The main goal of this work is an efficient numerical solution of a set of coupled differential equations. A lithium ion battery model, supported by Kompetenzzentrum - Das Virtuelle Fahrzeug Forschungsgesellschaft mbH (ViF), that consists of four coupled nonlinear elliptic and parabolic partial differential equations, is reformulated in a weak formulation. That weak formulation is discretised in space, using the finite element method. In this method, Lagrange elements of arbitrary order are used. The system of differential algebraic equations, which arises that way, is discretised in time, using a two layer method. Furthermore, an initialisation strategy, to achieve consistent initial values of the unknowns, is designed. That way discretised in space and time, a solution to the resulting finite dimensional nonlinear problem is numerically approximated. For this purpose several methods are used, such as a damped Newton method, the Gaussian quadrature and an adaptive time stepping strategy. A program, which supports all the needed functionality, is implemented in an object oriented manner. The equations are solved fully coupled and an additionally implemented control object supports the time dependent boundary values for the equations. Special effort is put into an efficient implementation. Therefore the assembling procedure of the matrices, that occur in using the finite element method, is stated in a tensor formalism. This allows for the use of existing and very efficient program libraries. Furthermore, all floating point operations that occur with the same data again and again are calculated in advance in order to minimize the quantity of these operations. The implementation is tested via the method of manufactured solutions. In the last chapter some convergence results of this test are shown. At last some typical simulation results are shown.