In this paper, we propose the Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, residual minimization, and alternating direction solver. Namely, we utilize tensor product B-spline basis functions and alternating direction methods. We apply a stabilized mixed method based on residual minimization. We propose a preconditioned conjugate gradients method with a linear computational cost resulting from a Kronecker product structure of the system of linear equations. We test our method on two-dimensional simulations of advection-diffusion problems, including the problem with the manufactured solution, the Eriksson-Johnson problem, and a rotating flow problem. We compare our method to the Discontinuous Petrov-Galerkin and the Streamline Upwind Petrov-Galerkin (SUPG) stabilization methods. The resulting method is not restricted to a Kronecker product structure of the diffusion or advection data.

中文翻译：

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等几何残差最小化方法 (iGRM) 与方向分裂预处理器，用于平稳对流主导的扩散问题

在本文中，我们提出了具有方向分裂的等几何残差最小化 (iGRM)。该方法混合了等几何分析、残差最小化和交替方向求解器带来的好处。即，我们利用张量积 B 样条基函数和交替方向方法。我们应用基于残差最小化的稳定混合方法。我们提出了一种预处理共轭梯度方法，其线性计算成本由线性方程组的克罗内克积结构产生。我们在对流扩散问题的二维模拟上测试我们的方法，包括制造解决方案的问题、Eriksson-Johnson 问题和旋转流问题。我们将我们的方法与 Discontinuous Petrov-Galerkin 和 Streamline Upwind Petrov-Galerkin (SUPG) 稳定方法进行比较。所得方法不限于扩散或对流数据的 Kronecker 积结构。