International audienceThe complex transverse water proton magnetization subject to diffusion-encoding magneticfield gradient pulses in a heterogeneous medium can be modeled by themultiple compartment Bloch-Torrey partial differential equation (PDE).In addition, steady-state Laplace PDEs can be formulated to produce the homogenized diffusion tensor that describes the diffusion characteristicsof the medium in the long time limit.In spatial domains that model biological tissues at the cellular level, these two types of PDEs have to be completed with permeability conditions on the cellular interfaces.To solve these PDEs, we implemented a finite elements method thatallows jumps in the solution at the cell interfaces by using doublenodes. Using a transformation of the Bloch-Torrey PDE we reduced oscillations in the searched-for solution and simplified the implementationof the boundary conditions. The spatial discretizationwas then coupled to the adaptive explict Runge-Kutta-Chebychev time-stepping method. Our proposed method is second order accurate in space and second order accurate in time.We implemented this method on the FEniCSC++ platform and show time and spatial convergence results.Finally, this method is applied to study some relevant questions in diffusionMRI
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