Definition

Discrete Laplace operator

Definition

Let $d \ge 1$ and let $e_1, \dots, e_d$ be the standard unit vectors of $\Z^d$ . The discrete Laplace operator sends a function $u \colon \Z^d \to \R$ to the function $\Delta u \colon \Z^d \to \R$ given by $(\Delta u)(m) = \sum_{i=1}^{d} \big( u(m + e_i) - 2u(m) + u(m - e_i) \big) = \sum_{\abs{m' - m} = 1} \big( u(m') - u(m) \big).$ More generally, let a graph have vertex set $V$ , and let every vertex have finitely many neighbours. Then the operator sends $u \colon V \to \R$ to $(\Delta u)(v) = \sum_{w \sim v} \big( u(w) - u(v) \big), \qquad v \in V,$ where $w \sim v$ means that $w$ and $v$ are joined by an edge.

Remarks

Example: on $\Z^2$ , the function $u(m, n) = m^2 + n^2$ has $(\Delta u)(m, n) = 2 + 2 = 4$ everywhere, matching $\Delta(x^2 + y^2) = 4$ in the plane. The function $u(m, n) = mn$ has $\Delta u = 0$ , so it is discrete harmonic.

Non-example of a discrete harmonic function: $u(m) = 2^m$ on $\Z$ has $(\Delta u)(m) = 2^{m+1} - 2^{m+1} + 2^{m-1} = 2^{m-1} \ne 0$ .

On a grid of mesh $h > 0$ one uses $h^{-2} \Delta$ , which tends to the Laplacian of $u$ as $h \to 0$ when $u$ is a smooth function sampled on the grid. On a finite graph with Laplacian matrix $L = D - A$ , one has $\Delta u = -L u$ when $u$ is written as a column vector. Some authors use the opposite sign or divide by the degree.

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Tag	`TC6Y42YGQF`
URI	`m2://TC6Y42YGQF`
Kind	definition
Writing	W0 stub
MSC	39A12, 39A14, 05C50
Defines	Discrete Laplace operator
Uses	Function, Laplacian matrix

Discrete Laplace operator

Definition

Remarks

Reading path

Further reading

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