38,045 research outputs found
A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods
We investigate the connections between several recent methods for the
discretization of anisotropic heterogeneous diffusion operators on general
grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite
Volume scheme and the Mixed Finite Volume scheme are in fact identical up to
some slight generalizations. As a consequence, some of the mathematical results
obtained for each of the method (such as convergence properties or error
estimates) may be extended to the unified common framework. We then focus on
the relationships between this unified method and nonconforming Finite Element
schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit
lifting operator close to the ones used in some theoretical studies of the
Mimetic Finite Difference scheme. We also show that for isotropic operators, on
particular meshes such as triangular meshes with acute angles, the unified
method boils down to the well-known efficient two-point flux Finite Volume
scheme
A note on stress-driven anisotropic diffusion and its role in active deformable media
We propose a new model to describe diffusion processes within active
deformable media. Our general theoretical framework is based on physical and
mathematical considerations, and it suggests to use diffusion tensors directly
coupled to mechanical stress. A proof-of-concept experiment and the proposed
generalised reaction-diffusion-mechanics model reveal that initially isotropic
and homogeneous diffusion tensors turn into inhomogeneous and anisotropic
quantities due to the intrinsic structure of the nonlinear coupling. We study
the physical properties leading to these effects, and investigate mathematical
conditions for its occurrence. Together, the experiment, the model, and the
numerical results obtained using a mixed-primal finite element method, clearly
support relevant consequences of stress-assisted diffusion into anisotropy
patterns, drifting, and conduction velocity of the resulting excitation waves.
Our findings also indicate the applicability of this novel approach in the
description of mechano-electrical feedback in actively deforming bio-materials
such as the heart
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