In this paper, we introduce the multi-region discontinuous Galerkin composite finite element method (MRDGCFEM) with hp-adaptivity for the discretization of second-order elliptic partial differential equations with discontinuous coefficients. This method allows for the approximation of problems posed on computational domains where the jumps in the diffusion coefficient form a micro-structure. Standard numerical methods could be used for such problems but the computational effort may be extremely high. Small enough elements to represent the underlying pattern in the diffusion coefficient have to be used. In contrast, the dimension of the underlying MRDGCFE space is independent of the complexity of the diffusion coefficient pattern. The key idea is that the jumps in the diffusion coefficient are no longer resolved by the mesh where the problem is solved; instead, the finite element basis (or shape) functions are adapted to the diffusion pattern allowing for much coarser meshes. In this paper, we employ hp-adaptivity on a series of test cases highlighting the practical application of the proposed numerical scheme.

中文翻译：

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具有不连续系数的复杂域上椭圆问题的自适应复合不连续Galerkin方法

本文介绍了具有hp自适应性的多区域不连续Galerkin复合有限元方法（MRDGCFEM），用于离散具有不连续系数的二阶椭圆型偏微分方程。这种方法可以近似计算域上的问题，在这些域中，扩散系数的跃迁形成了一个微结构。可以使用标准数值方法来解决此类问题，但计算量可能会非常大。必须使用足够小的元素来表示扩散系数中的基础图案。相反，下层MRDGCFE空间的尺寸与扩散系数模式的复杂性无关。关键思想是解决问题的网格不再解决扩散系数的跳跃问题。取而代之的是，将有限元基础（或形状）函数与扩散模式相适应，从而允许更粗糙的网格。在本文中，我们在一系列测试案例中采用了hp-adaptivity，突出了所提出数值方案的实际应用。