A Discontinuous Galerkin Scheme for the Cahn-Hilliard Equations with Discrete Maximum Principle for Arbitrary Polynomial Order
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Gunnarsson, Jimmy Kornelije Klöfkorn, Robert |
| author_facet | Gunnarsson, Jimmy Kornelije Klöfkorn, Robert |
| contents | We propose a structure-preserving discontinuous Galerkin scheme for the Cahn--Hilliard equations with degenerate mobility based on the Symmetric Weighted Interior Penalty formulation. By evaluating the mobility at cell averages rather than as a piecewise polynomial, the proposed scheme preserves strict degeneracy and yields a coercivity constant that is independent of the mobility, removing the need for regularisation. Moreover, we establish existence of discrete solutions even with degeneracy via a Leray--Schauder fixed-point argument, and show that the scheme satisfies a provable discrete maximum principle at arbitrary polynomial order $p$ when combined with the Zhang--Shu scaling limiter for $p > 0$ and from the scheme alone for $p = 0$. Mass conservation and energy dissipation are established for the unlimited scheme; for the limited variant, we discuss observed energy dissipation for $p \geq 1$ and potential theoretical solutions. Numerical experiments confirm optimal convergence rates of order $p+1$ in $L^2$ and validate structure-preserving properties with numerical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00988 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Discontinuous Galerkin Scheme for the Cahn-Hilliard Equations with Discrete Maximum Principle for Arbitrary Polynomial Order Gunnarsson, Jimmy Kornelije Klöfkorn, Robert Numerical Analysis 65M12, 65M60, 35K65 We propose a structure-preserving discontinuous Galerkin scheme for the Cahn--Hilliard equations with degenerate mobility based on the Symmetric Weighted Interior Penalty formulation. By evaluating the mobility at cell averages rather than as a piecewise polynomial, the proposed scheme preserves strict degeneracy and yields a coercivity constant that is independent of the mobility, removing the need for regularisation. Moreover, we establish existence of discrete solutions even with degeneracy via a Leray--Schauder fixed-point argument, and show that the scheme satisfies a provable discrete maximum principle at arbitrary polynomial order $p$ when combined with the Zhang--Shu scaling limiter for $p > 0$ and from the scheme alone for $p = 0$. Mass conservation and energy dissipation are established for the unlimited scheme; for the limited variant, we discuss observed energy dissipation for $p \geq 1$ and potential theoretical solutions. Numerical experiments confirm optimal convergence rates of order $p+1$ in $L^2$ and validate structure-preserving properties with numerical results. |
| title | A Discontinuous Galerkin Scheme for the Cahn-Hilliard Equations with Discrete Maximum Principle for Arbitrary Polynomial Order |
| topic | Numerical Analysis 65M12, 65M60, 35K65 |
| url | https://arxiv.org/abs/2604.00988 |