An Improved Iterative HDG Approach for Partial Differential Equations
arXiv:1711.01175 · doi:10.1016/j.jcp.2018.04.033
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 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.
arXiv admin note: text overlap with arXiv:1605.03228
References in corpus (2)
Cited by in corpus (6)
- HDGlab: An open-source implementation of the hybridisable discontinuous Galerkin method in MATLAB
- A Multilevel Approach for Trace System in HDG Discretizations
- Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
- A high-order hybridizable discontinuous Galerkin method with fast convergence to steady-state solutions of the gas kinetic equation
- Preconditioning for a pressure-robust HDG discretization of the Stokes equations
- Uniform block-diagonal preconditioners for divergence-conforming HDG Methods for the generalized Stokes equations and the linear elasticity equations