  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.
