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In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior. For example, many basic relaxation methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid. MG methods can be used as solvers as well as preconditioners.

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  • Mehrgitterverfahren (de)
  • マルチグリッド法 (ja)
  • Multigrid method (en)
  • Método Multigrid (pt)
  • Многосеточный метод (ru)
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  • Mehrgitterverfahren bilden in der numerischen Mathematik eine Klasse von effizienten Algorithmen zur näherungsweisen Lösung von Gleichungssystemen, die aus der Diskretisierung partieller Differentialgleichungen stammen. Elliptische Probleme wie die Poisson-Gleichung können damit bei Unbekannten mit einem Rechenaufwand von der Ordnung gelöst werden. Die Konvergenzordnung ist dabei nicht von der Feinheit der Gitter abhängig, im Gegensatz zu den meisten anderen numerischen Verfahren, die mit kleiner werdender Diskretisierungsfeinheit langsamer werden. Mehrgitterverfahren sind in dieser Hinsicht „optimal“. Die wesentliche Alternative zu Mehrgitterverfahren sind vorkonditionierte Krylow-Unterraum-Verfahren. (de)
  • マルチグリッド（MG）法は、複数階層で離散化を行うことにより、微分方程式を解くための数値アルゴリズムの一種である。間隔の異なる格子間での補外と考えることもできる。マルチグリッド法は、主に多次元の楕円型偏微分方程式の数値計算に用いられる。 マルチグリッド法は任意の離散化手法と組み合わせることができ、現在知られているものの中でも最速な解法の一つである。他の手法と異なり、マルチグリッド法は任意の領域・境界条件を扱うことができる。これは微分方程式の性質（変数分離可能かどうか等）には依存しない。MG法は、弾性に関するラメの微分方程式やナビエ・ストークス方程式などの、より複雑な非対称・非線形問題にもそのまま適用することができる。 (ja)
  • Многосеточный метод (МС, англ. multigrid) — метод решения системы линейных алгебраических уравнений, основанный на использовании последовательности уменьшающихся и операторов перехода от одной сетки к другой. Сетки строятся на основе больших значений в матрице системы, что позволяет использовать этот метод при решении эллиптических уравнений даже на нерегулярных сетках. (ru)
  • In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior. For example, many basic relaxation methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid. MG methods can be used as solvers as well as preconditioners. (en)
  • Métodos Multigrid em análise numérica são um grupo de algoritmos para solução de equações diferenciais usando hierarquia de discretizações. A ideia é similar à extrapolação entre malhas mais grossas e mais finas. A aplicação típica para o multigrid é na solução de equações diferenciais parciais elípticas em duas ou mais direções. O multigrid pode ser aplicado junto com qualquer técnica comum de discretização. Nesses casos, o multigrid está entre as soluções mais rápidas conhecidas hoje. Em contraste com outros métodos, o multigrid pode ser aplicado em regiões arbitrárias e condições de contorno. Ele não depende da separabilidade das equações ou de outras propriedades da equação. O multigrid é também aplicável a sistemas de equações mais complicados não-lineares e não-simétricos, como as eq (pt)
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