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 Γ(t). The partial differential equations are solved on a stationary quadrilateral, resp., hexahedral mesh. The zero level set of the time dependent indicator function ϕ(t) implicitly describes the position of Γ(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.