Permeability-regulated reaction–diffusion patterns across semipermeable membranes: Structure-preserving numerical simulation

We develop a structure-preserving fully discrete method for reaction–diffusion systems separated by semipermeable membranes and ask whether increasing permeability primarily reduces interfacial mismatch or also suppresses bulk pattern formation. A broken finite element space duplicates the interface degrees of freedom and therefore preserves the two membrane traces. It is combined with a linearly implicit variable-step BDF2 discretization that treats diffusion and membrane exchange implicitly and extrapolates the nonlinear reaction. The scheme preserves species-wise membrane mass balance and the nonnegative membrane-jump term in the discrete energy identity. For the reaction-free problem, an increment inequality and discrete convolution kernels give complementary L 2 -dissipation and modified diffusion–membrane energy laws, including permeability-dependent jump control. For locally smooth reactions, a local mean-value estimate and bootstrap argument give optimal-order fully discrete error bounds. Numerical tests confirm the convergence and structural identities. In a Schnakenberg experiment, increasing permeability reduces the combined membrane mismatch by nearly two orders of magnitude, while the bulk pattern amplitudes remain robust and the visible onset time changes only mildly. Thus semipermeable exchange can regulate interfacial mismatch without numerically suppressing nonlinear pattern formation.