In this work we introduce and analyse a new multiscale method for strongly nonlinear monotone equations in the spirit of the localized orthogonal decomposition. A problem-adapted multiscale space is constructed by solving linear local fine-scale problems, which is then used in a generalized finite element method. The linearity of the fine-scale problems allows their localization and, moreover, makes the method very efficient to use. The new method gives optimal a priori error estimates up to linearization errors. The results neither require structural assumptions on the coefficient such as periodicity or scale separation nor higher regularity of the solution. The effect of different linearization strategies is discussed in theory and practice. Several numerical examples including the stationary Richards equation confirm the theory and underline the applicability of the method.

中文翻译：

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非线性强单调问题的数值均匀化

在这项工作中，我们本着局部正交分解的精神，介绍和分析了一种新的强非线性单调方程的多尺度方法。通过求解线性局部精细尺度问题来构建适应问题的多尺度空间，然后将其用于广义有限元方法。精细问题的线性允许它们的定位，此外，使该方法使用起来非常有效。新方法给出了线性化误差的最佳先验误差估计。结果既不需要对系数进行结构假设，例如周期性或尺度分离，也不需要更高的解规律性。在理论和实践中讨论了不同线性化策略的效果。