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Numerical analysis of an evolving bulk--surface model of tumour growth

Dominik Edelmann, Balázs Kovács, Christian Lubich

TL;DR

This work analyzes a numerically tractable bulk–surface tumour-growth model obtained by a generalized Robin boundary condition with a stabilizing term $\mu>0$, enabling a rigorous stability and convergence theory for an evolving bulk–surface finite element method. The authors formulate four coupled subsystems (Robin boundary value problem for the tissue pressure, forced mean curvature flow on the moving surface, harmonic extension of surface velocity into the bulk, and an ODE for domain evolution) and provide a matrix–vector discretization that yields optimal-order $H^1$-norm error bounds for $u$ and geometric quantities $(x,n,H)$. A central contribution is the stability analysis (via defects and consistency) showing that discretization errors remain controlled by the defects, together with comprehensive consistency estimates for all subproblems; this culminates in a global error bound of order $h^k$ for $k\ge 2$. Numerical experiments corroborate the theory, demonstrate convergence rates, and reveal the impact (small) of regularization on the solution, while the analysis clarifies why the original $\mu=0$ EKS model resists such stability proofs. Overall, the paper provides a rigorous, implementable framework for simulating coupled bulk–surface evolution in tissue growth with reliable error control.

Abstract

This paper studies an evolving bulk--surface finite element method for a model of tissue growth, which is a modification of the model of Eyles, King and Styles (2019). The model couples a Poisson equation on the domain with a forced mean curvature flow of the free boundary, with nontrivial bulk--surface coupling in both the velocity law of the evolving surface and the boundary condition of the Poisson equation. The numerical method discretizes evolution equations for the mean curvature and the outer normal and it uses a harmonic extension of the surface velocity into the bulk. The discretization admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The stability of the discretized bulk--surface coupling is a major concern. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed tissue pressure and for the surface position, velocity, normal vector and mean curvature. Numerical experiments illustrate and complement the theoretical results.

Numerical analysis of an evolving bulk--surface model of tumour growth

TL;DR

This work analyzes a numerically tractable bulk–surface tumour-growth model obtained by a generalized Robin boundary condition with a stabilizing term , enabling a rigorous stability and convergence theory for an evolving bulk–surface finite element method. The authors formulate four coupled subsystems (Robin boundary value problem for the tissue pressure, forced mean curvature flow on the moving surface, harmonic extension of surface velocity into the bulk, and an ODE for domain evolution) and provide a matrix–vector discretization that yields optimal-order -norm error bounds for and geometric quantities . A central contribution is the stability analysis (via defects and consistency) showing that discretization errors remain controlled by the defects, together with comprehensive consistency estimates for all subproblems; this culminates in a global error bound of order for . Numerical experiments corroborate the theory, demonstrate convergence rates, and reveal the impact (small) of regularization on the solution, while the analysis clarifies why the original EKS model resists such stability proofs. Overall, the paper provides a rigorous, implementable framework for simulating coupled bulk–surface evolution in tissue growth with reliable error control.

Abstract

This paper studies an evolving bulk--surface finite element method for a model of tissue growth, which is a modification of the model of Eyles, King and Styles (2019). The model couples a Poisson equation on the domain with a forced mean curvature flow of the free boundary, with nontrivial bulk--surface coupling in both the velocity law of the evolving surface and the boundary condition of the Poisson equation. The numerical method discretizes evolution equations for the mean curvature and the outer normal and it uses a harmonic extension of the surface velocity into the bulk. The discretization admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The stability of the discretized bulk--surface coupling is a major concern. The error analysis combines stability estimates and consistency estimates to yield optimal-order -norm error bounds for the computed tissue pressure and for the surface position, velocity, normal vector and mean curvature. Numerical experiments illustrate and complement the theoretical results.
Paper Structure (63 sections, 8 theorems, 203 equations, 4 figures)

This paper contains 63 sections, 8 theorems, 203 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Eyles--King--Styles model of tumour growth
  3. Basic notions and notation
  4. Model equations and more equations
  5. A 'tractable' model for tumour growth
  6. Forced mean curvature flow
  7. Harmonic extension of the surface velocity into the bulk
  8. Regularization
  9. Collected equations used for the discretization
  10. Weak formulation
  11. Generalized Robin boundary value problem
  12. Forced mean curvature flow
  13. Harmonic velocity extension
  14. Spatial semi-discretization with bulk--surface finite elements
  15. Evolving bulk--surface finite elements
  16. ...and 48 more sections

Key Result

Theorem 1

Consider the space discretization of Section subsec:semi-disc-bs of the coupled bulk--surface problem robin-bvp--ODE, using evolving bulk--surface finite elements of polynomial degree $k\ge 2$. Suppose that the problem admits an exact solution $(X,v,\textnormal{n},H,u)$ that is sufficiently differen where $x_h^L$ is defined in xhL. Furthermore, $\|X_h^\ell(\cdot,t) - X(\cdot,t)\|_{H^1(\varGamma_0)

Figures (4)

  • Figure 1: Spatial and temporal convergence of the BDF2 / quadratic ESFEM discretization of the coupled bulk--surface problem with $\alpha = 1$ and $\beta = 1$.
  • Figure 2: Numerical solution of the coupled bulk--surface problem with $\alpha = 10$ and $\beta = 1$.
  • Figure 3: Numerical solution of the coupled bulk--surface problem with $\alpha = 1$ and $\beta = 0.1$.
  • Figure 4: Numerical solution ($\varGamma_h^n$ and $u_h^n$) of the coupled bulk--surface problem with varying regularization parameter $\mu = 0$, $0.01$, $0.1$, $1$ (column-wise from left to right) $\alpha = 1$ and $\beta = 1$.

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • Proposition 4
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • ...and 3 more