We construct solvers for an isogeometric multi-patch discretization, where the patches are coupled via a discontinuous Galerkin approach, which allows for the consideration of discretizations that do not match on the interfaces. We solve the resulting linear system using a Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We are interested in solving the arising patch-local problems using iterative solvers since this allows for the reduction of the memory footprint. We solve the patch-local problems approximately using the Fast Diagonalization method, which is known to be robust in the grid size and the spline degree. To obtain the tensor structure needed for the application of the Fast Diagonalization method, we introduce an orthogonal splitting of the local function spaces. We present a convergence theory for two-dimensional problems that confirms that the condition number of the preconditioned system only grows poly-logarithmically with the grid size. The numerical experiments confirm this finding. Moreover, they show that the convergence of the overall solver only mildly depends on the spline degree. We observe a mild reduction of the computational times and a significant reduction of the memory requirements in comparison to standard IETI-DP solvers using sparse direct solvers for the local subproblems. Furthermore, the experiments indicate good scaling behavior on distributed memory machines. Additionally, we present an extension of the solver to three-dimensional problems and provide numerical experiments assessing good performance also in that setting.

我们为等几何多面片离散化构建求解器，其中面片通过不连续伽辽金方法耦合，这允许考虑接口上不匹配的离散化。我们使用双原始 IsogEometric 撕裂和互连 (IETI-DP) 方法求解所得线性系统。我们有兴趣使用迭代求解器解决出现的补丁局部问题，因为这可以减少内存占用。我们使用快速对角化方法近似解决局部补丁问题，该方法在网格大小和样条度方面具有鲁棒性。为了获得应用快速对角化方法所需的张量结构，我们引入了局部函数空间的正交分裂。我们提出了二维问题的收敛理论，证实预处理系统的条件数仅随网格大小呈多对数增长。数值实验证实了这一发现。此外，他们表明整个求解器的收敛仅轻微依赖于样条度。与使用稀疏直接求解器解决局部子问题的标准 IETI-DP 求解器相比，我们观察到计算时间略有减少，内存需求显着减少。此外，实验表明分布式内存机器上具有良好的扩展行为。此外，我们还提出了求解器对三维问题的扩展，并提供了评估该设置中良好性能的数值实验。