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

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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.

Keywords

Biharmonic equation Discontinuous Galerkin method \(L^2\) error estimates Swift–Hohenberg equation 

Mathematics Subject Classification

65N15 65N30 35J40 

Notes

Acknowledgements

This research was partially supported by the National Science Foundation under Grant DMS1312636.

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Authors and Affiliations

  1. 1.Mathematics DepartmentIowa State UniversityAmesUSA

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