A multilevel approach for trace system in HDG discretizations Abstract   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 highorder characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating andor limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or highorder polynomials on a set of increasingly hxe2x88x92coarser edgesfaces. We then combine the coarse solver with a blockJacobi fine scale solver to form a twolevel solverpreconditioner. Numerical experiments indicate that the performance of the resulting twolevel solverpreconditioner 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) highorder 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 convectiondiffusion 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.

A New Hybrid Staggered Discontinuous Galerkin Method on General Meshes A novel highorder hybrid staggered discontinuous Galerkin method for general meshes is proposed to solve general second order elliptic problems. Our new formulation is related to standard staggered discontinuous Galerkin method, but more flexible and cost effective: rough grids are allowed and the size of the final system is remarkably reduced thanks to the partial hybridization. Optimal convergence estimates for both the scalar and vector variables are developed. Moreover, superconvergent results with respect to discrete H1 norm and L2 norm for the scalar variable are proved and negative norm error estimates for both the scalar and vector variables are also developed. On the other hand, mesh adaptation is particularly simple since hanging nodes are allowed, which makes the proposed method well suited for adaptive mesh refinement.  Therefore, we design a residual type a posteriori error estimator, and the reliability and local efficiency of the error estimator are proved. Numerical experiments confirm the theoretical findings.