The study of the dynamics of membranes subject to electric fields is motivated by several applications in micro-scale biomechanics; one example is the characterization of red blood cells in common blood analysis systems. A numerical simulation is put in place to represent the transition of a cell undergoing an electric drive. The membrane of the cell is modeled as a non stationary closed interface immersed in the domain, which is partitioned by the former into the union of an internal and an external subdomain. The flow in the two subdomains is described as an incompressible Stokes system with matching interfacial velocity. The coupling between the flow and the membrane is realised by imposing a balance of tension at the interface and pure advection of the interface by the flow. To complete the model, interfacial Maxwell stress acting on the membrane is to be estimated by solving an elliptic equation with interface conditions. To represent the movement of the membrane, a polygonal discretization of the interface is introduced, such that the position of nodes is updated at the end of each time iteration. Given the updated position of the interface, an iteration-dependent mesh is computed by cutting a background simplicial mesh. By this construction, each one the elements of the cut mesh belongs to only one of the two subdomains but polygonal elements can be generated. This motivates the introduction of polytopal schemes. A low order HHO method is adopted to discretize the Stokes system and a DDR scheme is employed for the elliptic interface problem. Both schemes enjoy a-priori convergence estimates and support generic meshes.