In this article, we develop and analyze two-grid/multi-level algorithms via mesh refinement in the abstract framework of Brezzi, Rappaz, and Raviart for approximation of branches of nonsingular solutions. Optimal fine grid accuracy of two-grid/multi-level algorithms can be achieved via the proper scaling of relevant meshes. An important aspect of the proposed algorithms is the use of mesh refinement in conjunction with Newton-type methods for system solution in contrast to the usual Newton’s method on a fixed mesh. The pseudostress-velocity formulation of the stationary, incompressible Navier–Stokes equations is considered as an application and the Raviart–Thomas mixed finite element spaces are used for the approximation. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.

在本文中，我们在Brezzi，Rappaz和Raviart的抽象框架中通过网格细化来开发和分析两层/多层算法，用于近似非奇异解的分支。可以通过适当缩放相关网格来实现两网格/多级算法的最佳精细网格精度。与固定网格上的常规牛顿方法相比，所提出算法的一个重要方面是将网格细化与牛顿型方法结合用于系统求解。静态不可压缩的Navier–Stokes方程的拟应力速度公式被认为是一种应用，并且将Raviart–Thomas混合有限元空间用于逼近。最后，给出了几个数值示例来测试算法的性能和所开发理论的有效性。