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Journal of Scientific Computing

Erschienen in:

13.06.2018

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

verfasst von: Hailiang Liu, Peimeng Yin

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2018

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Abstract

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 \(L^2\) stable even without interior penalty. For time discretization, we use Crank–Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal \(L^2\) error estimate of \(O(h^{k+1})\) for polynomials of degree k for semi-discrete DG schemes, and the \(L^2\) error of \(O(h^{k+1} +(\Delta t)^2)\) for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift–Hohenberg equation endowed with a decay free energy is presented.

Vorheriger Artikel Fully Computable a Posteriori Error Bounds for Hybridizable Discontinuous Galerkin Finite Element Approximations

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Titel: A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems
verfasst von: Hailiang Liu
Peimeng Yin
Publikationsdatum: 13.06.2018
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2018
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0756-0

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