We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a common extension of both the full approximation scheme (FAS) multigrid technique for nonlinear partial differential equations, due to A.~Brandt, and the constraint decomposition (CD) method introduced by X.-C.~Tai for VIs arising in optimization. We extend the CD idea by exploiting the telescoping nature of certain function space subset decompositions arising from multilevel mesh hierarchies. When a reduced-space (active set) Newton method is applied as a smoother, with work proportional to the number of unknowns on a given mesh level, FASCD V-cycles exhibit nearly mesh-independent convergence rates, and full multigrid cycles are optimal solvers. The example problems include differential operators which are symmetric linear, nonsymmetric linear, and nonlinear, in unilateral and bilateral VI problems.

翻译：我们提出了用于求解变分不等式（VI）的完全逼近格式约束分解（FASCD）多层方法。FASCD 是 A.~Brandt 针对非线性偏微分方程提出的完全逼近格式（FAS）多重网格技术，与 X.-C.~Tai 针对优化问题中的变分不等式引入的约束分解（CD）方法的一种共同扩展。通过利用多重网格层次结构中特定函数空间子集分解的伸缩性质，我们扩展了 CD 思想。当采用缩减空间（活动集）牛顿方法作为光滑子时，其工作量与给定网格层上的未知量数目成正比，FASCD V 循环表现出近乎网格无关的收敛速度，且完全多重网格循环是最优求解器。示例问题包括单侧和双侧变分不等式问题中涉及的对称线性、非对称线性和非线性微分算子。