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Model Order Reduction for Pattern Formation in FitzHugh-Nagumo Equation

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FitzHugh–Nagumo equations, Gradient systems, Traveling fronts and pulses, Turing patterns , Energy preservation, Discontinuous Galerkin, Model order reduction, Discrete empirical interpolation

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      We developed a reduced order model (ROM) using the proper orthogonal decomposition (POD) to compute efficiently the labyrinth and spot like patterns of the FitzHugh-Nagumo (FNH) equation. The FHN equation is discretized in space by the discontinuous Galerkin (dG) method and in time by the backward Euler method. Applying POD-DEIM (discrete empirical interpolation method) to the full order model (FOM) for different values of the parameter in the bistable nonlinearity, we show that using few POD and DEIM modes, the patterns can be computed accurately. Due to the local nature of the dG discretization, the PODDEIM requires less number of connected nodes than continuous finite element for the nonlinear terms, which leads to a significant reduction of the computational cost for dG POD-DEIM.

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