A posteriori error analysis for a reduced-basis approximation of two parabolic problems for tumour growth

Abstract

This work presents a theoretical analysis for an a posteriori error estimate for a reduced-basis approximation of the solution for two parametrised parabolic problems. Their motivation is in mathematical oncology and they describe a) a model for brain tumour growth [14] and b) a model for phenotype evolution of a tumour based on [3, 13], with the free parameter being the therapeutic dose. The discretisation of the problems in space is realised by the finite element method, and the numerical integration uses a first-order IMEX scheme due to model b) being a integro-differental problem. The obtained a posteriori error estimate for the approximation error between the numerical solution and the reduced basis solution gives the opportunity for constructing a reduced basis for the solution manifold by combining a proper orthogonal decomposition of the temporal trajectories and a greedy algorithm over the parameter range, as has been done in the case of explicit or implicit integration of linear parabolic problems [8, 10].