# A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems

@article{Liu2018AMD,
title={A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems},
author={Hailiang Liu and Peimeng Yin},
journal={Journal of Scientific Computing},
year={2018},
volume={77},
pages={467-501}
}
• Published 13 June 2018
• Computer Science, Mathematics
• Journal of Scientific Computing
A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are $$L^2$$L2 stable even without interior penalty. For time discretization, we use Crank–Nicolson so that the resulting scheme is unconditionally stable and second order… Expand
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#### References

SHOWING 1-10 OF 45 REFERENCES
The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems
• Mathematics, Computer Science
• SIAM J. Numer. Anal.
• 2008
A general numerical flux formula for the solution derivative is proposed, which is consistent and conservative; and a concept of admissibility is introduced to identify a class of numerical fluxes so that the nonlinear stability for both one-dimensional (1D) and multidimensional problems are ensured. Expand
A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives
• Computer Science, Mathematics
• Math. Comput.
• 2008
A new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives that can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Expand
Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs
• Mathematics
• 2017
The main purpose of this paper is to give stability analysis and error estimates of the local discontinuous Galerkin (LDG) methods coupled with three specific implicit-explicit (IMEX) Runge–KuttaExpand
Adaptive discontinuous Galerkin approximations to fourth order parabolic problems
• Computer Science, Mathematics
• Math. Comput.
• 2015
The a posteriori estimates are used within an adaptive algorithm, highlighting their relevance in practical computations, which results into substantial reduction of computational effort. Expand
THE DIRECT DISCONTINUOUS GALERKIN (DDG) METHOD FOR DIFFUSION WITH INTERFACE CORRECTIONS
• Mathematics
• 2010
Based on a novel numerical ∞ux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Gallerkin(DDG) method for difiusion problems was introduced in (H. Liu andExpand
hp-Version a priori Error Analysis of Interior Penalty Discontinuous Galerkin Finite Element Approximations to the Biharmonic Equation
• Mathematics, Computer Science
• J. Sci. Comput.
• 2007
For a shape-regular family of meshes consisting of parallelepipeds, the symmetric formulation of the interior penalty discontinuous Galerkin finite element method for the numerical solution of the biharmonic equation with Dirichlet boundary conditions in a bounded polyhedral domain is considered. Expand
Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation
• Mathematics, Computer Science
• J. Sci. Comput.
• 2008
In this paper, we first split the biharmonic equation Δ2u=f with nonhomogeneous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v=Δu andExpand
Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives
• Mathematics, Computer Science
• J. Sci. Comput.
• 2002
This paper develops new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions and presents new results on a post-processing technique, originally designed for methods with good negative-order error estimates, that works as well for the new higher derivative cases. Expand
A priori and a posteriori error analysis for the mixed discontinuous Galerkin finite element approximations of the biharmonic problems
• Mathematics
• 2017
In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three-dimension space. It is amenable to an efficient implementationExpand
Superconvergence of the local discontinuous Galerkin method for linear fourth-order time-dependent problems in one space dimension
• Mathematics
• 2012
In this paper we investigate the superconvergence of local discontinuous Galerkin (LDG) methods for solving one-dimensional linear time-dependent fourth-order problems. We prove that the errorExpand