Convergence and positivity of finite element methods for a haptotaxis model of tumoral invasion

Abstract: In this paper, we consider a mathematical model for the invasion of host tissue by tumour cells in a $d$-dimensional bounded domain, $d\leq 3$. This model consists of a system of differential equations describing the evolution of cancer cell density, the extracellular matrix protein density and the matrix degrading enzyme concentration. We develop two fully discrete schemes for approximating the solutions based on the Finite Element (FE) method. For the first numerical scheme, we use a splitting technique to deal with the haptotaxis term, leading to introduce an equivalent system with a new variable given by the gradient of extracellular matrix. This scheme is well-posed and preserves the non-negativity of extracellular matrix and the degrading enzyme. We analyze error estimates and convergence towards regular solutions. The second numerical scheme is based on an equivalent formulation in which the cancer cell density equation is expressed in a divergence form through a suitable change of variables. This second numerical scheme preserves the non-negativity of all the discrete variables. Finally, we present some numerical simulations in agreement with the theoretical analysis.