Multigrid solvers face multiple challenges on parallel computers. Two fundamental ones read as follows: Multiplicative solvers issue coarse grid solves which exhibit low concurrency and many multigrid implementations suffer from an expensive coarse grid identification phase plus adaptive mesh refinement overhead. We propose a new additive multigrid variant for spacetrees, that is, meshes as they are constructed from octrees and quadtrees: It is an additive scheme, that is, all multigrid resolution levels are updated concurrently. This ensures a high concurrency level, while the transfer operators between the mesh levels can still be constructed algebraically. The novel flavor of the additive scheme is an augmentation of the solver with an additive, auxiliary damping parameter per grid level per vertex that is in turn constructed through the next coarser level—an idea which utilizes smoothed aggregation principles or the motivation behind AFACx: Per level, we solve an additional equation whose purpose is to damp too aggressive solution updates per vertex which would otherwise, in combination with all the other levels, yield an overcorrection and, eventually, oscillations. This additional equation is constructed additively as well, that is, is once more solved concurrently to all other equations. This yields improved stability, closer to what is seen with multiplicative schemes, while pipelining techniques help us to write down the additive solver with single‐touch semantics for dynamically adaptive meshes.

中文翻译：

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使用加性阻尼的稳定异步快速自适应复合多重电网

多网格求解器在并行计算机上面临多个挑战。两个基本的解释如下：乘法求解器发出粗糙的网格求解，并发性低，许多多重网格实现都经历了昂贵的粗糙网格识别阶段以及自适应网格细化开销。我们为空间树（即从八叉树和四叉树构造的网格）提出了一种新的加性多网格变体：这是一种加性方案，即所有多网格分辨率级别均会同时更新。这样可以确保较高的并发级别，而网格级别之间的转移算子仍可以代数构造。加法方案的新颖风味是增加了加法器的求解器，每个顶点的每个网格级别的辅助阻尼参数，该参数依次通过下一个更粗糙的级别构建-一种想法，该思想利用了平滑的聚合原理或AFACx背后的动机：对于每个级别，我们解决了一个附加方程，其目的是阻尼每个节点的过于激进的解决方案更新否则将与所有其他级别结合使用的顶点会产生过度校正，并最终产生振荡。这个附加的方程式也被累加地构造，也就是说，再次与所有其他方程式同时求解。这产生了改进的稳定性，接近乘法方案所看到的稳定性，而流水线技术则帮​​助我们写下具有单触语义的加法求解器，以实现动态自适应网格。