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A $C^1$ virtual element method for the Cahn-Hilliard equation with polygonal meshes
Paola F. Antonietti, Lourenco Beirao da Veiga, Simone Scacchi, Marco Verani
TL;DR
The paper addresses efficient approximation of the fourth-order nonlinear Cahn-Hilliard equation without the implementation difficulty of conventional C1 finite elements or the extra degrees of freedom of mixed methods. It develops a minimal-degree C1 VEM for general polygonal meshes, proves semi-discrete convergence, and evaluates the fully discrete scheme numerically.
Problem
Efficiently approximating the fourth-order nonlinear Cahn-Hilliard equation requires balancing C1 regularity, implementation complexity, mesh generality, and computational cost.
Method
The paper develops a modification of minimal-degree C1 virtual elements using enhancement techniques, three projection operators, and virtual spaces suited to general polygonal meshes.
Results
The semi-discrete scheme is proved convergent, while numerical tests show L2 convergence of order 2, H2 convergence of order 1, and H1 convergence of order 2.
Takeaways & Limitations
The approach provides a C1 virtual element discretization for the Cahn-Hilliard equation with general polygonal meshes and computable projection operators based on simple degrees of freedom.
Takeaways & Limitations
The analysis uses assumptions involving positive constants and allows error-bound constants to depend on the regularity of the continuous solution.
Abstract
from arXiv · showhide
In this paper we develop an evolution of the $C^1$ virtual elements of minimal degree for the approximation of the Cahn-Hilliard equation. The proposed method has the advantage of being conforming in $H^2$ and making use of a very simple set of degrees of freedom, namely 3 degrees of freedom per vertex of the mesh. Moreover, although the present method is new also on triangles, it can make use of general polygonal meshes. As a theoretical and practical support, we prove the convergence of the semi-discrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests.
1. Introduction.
The paper motivates a C1 virtual element method for the fourth-order nonlinear Cahn-Hilliard equation, targeting simple implementation, high regularity, and general polygonal meshes. It develops the method, proves semi-discrete convergence, and tests the fully discrete scheme numerically.
- C1 finite elements naturally address the fourth-order Cahn-Hilliard equation, but their implementation is difficult.
- Mixed methods avoid the implementation difficulty of C1 basis functions but increase degrees of freedom and computational cost.
- The proposed C1 VEM handles general polygonal or polyhedral meshes while avoiding explicit local basis construction and complex element integrations.
- The paper develops a modification of minimal-degree C1 virtual elements using enhancement techniques and three computable projection operators.
- The study proves convergence of the semi-discrete scheme and investigates the fully discrete scheme through numerical tests.
2. The continuous and discrete problems.
The paper formulates the Cahn–Hilliard problem and constructs an H^2-conforming C^1 virtual element discretization on polygonal meshes using computable projectors and vertex-based degrees of freedom.
- The continuous problem: The Cahn–Hilliard equation is posed variationally on a space V, with a0, a∆, and r representing the time, fourth-order, and nonlinear terms.The weak problem seeks u(·, t) ∈ V satisfying a0(∂tu, v) + γ^2a∆(u, v) + r(u; u, v) = 0 for all v ∈ V, with prescribed initial data.
- A C^1 Virtual Element space: The local virtual space is defined on polygonal elements and consists of functions with piecewise cubic boundary traces, continuous boundary gradients, and quadratic bilaplacians.The construction uses an augmented local space and an enhancement condition to support the required projectors.
- A C^1 Virtual Element space: The degrees of freedom evaluate the function and its gradient at element vertices, and they uniquely determine the function and gradient on each element boundary.The sets D1 and D2 form a complete set of local degrees of freedom for the virtual space.
- Virtual forms: Three projection operators are computable from the degrees of freedom, enabling construction of the discrete bilinear forms for the problem.The projectors include the operators associated with the Laplacian, gradient, and L2 forms.
- A C^1 Virtual Element space: The global space is contained in H^2(Ω), yielding an H^2-conforming method with three degrees of freedom per mesh vertex.The global degrees of freedom are vertex values and the two components of the vertex gradient.
- Virtual forms: The discrete bilinear forms are exact when one argument belongs to P2(E), while stability requires mesh regularity assumptions and provides element-independent positive bounds.The consistency and stability statements apply to all three discrete bilinear forms.
3. Error analysis of the semi-discretization scheme.
The semi-discrete analysis establishes convergence under regularity assumptions on the exact and semi-discrete solutions. Its central result is an O(h^2) L2(Ω) error estimate, obtained through projection-based error decomposition and stability estimates.
- Scope of the analysis: The analysis focuses on the semi-discrete Virtual Element formulation, while fully discrete error analysis is deferred to standard time-discretization arguments.The paper identifies spatial discretization as the main novelty of the analysis.
- Assumptions: The convergence proof assumes regularity of the exact solution and of the semi-discrete solution uh.These assumptions are invoked throughout the projection and nonlinear-term estimates.
- Error-estimation strategy: The proof uses an elliptic projection P_hu, interpolation and polynomial approximation estimates, and consistency properties of the continuous and discrete bilinear forms.The approximation argument introduces interpolants and discontinuous piecewise quadratic polynomials to control projection and consistency errors.
- Convergence result: For every t ∈ [0,T], the method achieves the error estimate ∥u − uh∥L2(Ω) ≲ h^2.The final argument decomposes the error into projection and discrete components and controls the latter with Gronwall’s lemma.
4. Numerical results.
The numerical tests assess convergence to an exact solution and simulate interface evolution on quadrilateral, triangular, and polygonal meshes. The method achieves the predicted convergence orders and produces curvature-driven evolutions and spinodal decomposition across mesh types.
- Test 1: convergence to exact solution: Order 2 L2, order 1 H2, and order 2 H1 convergence are observed on four quadrilateral meshes.The L2 rate agrees with Theorem 3.6; the H2 and H1 rates align with expected approximation properties.
- Test 2: evolution of an ellipse: An ellipse-shaped jump-set evolves to a steady circular interface on structured quadrilateral and unstructured triangular meshes.The interface stops moving once it has constant curvature.
- Test 3: evolution of a cross: A cross-shaped jump-set evolves to a steady circular interface on quadrilateral, triangular, and Voronoi polygonal meshes.The Voronoi mesh includes quadrilateral, pentagonal, and hexagonal elements and uses 59490 degrees of freedom.
- Test 4: spinoidal decomposition: Spinodal decomposition separates the mixture into bulk regions rapidly, followed by slower interface motion toward constant curvature.The three simulations use quadrilateral, triangular, and polygonal meshes with different initial random configurations.
- Test 4: spinoidal decomposition: Different initial random configurations produce different final configurations, consistent with prior results.Quadrilateral and triangular meshes reach two rectangles, whereas the polygonal mesh reaches a circle.