Application of weak Galerkin finite element method for nonlinear chemotaxis and haptotaxis models

In this paper, we introduce a weak Galerkin finite element method for 2D Keller-Segel chemotaxis models and closely related haptotaxis models to simulate tumor invasion into surrounding healthy tissue. The chemotaxis and haptotaxis models consist of a system of time-dependent nonlinear reaction-diffusion-taxis partial differential equations. It is well known that solutions of chemotaxis systems may blow-up in finite time in the cell density. This blow-up is a mathematical description of a cell concentration phenomenon, which mathematically results in rapidly growing solutions in small neighborhoods of concentration points or curves. Capturing such singular solutions numerically is a challenging problem. We applied the weak Galerkin finite element method (WGFEM) to approximate the spatial variables. Weak Galerkin finite element method can be considered as an extension of the standard finite element method where classical derivatives are replaced in the variational equation by the weakly defined derivatives on discontinuous weak functions. The weak derivatives of weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the WGFEM is impacted by the selection of such polynomials. We show that the proposed method produced numerical results that are nonnegative, oscillation-free and suitable for solving models that have a blow-up of the cell density. Error bounds for cell density and chemoattractant equation are obtained. The proposed method is applied to several Keller-Segel chemotaxis models and extended to the haptotaxis model for simulating the behavior of cancer cell invasion of surrounding healthy tissue. Numerical results demonstrate the accuracy and efficiency of the proposed scheme.