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A higher order nonconforming virtual element method for the Cahn-Hilliard equation

Andreas Dedner, Alice Hodson

TL;DR

A fully discrete scheme which uses a convex splitting Runge-Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization is presented.

Abstract

In this paper we develop a fully nonconforming virtual element method (VEM) of arbitrary approximation order for the two dimensional Cahn-Hilliard equation. We carry out the error analysis for the semidiscrete (continuous-in-time) scheme and verify the theoretical convergence result via numerical experiments. We present a fully discrete scheme which uses a convex splitting Runge-Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization.

A higher order nonconforming virtual element method for the Cahn-Hilliard equation

TL;DR

A fully discrete scheme which uses a convex splitting Runge-Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization is presented.

Abstract

In this paper we develop a fully nonconforming virtual element method (VEM) of arbitrary approximation order for the two dimensional Cahn-Hilliard equation. We carry out the error analysis for the semidiscrete (continuous-in-time) scheme and verify the theoretical convergence result via numerical experiments. We present a fully discrete scheme which uses a convex splitting Runge-Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization.
Paper Structure (21 sections, 12 theorems, 125 equations, 10 figures, 7 tables)

This paper contains 21 sections, 12 theorems, 125 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. The continuous problem
  4. Basic spaces and notation
  5. Virtual element discretization
  6. Local VEM space
  7. Global spaces and the discrete forms
  8. The semidiscrete problem
  9. Error analysis of the semidiscrete scheme
  10. The elliptic projection
  11. Technical results
  12. Error estimate for the semidiscrete scheme
  13. Numerical Results
  14. Test 1: VEM convergence
  15. Test 1: Convergence to an exact solution
  16. ...and 6 more sections

Key Result

Theorem 2.4

Under Assumption assumption: mesh regularity, for $\ell\geq 0$ and for any $w \in H^m (K)$ with $1 \leq m \leq \ell +1$, it follows that for $s = 0,1,2$ with $s\leq m$. Further, for any edge $e$ shared by $K^{+}$,$K^{-} \in \mathcal{T}_h$ and for any $w \in H^m(K^{+} \cup K^{-})$, with $1 \leq m \leq \ell+1$, it follows that for $s=0,1,2$ with $s\leq m$.

Figures (10)

  • Figure 1: Degrees of freedom for polynomial orders $\ell=2,3,4,5$ on triangles for the local VEM space. Circles at vertices represent vertex dofs, arrows represent edge normal dofs, circles on edges represent edge value moments and interior squares represent inner dofs.
  • Figure 2: Test 2: Two interacting bubbles. Screenshots of the zero-level sets at times $t=0,0.004,0.016,0.048,0.144,0.3$ with $\varepsilon=3/100$ are displayed on the $25 \times 25$ Voronoi polygonal grid.
  • Figure 3: Test 3: evolution of a cross on the $25 \times 25$ grids displayed at three different time frames from left to right $(t=0,0.004,0.8)$ with $\varepsilon=1/100$.
  • Figure 4: Test 3: evolution of a cross on the $25 \times 25$ grids displayed at three different time frames from left to right $(t=0,0.004,0.8)$ with $\varepsilon=1/25$.
  • Figure 5: Test 3: evolution of a cross displayed at the end time frame $t=0.8$ on the grid sizes from left to right $15 \times 15$, $25 \times 25$, and $45 \times 45$ with $\varepsilon=1/100$.
  • ...and 5 more figures

Theorems & Definitions (32)

  • Definition 2.1
  • Remark 2.3
  • Theorem 2.4
  • Definition 3.1: Local enlarged space
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6: Local virtual space
  • Lemma 3.7
  • Definition 3.8: Global virtual space
  • ...and 22 more