A parallel implementation of a compatible discretization scheme for steady-state Stokes problems is presented in this work. The scheme uses generalized moving least squares to generate differential operators and apply boundary conditions. This meshless scheme allows a high-order convergence for both the velocity and pressure, while also incorporates finite-difference-like sparse discretization. Additionally, the method is inherently scalable: the stencil generation process requires local inversion of matrices amenable to GPU acceleration, and the divergence-free treatment of velocity replaces the traditional saddle point structure of the global system with elliptic diagonal blocks amenable to algebraic multigrid. The implementation in this work uses a variety of Trilinos packages to exploit this local and global parallelism, and benchmarks demonstrating high-order convergence and weak scalability are provided.

中文翻译：

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斯托克斯方程的兼容高阶无网格方法的并行实现

在这项工作中，提出了针对稳态Stokes问题的兼容离散化方案的并行实现。该方案使用广义移动最小二乘法生成差分算子并应用边界条件。这种无网格方案允许对速度和压力进行高阶收敛，同时还包含类似于有限差分的稀疏离散化。此外，该方法具有固有的可扩展性：模版生成过程需要对适合GPU加速的矩阵进行局部求逆，并且速度的无散度处理用适合于代数多重网格的椭圆对角线块代替了全局系统的传统鞍点结构。本工作中的实现使用各种Trilinos软件包来利用这种本地和全局并行性，