This thesis introduces a new computational methodology to solve partial differential equations within cracked domains, focusing on coupling in-crack physics with the adjacent environment. The crack is represented discretely, allowing to describe explicitly the crack opening, which is a key parameter in the description of the in-crack physics. The governing equations are projected onto the tangential direction of the crack to obtain a hybrid formulation of the weak form. Because the fracture is reduced to a lower dimensional domain, this methodology leads to a discontinuity in the primal solution field. Therefore, a discontinuous Galerkin approach is employed in discretizing the equations, yielding results that align well with the analytical model across various configurations and increasing the model’s possibilities. The results obtained for the electric governing equations show potential for developing a model incorporating mechanical and chemical aspects, enabling a fully coupled model for dendrite propagation in solid-state batteries.