Investigation of phase-field models of tumor growth based on a reduced-order meshless Galerkin method

The current paper concerns to develop a new numerical formulation to simulate the tumor growth. The used numerical method is based on the meshless Galerkin technique in which the test and trial functions have been selected from the shape functions of moving Taylor approximation. The main mathematical model to describe the tumor growth is defined as a nonlinear system of equations. Thus, to get acceptable results from the Galerkin weak form, a two-grid algorithm is employed. The first step of the two-grid algorithm computes the corresponding approximated scheme in a coarse mesh by solving a nonlinear algebraic system of equations. Then, the obtained solution in the previous step has been used to solve the corresponding approximated scheme in a fine mesh, such that in the second step, a linear algebraic system of equations is solved. On the other hand, to access more accurate results, the number of nodes in the computational domain must be increased which causes the matrix to become larger. Therefore, the proper orthogonal decomposition is used to reduce size of the algebraic system of equations. Finally, some test problems are tested to confirm the efficiency and accuracy of the proposed numerical formulation.