Summary: We introduce a new phase field model for tumor growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn-Hilliard equation with source terms for the tumor cells and a convected reaction-diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behavior of tumors, are taken into account. The model is completed by a viscoelastic system consisting of the Navier-Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy-Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumor growth in two space dimensions \(d=2\) by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. \(\Delta t\le Ch^2\), is required.
Moreover, in arbitrary dimensions \(d\in\{2,3\}\), we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy-Green tensor is proved with a regularization technique that was first introduced by J. W. Barrett and S. Boyaval [Math. Models Methods Appl. Sci. 21, No. 9, 1783–1837 (2011; Zbl 1256.35048)]. After that, we improve the regularity results in arbitrary dimensions \(d\in\{2,3\}\) and in two dimensions \(d=2\), where a CFL condition is required. Then, in two dimensions \(d=2\), we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results in two dimensions \(d=2\).