Adaptive approximations of inclusions in a semilinear elliptic problem related to cardiac electrophysiology

In this work, we investigate the numerical reconstruction of inclusions in a semilinear elliptic equation arising in the mathematical modeling of cardiac ischemia. We propose an adaptive finite-element method for the resulting constrained minimization problem that is relaxed by a phase-field approach. The a posteriori error estimators of the adaptive algorithm consist of three components, i.e., the state variable, the adjoint variable and the complementary relation. Moreover, using tools from adaptive finite-element analysis and nonlinear optimization, we establish the strong convergence for a subsequence of adaptively generated discrete solutions to a solution of the continuous optimality system. Several numerical examples are presented to illustrate the convergence and efficiency of the adaptive algorithm.