A posteriori error estimates and adaptive stopping criteria for cardiac monodomain model
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In this work, we consider the monodomain problem to simulate the electrical activity within the heart tissue. From a mathematical standpoint, it is represented by a reaction diffusion equation coupled with a system of ODEs whose numerical resolution is often challenging and highly expensive. We employ the implicit Euler scheme for the time discretization and the finite element method or order p≥1 for the discretization in space. We derive a guaranteed a posteriori error estimate on the error which is valid at each time step and each step of the linearization solver. Our estimate, based on equilibrated flux reconstructions, also distinguishes the temporal and the spatial error; the later being composed of the discretization and the linearization error. Thanks to an adaptive procedure, we show that it is possible to reduce the number of iterations while preserving the accuracy of the solution. Numerical experiments are performed to show the strength of the proposed approach.