The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed and optimally convergent under local regularity conditions. Also, a multi-level error analysis demonstrates that the MHM method achieves convergence without refining the coarse partition. The upshot is that the Poincaré and Korn’s inequalities do not degenerate, and then the convergence arises on general meshes. Two- and three-dimensional numerical tests assess theoretical results and verify the robustness of the method on a multi-layer media case.

中文翻译：

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多面体网格上线性弹性的 MHM 方法

多尺度混合混合 (MHM) 方法由多级策略组成，用于逼近具有异质系数的边值问题的解决方案。在这种情况下，我们为定义在域的粗多面体分区上的线性弹性方程提出了一个新的有限元族。有限元依赖于与从局部诺依曼问题获得的多尺度基相关的面自由度，并在面上进行分段多项式插值。我们在细尺度插值上建立了充分条件，使得 MHM 方法在局部规律性条件下是适定的并且最优收敛。此外，多级误差分析表明，MHM 方法无需细化粗划分即可实现收敛。结果是 Poincaré 和 Korn 的不等式不会退化，然后在一般网格上出现收敛。二维和三维数值测试评估理论结果并验证该方法在多层介质案例上的稳健性。