A fully-decoupled discontinuous Galerkin approximation of the Cahn–Hilliard–Brinkman–Ohta–Kawasaki tumor growth model

¹ School of Mathematics and Statistics, Henan University, Kaifeng 475004, P.R. China
² Henan Engineering Research Center for Artificial Intelligence Theory and Algorithms, Henan University, Kaifeng 475004, P.R. China
³ Henan Key Laboratory of Earth System Observation and Modeling, Henan University, Kaifeng 475004, P.R. China
⁴ Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

^* Corresponding author: xfyang@math.sc.edu

Received: 18 December 2021
Accepted: 6 September 2022

Abstract

In this article, we consider the Cahn–Hilliard–Brinkman–Ohta–Kawasaki tumor growth system, which couples the Brinkman flow equations in the porous medium and the Cahn–Hilliard type equation with the nonlocal Ohta–Kawasaki term. We first construct a fully-decoupled discontinuous Galerkin method based on a decoupled, stabilized energy factorization approach and implicit-explicit Euler method in the time discretization, and strictly prove its unconditional energy stability. The optimal error estimate for the tumor interstitial fluid pressure is further obtained. Numerical results are also carried out to demonstrate the effectiveness of the proposed numerical scheme and verify the theoretical results. Finally, we apply the scheme to simulate the evolution of brain tumors based on patient-specific magnetic resonance imaging, and the obtained computational results show that the proposed numerical model and scheme can provide realistic calculations and predictions, thus providing an in-depth understanding of the mechanism of brain tumor growth.