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A Discontinuous Galerkin Scheme for the Cahn-Hilliard Equations with Discrete Maximum Principle for Arbitrary Polynomial Order

Jimmy Kornelije Gunnarsson, Robert Klöfkorn

Abstract

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.

A Discontinuous Galerkin Scheme for the Cahn-Hilliard Equations with Discrete Maximum Principle for Arbitrary Polynomial Order

Abstract

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 when combined with the Zhang--Shu scaling limiter for and from the scheme alone for . Mass conservation and energy dissipation are established for the unlimited scheme; for the limited variant, we discuss observed energy dissipation for and potential theoretical solutions. Numerical experiments confirm optimal convergence rates of order in and validate structure-preserving properties with numerical results.

Paper Structure

This paper contains 15 sections, 12 theorems, 73 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Discretization
  3. Notation
  4. Discontinuous Galerkin
  5. Scheme
  6. Energy formulation and a priori bounds
  7. Existence
  8. Discrete maximum principle
  9. Scaling limiter
  10. Cell-averaged bounds
  11. Comparison to other schemes
  12. Numerics
  13. Software implementation
  14. Experiments
  15. Summary and Outlook

Key Result

Lemma 2.2

Suppose that we have an orthogonal basis $\{\varphi_j\}_{j = 0}^N$ with respect to the L$^2$ inner product for $\mathbb{P}^p(K)$ for each $K \in {\mathcal{T}_h}$ such that $\left\langle \varphi_i, \varphi_j\right\rangle_K = \delta_{ij} |K|$ and $\varphi_0 = 1$. Then for any $\phi \in \mathbb{P}^p(K)

Figures (4)

  • Figure 1: Example \ref{['ex:spinodal']}: Monotone energy dissipation, mass change (black line is $n\text{TOL}$), and bounds.
  • Figure 2: Example \ref{['ex:spinodal']}: Discrete maximum principle violation, we make a linear fit to show that the error scales with computer precision in the floating point interval around $1$.
  • Figure 3: Example \ref{['ex:spinodal']}: Snapshots of the phase-field at time steps $t = \frac{T}{2}$, $t = \frac{3T}{4}$, and $t = T$ (left to right).
  • Figure 4: Example \ref{['ex:merging']}: Monotone energy dissipation, mass change (black line is $n\text{TOL}$), and bounds.

Theorems & Definitions (37)

  • Definition 2.1: Cell--averaged projection
  • Lemma 2.2: Orthogonality
  • Proof 1
  • Remark 2.3: Zero-mean decomposition
  • Theorem 2.4: Coercivity
  • Proof 2
  • Remark 2.5: Semi-positivity of $\tilde{b}(\cdot,\cdot)$ over $V_h^0$
  • Definition 3.1: Phase-field mass
  • Lemma 3.2: Mass conservation
  • Proof 3
  • ...and 27 more