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A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation. (English) Zbl 1453.65338

Summary: We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard equation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The latter permits us to show that the discrete free-energy is bounded, and as a result, the scheme is provably stable. The scheme and the stability proof are presented for general curvilinear three-dimensional hexahedral meshes. We use the Bassi-Rebay 1 (BR1) scheme to compute interface fluxes, and a first order IMplicit-EXplicit (IMEX) scheme to integrate in time. We provide a semi-discrete stability study, and a fully-discrete proof subject to a positivity condition on the solution. Lastly, we test the theoretical findings using numerical cases that include two and three-dimensional problems.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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