On the superconvergence of a hydridizable discontinuous Galerkin method for the Cahn-Hilliard equation

We propose a hydridizable discontinuous Galerkin (HDG) method for solving the Cahn-Hilliard equation. The temporal discretization can be based on either the backward Euler method or the convex-splitting method. We show that the fully discrete scheme admits a unique solution, and we establish optimal...

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Bibliographic Details
Published in:arXiv.org 2019-04
Main Authors: Chen, Gang, Han, Daozhi, Singler, John, Zhang, Yangwen
Format: Article
Language:English
Subjects:
Economic models
Free energy
Galerkin method
Norms
Online Access:Get full text
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Summary:We propose a hydridizable discontinuous Galerkin (HDG) method for solving the Cahn-Hilliard equation. The temporal discretization can be based on either the backward Euler method or the convex-splitting method. We show that the fully discrete scheme admits a unique solution, and we establish optimal convergence rates for all variables in the \(L^2\) norm for arbitrary polynomial orders. In terms of the globally coupled degrees of freedom, the scalar variables are superconvergent. Another theoretical contribution of this work is a novel HDG Sobolev inequality that is useful for HDG error analysis of nonlinear problems. Numerical results are reported to confirm the theoretical convergence rates.
ISSN:2331-8422