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Mehrgitterverfahren bilden in der numerisc … 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.konditionierte Krylow-Unterraum-Verfahren.
, Métodos Multigrid em análise numérica são … 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 equações de Navier-Stokes. O multigrid pode ser generalizado numa variedade de formas diferentes e pode ser aplicado diretamente para equações diferenciais parciais dependentes do tempo. Algoritmo Existem muitas variações de algoritmos de multigrid. A característica mais representativa e comum a todos os algoritmos é que existe uma hierarquia de discretizações, ou seja, uma hierarquia de malhas. Os passos importantes são:
* Suavização – redução de erros de alta frequência, por exemplo, através de algumas iterações do método de Gauss-Seidel ou Jacobi
* Restricão – passagem do erro residual para uma malha mais grossa
* Correção – interpolação de valores do resíduo calculados na malha mais grossa para uma malha mais finamalha mais grossa para uma malha mais fina
, 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 … 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。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。
, In numerical analysis, a multigrid method … 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. The main idea of multigrid is to accelerate the convergence of a basic iterative method (known as relaxation, which generally reduces short-wavelength error) by a global correction of the fine grid solution approximation from time to time, accomplished by solving a coarse problem. The coarse problem, while cheaper to solve, is similar to the fine grid problem in that it also has short- and long-wavelength errors. It can also be solved by a combination of relaxation and appeal to still coarser grids. This recursive process is repeated until a grid is reached where the cost of direct solution there is negligible compared to the cost of one relaxation sweep on the fine grid. This multigrid cycle typically reduces all error components by a fixed amount bounded well below one, independent of the fine grid mesh size. The typical application for multigrid is in the numerical solution of elliptic partial differential equations in two or more dimensions. Multigrid methods can be applied in combination with any of the common discretization techniques. For example, the finite element method may be recast as a multigrid method. In these cases, multigrid methods are among the fastest solution techniques known today. In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions and boundary conditions. They do not depend on the separability of the equations or other special properties of the equation. They have also been widely used for more-complicated non-symmetric and nonlinear systems of equations, like the Lamé equations of elasticity or the Navier-Stokes equations.elasticity or the Navier-Stokes equations.
, Многосеточный метод (МС, англ. multigrid) … 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.ких уравнений даже на нерегулярных сетках.
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Métodos Multigrid em análise numérica são … 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 não-lineares e não-simétricos, como as eq
, In numerical analysis, a multigrid method … 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.sed as solvers as well as preconditioners.
, Mehrgitterverfahren bilden in der numerisc … 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.konditionierte Krylow-Unterraum-Verfahren.
, 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 … 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。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。
, Многосеточный метод (МС, англ. multigrid) … 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.ких уравнений даже на нерегулярных сетках.
|
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Mehrgitterverfahren
, Многосеточный метод
, マルチグリッド法
, Método Multigrid
, Multigrid method
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