Dryja Maksymilian, Krzyżanowski Piotr
University of Warsaw, Warsaw, Poland.
Numer Math (Heidelb). 2016;132(2):347-367. doi: 10.1007/s00211-015-0718-5. Epub 2015 Apr 1.
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.
考虑二维或三维中具有间断系数的二阶椭圆问题。该问题通过具有非匹配单纯形单元和分段线性函数的对称加权内部罚间断伽辽金有限元方法进行离散化。所得离散问题通过具有相对粗网格的两级加法施瓦茨方法求解，且局部求解限制在可小至单个单元的子域上。通过这种方式，该方法具有实现非常高水平细粒度并行性的潜力。给出了依赖于基础网格相对大小的条件数估计。如果系数在粗网格子结构内部或局部求解器的子域边界上的变化适度，则该方法的收敛速率与系数的跳跃无关。报告了证实理论结果的数值实验。
