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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\R^d$ be a domain and let $X = \\set{x_1, \\ldots, x_n} \\subset \\overline{\\Omega}$ be a finite set of nodes.\nA *meshfree method* for a differential equation on $\\Omega$ approximates the solution by\n$$u_h(x) = \\sum_{i=1}^{n} \\varphi_i(x) \\, c_i,$$\nwhere the functions $\\varphi_i$ and the method satisfy two conditions:\n\n1. each $\\varphi_i$ is determined by the node set $X$ and fixed weight or kernel functions alone, with no elements or edges between nodes;\n2. the coefficients $c_i$ are found by a [[Galerkin|Y5B0PEBKE8]] condition, by collocation at points of $\\Omega$, or by a particle evolution law.\n\nUsually each $\\varphi_i$ has compact support, a ball of radius $\\rho$ around $x_i$.\n\n## Remarks\n\nCommon choices of $\\varphi_i$ are [[moving least squares|T0JZN6WP5X]] shape functions and [[radial basis functions|ZBSXSTQM2J]] $\\varphi_i(x) = \\phi(\\abs{x - x_i})$.\nMoving least squares shape functions of order $m$ reproduce polynomials of degree at most $m$; since they reproduce constants, they form a [[partition of unity|NH739XXVXK]].\nEssential boundary conditions need extra care, because these $\\varphi_i$ usually do not satisfy $\\varphi_i(x_j) = \\delta_{ij}$.\n\nExample: Gaussian radial basis function collocation for $-\\Delta u = f$ uses $\\varphi_i(x) = e^{-\\varepsilon^2 \\abs{x - x_i}^2}$ with a shape parameter $\\varepsilon \u003e 0$.\nThese functions depend only on the nodes and on $\\varepsilon$, which satisfies clause 1.\nThe $c_i$ solve $-\\Delta u_h(x_j) = f(x_j)$ at interior nodes and the boundary condition at boundary nodes, which is clause 2.\n\nNon-example: the hat functions of linear finite elements on a triangulation are not meshfree.\nEach hat function depends on which triangles contain its node, so clause 1 fails.\n",
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