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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\R^d$ be a bounded domain and $L$ a linear elliptic differential operator with constant coefficients.\nLet $G$ be a [[fundamental solution|K2NK9A8M5F]] of $L$, so that $L G = \\delta$; it is smooth on $\\R^d \\setminus \\set{0}$.\nConsider $L u = 0$ in $\\Omega$ with the condition $u = g$ on $\\partial \\Omega$.\nThe *method of fundamental solutions* has three ingredients:\n\n1. source points $y_1, \\ldots, y_n$ in $\\R^d \\setminus \\overline{\\Omega}$;\n2. the trial function $u_n(x) = \\sum_{j=1}^{n} c_j \\, G(x - y_j)$, which satisfies $L u_n = 0$ in $\\Omega$;\n3. boundary points $x_1, \\ldots, x_m \\in \\partial \\Omega$, with $m \\ge n$.\n\nThe coefficients $c_j$ solve $u_n(x_i) = g(x_i)$ for all $i$, exactly if $m = n$ and in the least squares sense if $m \u003e n$.\n\n## Remarks\n\nThe sources lie outside $\\overline{\\Omega}$, so each $G(\\cdot - y_j)$ is smooth on $\\overline{\\Omega}$ and solves the equation there.\nThe error comes only from the boundary condition.\nThe sources are often placed on a curve or surface surrounding $\\Omega$ at some distance.\nThe linear system becomes very ill-conditioned as $n$ grows, which limits the accuracy reached in floating point arithmetic.\n\nExample: the [[Dirichlet problem|8D56GQ4AG8]] for the Laplace equation $\\Delta u = 0$, with $\\Delta$ the [[Laplacian|0E0B6RA8K0]], in the unit disk of $\\R^2$.\nTake $L = -\\Delta$, with fundamental solution $G(x) = -\\frac{1}{2\\pi} \\ln \\abs{x}$.\nTake sources $y_j = R (\\cos \\theta_j, \\sin \\theta_j)$ with $\\theta_j = 2\\pi j / n$ and $R \u003e 1$.\nTake as many boundary points $x_k = (\\cos \\alpha_k, \\sin \\alpha_k)$, with $\\alpha_k = 2\\pi (k - \\tfrac12) / n$.\nEvery $u_n$ is harmonic in the disk, and the $c_j$ are fixed by matching $g$ at the $x_k$.\n\nNon-example: with a source point inside $\\Omega$, clause 1 fails.\nThen $u_n$ has a logarithmic singularity in $\\Omega$ and does not solve the equation there.\n",
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