8 research outputs found
Augmented saddle point formulation of the steady-state Stefan--Maxwell diffusion problem
We investigate structure-preserving finite element discretizations of the
steady-state Stefan--Maxwell diffusion problem which governs diffusion within a
phase consisting of multiple species. An approach inspired by augmented
Lagrangian methods allows us to construct a symmetric positive definite
augmented Onsager transport matrix, which in turn leads to an effective
numerical algorithm. We prove inf-sup conditions for the continuous and
discrete linearized systems and obtain error estimates for a phase consisting
of an arbitrary number of species. The discretization preserves the
thermodynamically fundamental Gibbs--Duhem equation to machine precision
independent of mesh size. The results are illustrated with numerical examples,
including an application to modelling the diffusion of oxygen, carbon dioxide,
water vapour and nitrogen in the lungs.Comment: 27 pages, 5 figure
A convergent entropy diminishing finite volume scheme for a cross-diffusion system
We study a two-point flux approximation finite volume scheme for a cross-diffusion system. The scheme is shown to preserve the key properties of the continuous systems, among which the decay of the entropy. The convergence of the scheme is established thanks to compactness properties based on the discrete entropy-entropy dissipation estimate. Numerical results illustrate the behavior of our scheme
An entropy structure preserving space-time Galerkin method for cross-diffusion systems
Cross-diffusion systems are systems of nonlinear parabolic partial
differential equations that are used to describe dynamical processes in several
application, including chemical concentrations and cell biology. We present a
space-time approach to the proof of existence of bounded weak solutions of
cross-diffusion systems, making use of the system entropy to examine long-term
behavior and to show that the solution is nonnegative, even when a maximum
principle is not available. This approach naturally gives rise to a novel
space-time Galerkin method for the numerical approximation of cross-diffusion
systems that conserves their entropy structure. We prove existence and
convergence of the discrete solutions, and present numerical results for the
porous medium, the Fisher-KPP, and the Maxwell-Stefan problem.Comment: 33 page
A convergent entropy diminishing finite volume scheme for a cross-diffusion system
International audienceWe study a two-point flux approximation finite volume scheme for a cross-diffusion system. The scheme is shown to preserve the key properties of the continuous systems, among which the decay of the entropy. The convergence of the scheme is established thanks to compactness properties based on the discrete entropy-entropy dissipation estimate. Numerical results illustrate the behavior of our scheme