This work extends the general form of the multiscale hybrid-mixed (MHM) method for the second-order Laplace (Darcy) equation to general non-conforming polygonal meshes. The main properties of the MHM method, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometry of the elements used for the first level mesh. More precisely, it is proven that piecewise polynomials of degree k and $$k+1$$ k + 1 , $$k \ge 0$$ k ≥ 0 , for the Lagrange multipliers (flux), along with continuous piecewise polynomial interpolations of degree $$k+1$$ k + 1 posed on second-level sub-meshes are stable if the latter is fine enough with respect to the mesh for the Lagrange multiplier. We provide an explicit sufficient condition for this restriction. Also, we prove that the error converges with order $$k+1$$ k + 1 and $$k+2$$ k + 2 in the broken $$H^1$$ H 1 and $$L^2$$ L 2 norms, respectively, under usual regularity assumptions, and that such estimates also hold for non-convex; or even non-simply connected elements. Numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides.

中文翻译：

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一般多边形网格中的多尺度混合混合方法

这项工作将二阶拉普拉斯 (Darcy) 方程的多尺度混合混合 (MHM) 方法的一般形式扩展到一般的非一致多边形网格。MHM 方法的主要特性，即稳定性、最佳收敛性和局部守恒，被证明独立于用于第一级网格的元素的几何形状。更准确地说，对于拉格朗日乘子（通量），证明了 k 阶和 $$k+1$$ k + 1 的分段多项式， $$k \ge 0$$ k ≥ 0 ，以及连续分段多项式插值如果二级子网格相对于拉格朗日乘子的网格足够精细，则在二级子网格上构成的度数 $$k+1$$ k + 1 是稳定的。我们为这个限制提供了一个明确的充分条件。还，我们证明了错误在 $$H^1$$ H 1 和 $$L^2$$ L 中收敛于 $$k+1$$ k + 1 和 $$k+2$$ k + 2 的阶次在通常的规律性假设下，分别有 2 个范数，并且这种估计也适用于非凸；甚至非简单连接的元素。数值结果证实了理论发现，并说明了使用多尺度函数提供的收益。