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Error analysis of a fully discrete structure-preserving finite element scheme for a diffuse-interface model of tumour growth

Agus L. Soenjaya, Ping Lin, Thanh Tran

TL;DR

This work addresses accurate and stable numerical simulation of a diffuse-interface tumor growth model described by a Cahn–Hilliard–type equation coupled to nutrient dynamics. It develops a fully discrete, linear, structure-preserving scheme using a scalar auxiliary variable (SAV) formulation together with a mixed finite element discretization for the Cahn–Hilliard part and standard FE for the reaction-diffusion component, with backward Euler time stepping. The main contributions are unconditional energy stability, mass conservation, and rigorous error analysis yielding first-order temporal and optimal spatial convergence, along with $L^\infty$ bounds for the numerical solution. Numerical experiments in three dimensions validate the theory and demonstrate robustness in capturing aggregation and chemotactic tumour growth, underscoring the method’s potential for reliable, physics-informed simulations of complex tumour phenomena.

Abstract

We develop a linear fully discrete structure-preserving finite element method for a diffuse-interface model of tumour growth. The system couples a Cahn--Hilliard type equation with a nonlinear reaction-diffusion equation for nutrient concentration and admits a dissipative energy law at the continuous level. For the discretisation, we employ a scalar auxiliary variable (SAV) formulation together with a mixed finite element method for the Cahn--Hilliard part and standard conforming finite elements for the reaction-diffusion equation in space, combined with a first-order Euler time-stepping scheme. The resulting method is unconditionally energy-stable, mass-preserving, and inherits a discrete energy dissipation law associated with the SAV-based approximate energy functional, while requiring the solution of only linear systems at each time step. Under suitable regularity assumptions on the exact solution, we derive rigorous error estimates in $L^2$, $H^1$, and $L^\infty$ norms, establishing first-order accuracy in time and optimal-order accuracy in space. A key step in this analysis is the proof of boundedness of the numerical solutions in $L^\infty$. Numerical experiments validate the theoretical convergence rates and demonstrate the robustness of the method in capturing characteristic phenomena such as aggregation and chemotactic tumour growth.

Error analysis of a fully discrete structure-preserving finite element scheme for a diffuse-interface model of tumour growth

TL;DR

This work addresses accurate and stable numerical simulation of a diffuse-interface tumor growth model described by a Cahn–Hilliard–type equation coupled to nutrient dynamics. It develops a fully discrete, linear, structure-preserving scheme using a scalar auxiliary variable (SAV) formulation together with a mixed finite element discretization for the Cahn–Hilliard part and standard FE for the reaction-diffusion component, with backward Euler time stepping. The main contributions are unconditional energy stability, mass conservation, and rigorous error analysis yielding first-order temporal and optimal spatial convergence, along with bounds for the numerical solution. Numerical experiments in three dimensions validate the theory and demonstrate robustness in capturing aggregation and chemotactic tumour growth, underscoring the method’s potential for reliable, physics-informed simulations of complex tumour phenomena.

Abstract

We develop a linear fully discrete structure-preserving finite element method for a diffuse-interface model of tumour growth. The system couples a Cahn--Hilliard type equation with a nonlinear reaction-diffusion equation for nutrient concentration and admits a dissipative energy law at the continuous level. For the discretisation, we employ a scalar auxiliary variable (SAV) formulation together with a mixed finite element method for the Cahn--Hilliard part and standard conforming finite elements for the reaction-diffusion equation in space, combined with a first-order Euler time-stepping scheme. The resulting method is unconditionally energy-stable, mass-preserving, and inherits a discrete energy dissipation law associated with the SAV-based approximate energy functional, while requiring the solution of only linear systems at each time step. Under suitable regularity assumptions on the exact solution, we derive rigorous error estimates in , , and norms, establishing first-order accuracy in time and optimal-order accuracy in space. A key step in this analysis is the proof of boundedness of the numerical solutions in . Numerical experiments validate the theoretical convergence rates and demonstrate the robustness of the method in capturing characteristic phenomena such as aggregation and chemotactic tumour growth.

Paper Structure

This paper contains 13 sections, 17 theorems, 200 equations, 15 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Assumptions
  5. SAV formulations
  6. Finite element approximation
  7. SAV-FEM with backward Euler time discretisation
  8. Numerical simulations
  9. Simulation 1 (growth of a single tumour)
  10. Simulation 2 (aggregation of multiple tumours)
  11. Simulation 3 (temporal rates of convergence)
  12. Simulation 4 (chemotactic growth of a three-dimensional tumour)
  13. Conclusion

Key Result

Proposition 3.2

Let $(u_h^k,n_h^k)\in V_h\times V_h$ and $r_h^k\in \mathbb{R}$ be given, with $P(u_h^k)>0$ on a subset of $\mathscr{D}$ of positive measure. For sufficiently small $\tau>0$, there exists a unique $(u_h^{k+1},\mu_h^{k+1},n_h^{k+1})\in V_h\times V_h\times V_h$ and $r_h^{k+1}\in \mathbb{R}$ solving the

Figures (15)

  • Figure 1: Snapshots of the tumour volume fraction $u$ in simulation 1.
  • Figure 2: Snapshots of the nutrient volume fraction $n$ in simulation 1.
  • Figure 3: Spatial convergence order of tumour volume fraction and chemical potential in simulation 1.
  • Figure 4: Spatial convergence order of nutrient volume fraction $n$ in simulation 1.
  • Figure 5: Conservation of mass and energy dissipation in simulation 1.
  • ...and 10 more figures

Theorems & Definitions (37)

  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 27 more