Numerical analysis of a parabolic variational inequality system modeling biofilm growth at the porescale

Abstract

In this paper we consider a system of two coupled nonlinear diffusion-reaction partial differential equations (PDEs) which model the growth of biofilm and consumption of the nutrient. At the scale of interest the biofilm density is subject to a pointwise constraint, thus the biofilm PDE is framed as a parabolic variational inequality. We derive rigorous error estimates for a finite element (FE) approximation to the coupled nonlinear system and confirm experimentally that the numerical approximation converges at the predicted rate. We also show simulations in which we track the free boundary in the domains which resemble the pore scale geometry and in which we test the different modeling assumptions. KeywordsParabolic variational inequality, coupled system, biofilm growth, finite elements, error estimates, semismooth Newton solver.

Analysis of a finite-volume scheme for a single-species biofilm model

“…where deg is the Brouwer topological degree. Since deg is invariant by homotopy, it is sufficient to show that any solution (S ε , W ε , ρ) 1] for sufficiently large values of R > 0. Let (S ε , W ε , ρ) be a fixed point and assume that ρ = 0, the case ρ = 0 being clear.…”

“…Some properties of the semi-implicit Euler finitedifference scheme were shown in [12]. A finite-element approximation for (2) with linear diffusion f (M ) = 1 but a constraint on the upper bound for the biomass was suggested in [1].…”

“…With the exception of [1,8], these mentioned works do not contain any analysis of the numerical scheme. The paper [1] is concerned with a finite-element method and assumes linear diffusion, while [8] does not contain an equation for the nutrient. In this paper, we provide a numerical analysis of a finite-volume scheme to (1)-(2) for the first time.…”

Numerical Analysis of a Mixed Finite Element Approximation of a Coupled System Modeling Biofilm Growth in Porous Media with Simulations

“…We approximate the model with mixed finite element method (MFEM). We believe this choice is better for the problem than the finite element method (FEM) we considered in our earlier work in [1], because of the conservative property of MFEM and its natural way of handling Neumann boundary conditions. (We remark that MFEM works also very well theoretically and computationally when Dirichlet condition is imposed unlike FEM that we succeeded in [1] in deriving an error estimate with Dirichlet conditions only.)…”

“…We believe this choice is better for the problem than the finite element method (FEM) we considered in our earlier work in [1], because of the conservative property of MFEM and its natural way of handling Neumann boundary conditions. (We remark that MFEM works also very well theoretically and computationally when Dirichlet condition is imposed unlike FEM that we succeeded in [1] in deriving an error estimate with Dirichlet conditions only.) Moreover, the implementation of MFEM with the lowest order of Raviart Thomas elements on rectangles and cubes RT [0] as cell centered finite difference method (CCFD) is very easy to implement and to use for voxel grids.…”