Andriy Sokolov

A positivity-preserving finite element method for chemotaxis problems in 3D

We present an implicit finite element method for a class of chemotaxis models in three spatial dimensions. The proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density. To enforce the discrete maximum principle, the standard Galerkin discretization is constrained using a local extremum diminishing flux limiter. To demonstrate the efficiency and robustness of this approach, we solve blow-up problems in a 3D chemostat domain. To give a flavor of more complex and realistic chemotactic applications, we investigate the pattern dynamics and aggregating behavior of the bacteria Escherichia coli and Salmonella typhimurium. The obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature.

A Flux-Corrected Finite Element Method for Chemotaxis Problems

Abstract An implicit flux-corrected transport (FCT) algorithm has been developed for a class of chemotaxis models. The coefficients of the Galerkin finite element discretization has been adjusted in such a way as to guarantee mass conservation and keep the cell density nonnegative. The numerical behaviour of the proposed highresolution scheme is tested on the blow-up problem for a minimal chemotaxis model with singularities. It has also been shown that the results for an Escherichia coli chemotaxis model are in good agreement with the experimental data reported in the literature.

An AFC-stabilized implicit finite element method for partial differential equations on evolving-in-time surfaces

In this article we present a new implicit numerical scheme for reaction-diffusion-advection equations on an evolving in time hypersurface ? ( t ) . The partial differential equations are solved on a stationary quadrilateral, resp., hexahedral mesh. The zero level set of the time dependent indicator function ? ( t ) implicitly describes the position of ? ( t ) . The dominating convective-like terms, which are due to the presence of chemotaxis, transport of the cell density and surface evolution may lead to the non-positiveness of a given numerical scheme and in such a way cause appearance of negative values and give rise of nonphysical oscillations in the numerical solution. The proposed finite element method is constructed to avoid this problem: implicit treatment of corresponding discrete terms in combination with the algebraic flux correction (AFC) techniques make it possible to obtain a sufficiently accurate solution for reaction-diffusion-advection PDEs on evolving surfaces.

Efficient, accurate and flexible finite element solvers for chemotaxis problems

In the framework of finite element discretizations, we introduce a fully nonlinear Newton-like method and a linearized second order approach in time applied to certain partial differential equations for chemotactic processes incorporating two entities, a chemical agent and the reacting population of certain biological organisms/species. We investigate the benefit of a corresponding monolithic approach and the decoupled variant. In particular, we analyze accuracy, efficiency and stability of different methods and their dependences on certain parameters in order to identify a well suited finite element solver for chemotaxis problems.

Flow Control on the Basis of a Featflow-Matlab Coupling

For the model-based active control of three-dimensional flows at high Reynolds numbers in real time, low-dimensional models of the flow dynamics and efficient actuator and sensor concepts are required. Numerous successful approaches to derive such models have been proposed in the literature.

Numerical study of the RBF-FD level set based method for partial differential equations on evolving-in-time surfaces

In this article we present a Radial Basis Function (RBF)-Finite Difference (FD) level set based method for the numerical solution of partial differential equations (PDEs) of the reaction-diffusion-convection type on an evolving-in-time hypersurface Γ(t). In a series of numerical experiments we study the accuracy and robustness of the proposed scheme and demonstrate that the method is applicable to practical models.