The electrophysiology of the heart can be described by models with distinct levels of detail, such as the bidomain and the extracellular-membrane-intracellular (EMI) model, respectively. Both models are formulated as differential-algebraic reaction-diffusion equations coupled with pointwise ordinary differ-ential equations for the state of the membrane’s ion channels and ion concentrations. The solution of the discretized elliptic algebraic constraint dominates the computational cost due to its much larger condition number compared to the parabolic equation. We propose a new approach by solving the elliptic constraint less accurately within an IMEX Euler scheme and compensating the resulting consistency error in the subsequent time step. This compensation can be performed at low cost and improves accuracy for a given iteration count. We interpret the scheme as the discretization of an extended time-continuous system. This interpretation allows the application of higher-order time integration schemes, such as the spectral deferred correction (SDC) method, and introduces a relaxation parameter. The locality of the travelling wave like solution calls for the use of adaptive methods. The iteratively correcting nature of SDC enables the identification of regions where solution updates are required in each time step. By restricting the linear solves to these regions, we significantly reduce the computational cost, resulting in a highly efficient method. Numerical results illustrate the impact of the approach on both efficiency and accuracy.