GPU上的自适应多级间断Galerkin方法
Adaptive Multilevel Discontinuous Galerkin Methods on GPUs
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中文总结 AI 辅助
本文提出GPU上适用于自适应加密笛卡尔网格的无矩阵对称内罚间断Galerkin方法，通过阴影单元处理非匹配界面，结合原始-对偶配对优化Krylov迭代，在NVIDIA A100上的三次单元实验验证了其稳定收敛性与高效GPU执行性能。
中文摘要 AI 辅助
本文提出一种无矩阵的对称内罚间断Galerkin方法，适用于GPU上自适应加密的笛卡尔网格。在非匹配界面处，辅助“阴影单元”表示细层级上相邻的粗多项式，该方法将非匹配界面转换为匹配面，允许在整个网格上进行统一的面评估。本文证明该构造等价于标准非匹配格式，且该表示可生成局部几何多重网格方法，其中冻结的阴影提供层级间边界数据并将残差贡献传递到更粗层级。原始-对偶配对消除了Krylov迭代中的阴影组装。在NVIDIA A100上采用三次单元（p=3）的数值实验表明，该方法在加密深度增加时具有稳定的多重网格收敛性，且GPU执行效率高。
英文摘要
I present a matrix-free symmetric interior penalty discontinuous Galerkin method for adaptively refined Cartesian meshes on GPUs. At non-matching interfaces, auxiliary \emph{shadow cells} represent the adjacent coarse polynomial on the fine level. This approach converts non-matching interfaces into matching faces, permitting uniform face evaluation throughout the mesh. I prove that this construction is equivalent to the standard non-matching formulation. Furthermore, this representation yields a local geometric multigrid method in which frozen shadows provide inter-level boundary data and carry residual contributions to coarser levels. A primal--dual pairing eliminates shadow assembly from the Krylov iteration. Numerical experiments with cubic elements ($p=3$) on an NVIDIA A100 show stable multigrid convergence under increasing refinement depth and efficient GPU execution.
