We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and high-order characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating and/or limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly h−coarser edges/faces. We then combine the coarse solver with a block-Jacobi fine scale solver to form a two-level solver/preconditioner. Numerical experiments indicate that the performance of the resulting two-level solver/preconditioner depends on the smoothness of the solution and can offer significant speedups and memory savings compared to the nested dissection direct solver. While the proposed algorithms are developed within the HDG framework, they are applicable to other hybrid(ized) high-order finite element methods. Moreover, we show that our multilevel algorithms can be interpreted as a multigrid method with specific intergrid transfer and smoothing operators. With several numerical examples from Poisson, pure transport, and convection-diffusion equations we demonstrate the robustness and scalability of the algorithms with respect to solution order. While scalability with mesh size in general is not guaranteed and depends on the smoothness of the solution and the type of equation, improving it is a part of future work.

我们为跟踪系统提出了一种多级方法，该系统是由混合不连续Galerkin（HDG）方法产生的。关键是要融合来自嵌套解剖，域分解和HDG离散化的高阶特征的思想。具体来说，我们首先通过消除和/或限制嵌套解剖中的前端生长来创建一个粗糙的求解器。这是通过将跟踪数据投影到一组越来越多的h上的一系列相同或高阶多项式序列中来实现的-粗边/脸部。然后，我们将粗解算器与块雅各比精细解算器组合在一起，以形成两级解算器/预处理器。数值实验表明，所产生的两级求解器/预处理器的性能取决于求解器的平滑度，与嵌套解剖直接求解器相比，可以显着提高速度并节省内存。虽然所提出的算法是在HDG框架内开发的，但它们也适用于其他混合（化）高阶有限元方法。此外，我们证明了我们的多级算法可以解释为具有特定网格间传递和平滑运算符的多网格方法。结合Poisson的几个数字示例，纯运输，和对流扩散方程式，我们证明了算法相对于解阶的鲁棒性和可扩展性。虽然通常不能保证网格大小的可伸缩性，并且取决于解决方案的平滑性和方程式的类型，但改进它是将来工作的一部分。