Toward generalized solutions of the Keller--Segel equations with singular sensitivity and signal absorption via an algebraic manipulation finite element algorithm

The paper that follows describes a numerical algorithm to solve the parabolic-parabolic Keller--Segel system characterized by singular sensitivity and signal absorption in such a manner that the numerical approximations converge towards a generalized solution on two-dimensional polygonal domains as the time and space discretization parameters tend to zero. The algorithm employs an algebraic manipulation finite element method for space, while time remains continuous, based on introducing a stabilized term. This term is constructed via a graph-Laplacian operator and a shock detector for detecting extrema in finite element functions. Furthermore, the cross-diffusion term also includes an algebraic manipulation, which is related to testing by a nodally interpolated, suitable nonlinear function involved in obtaining a discrete energy-like law leading to a priori estimates. This approach yields approximations that respect physical constraints at nodal points such as positivity and maximum principle, and maintaining mass properties as well. Compactness results are quite laborious due to the low regularity stemming from the a priori estimates and the discretization procedure itself. Finally, the passage to the limit rests on testing by the product of a positive test function and a renormalization of the numerical solution.