We investigate a macro-element variant of the hybridized discontinuous Galerkin (HDG) method, using patches of standard simplicial elements that can have non-matching interfaces. Coupled via the HDG technique, our method enables local refinement by uniform simplicial subdivision of each macro-element. By enforcing one spatial discretization for all macro-elements, we arrive at local problems per macro-element that are embarrassingly parallel, yet well balanced. Therefore, our macro-element variant scales efficiently to n-node clusters and can be tailored to available hardware by adjusting the local problem size to the capacity of a single node, while still using moderate polynomial orders such as quadratics or cubics. Increasing the local problem size means simultaneously decreasing, in relative terms, the global problem size, hence effectively limiting the proliferation of degrees of freedom. The global problem is solved via a matrix-free iterative technique that also heavily relies on macro-element local operations. We investigate and discuss the advantages and limitations of the macro-element HDG method via an advection-diffusion model problem.

翻译：我们研究了一种杂化间断伽辽金（HDG）方法的宏单元变体，该变体使用能够具有非匹配界面的标准单纯形单元补丁。通过HDG技术耦合，我们的方法通过对每个宏单元进行均匀单纯形细分来实现局部细化。通过为所有宏单元强制实施一种空间离散化，我们得到了每个宏单元的局部问题，这些问题具有令人尴尬的并行性且负载均衡良好。因此，我们的宏单元变体能够高效扩展到n节点集群，并可通过调整局部问题大小以适应单个节点的容量来定制硬件，同时仍使用如二次或三次这样的适中多项式阶次。增大局部问题大小意味着相对地同时减小全局问题大小，从而有效限制自由度的增加。全局问题通过一种也严重依赖宏单元局部操作的无矩阵迭代技术求解。我们通过对流-扩散模型问题研究并讨论了宏单元HDG方法的优势与局限性。