A second order elliptic problem with discontinuous coefficient in 2-D or 3-D is considered. The problem is discretized by a symmetric weighted interior penalty discontinuous Galerkin finite element method with nonmatching simplicial elements and piecewise linear functions. The resulting discrete problem is solved by a two-level additive Schwarz method with a relatively coarse grid and with local solves restricted to subdomains which can be as small as single element. In this way the method has a potential for a very high level of fine grained parallelism. Condition number estimate depending on the relative sizes of the underlying grids is provided. The rate of convergence of the method is independent of the jumps of the coefficient if its variation is moderate inside coarse grid substructures or on local solvers’ subdomain boundaries. Numerical experiments are reported which confirm theoretical results.

中文翻译：

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椭圆问题不连续伽辽金离散化的大规模并行非重叠加法Schwarz方法

考虑在二维或三维中具有不连续系数的二阶椭圆问题。该问题通过对称加权内罚不连续 Galerkin 有限元方法离散化，该方法具有不匹配的单纯元素和分段线性函数。由此产生的离散问题通过两级加法 Schwarz 方法解决，该方法具有相对粗糙的网格，并且局部求解仅限于可以小到单个元素的子域。通过这种方式，该方法具有实现非常高水平的细粒度并行性的潜力。提供了取决于底层网格的相对大小的条件数估计。如果该方法的收敛速度在粗网格子结构内或在本地求解器的子域边界上变化适中，则该方法的收敛速度与系数的跳跃无关。