An improved iterative HDG approach for partial differential equations

Abstract

We propose and analyze an iterative high-order hybridized discontinuous Galerkin (iHDG) discretization for linear partial differential equations. We improve our previous work (SIAM J. Sci. Comput. Vol. 39, No. 5, pp. S782-S808) in several directions: 1) the improved iHDG approach converges in a finite number of iterations for the scalar transport equation; 2) it is unconditionally convergent for both the linearized shallow water system and the convection-diffusion equation;3) it has improved stability and convergence rates; 4) we uncover a relationship between the number of iterations and time stepsize, solution order, meshsize and the equation parameters. This allows us to choose the time stepsize such that the number of iterations is approximately independent of the solution order and the meshsize; and 5) we provide both strong and weak scalings of the improved iHDG approach up to 16, 384 cores. A connection between iHDG and time integration methods such as parareal and implicit/explicit methods are discussed. Extensive numerical results are presented to verify the theoretical findings.

A multilevel approach for trace system in HDG discretizations

“…They concluded that both continuous and interior penalty discontinuous Galerkin algorithms with multigrid outperforms HDG with multigrid in terms of time to solution by a significant margin. One level Schwarz type domain decomposition algorithms in the context of HDG have been studied for elliptic equation [32,33], hyperbolic systems [36,37] and Maxwell's equations [34,35]. A balancing domain decomposition by constraints algorithm for HDG was introduced in [38] and studied for Euler and Navier-Stokes equations.…”

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…Starting from [80], several works also explored the capabilities of multigrid solvers for HDG formulations, including hierarchical scale separation [238], geometric multigrid [265], nested geometric multigrid on many-core processors [129], p-multigrid in the context of second-order elliptic problems [174] and compressible Navier-Stokes flows [139] and GPU-accelerated p-multigrid for linear elasticity [127]. Finally, iterative algorithms inspired by the Gauss-Seidel method were proposed in [196] and tested on massively parallel architectures up to 16,384 cores. A block symmetric Gauss-Seidel type preconditioner was also introduced in [224], whereas a multilevel solver coupled with a block-Jacobi fine scale solver is proposed in [195].…”

Spectral method for the Boltzmann equation for gases with viscosity

“…hương trình Boltzmann mô tả chuyển động của các phân tử của một chất khí trong đó sự tương tác của các phân tử là va chạm đàn hồi nhị phân (xem [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). Việc xây dựng các phương pháp tính cho phương trình Boltzmann có tầm quan trọng trong nhiều ứng dụng, từ động lực học khí hiếm (RGD) [27], vật lý plasma [28], dòng chảy dạng hạt [17,18], chất bán dẫn [32], và lý thuyết động lượng tử [29].…”