We consider the solution of \(-\nabla \cdot (\nu (x)\nabla u)=0\) by a non-overlapping optimized Schwarz domain decomposition method, where the subdomains do not align with jumps in the coefficient \(\nu (x)\). Such a decomposition can be of interest when the jumps are geometrically complex and/or an artifact of the measured data, in which case one would often prefer a simpler decomposition that disregards the location of the discontinuities. For analysis purposes, we focus on a model problem where the diffusivity is piecewise constant, and we analyze the convergence of optimized Schwarz for the two-subdomain case. We consider using either a constant Robin parameter along the whole interface, or a parameter that is scaled proportionally to the local diffusivity. We show that the convergence rate is not robust with respect to the heterogeneity ratio when a constant Robin parameter is used; however, using a scaled Robin parameter restores robustness. We then derive optimal scaling parameter and the corresponding convergence factor. Numerical examples show that this choice also leads to robust convergence behaviour for cases not covered by the analysis.

我们通过非重叠优化 Schwarz 域分解方法考虑\(-\nabla \cdot (\nu (x)\nabla u)=0\)的解决方案，其中子域不与系数\( \nu (x)\). 当跳跃在几何上是复杂的和/或测量数据的人工产物时，这种分解可能是有意义的，在这种情况下，人们通常更喜欢忽略不连续位置的更简单的分解。出于分析目的，我们专注于扩散率分段常数的模型问题，并分析优化 Schwarz 在两个子域情况下的收敛性。我们考虑沿整个界面使用恒定的 Robin 参数，或者与局部扩散率成比例缩放的参数。我们表明，当使用恒定的 Robin 参数时，收敛速度对于异质性比率并不稳健；然而，使用缩放的 Robin 参数可以恢复稳健性。然后我们推导出最优缩放参数和相应的收敛因子。