This dissertation presents techniques for high-order simulation of electromagnetic fields, particularly for problems involving ships with ferromagnetic hulls and active corrosion-protection systems. A set of numerically constrained hexahedral basis functions for volume integral equation discretization is presented in a method-of-moments context. Test simulations demonstrate the accuracy achievable with these functions as well as the improvement brought about in system conditioning when compared to other basis sets. A general method for converting between a locally-corrected Nyström discretization of an integral equation and a method-of-moments discretization is presented next. Several problems involving conducting and magnetic-conducting materials are solved to verify the accuracy of the method and to illustrate both the reduction in number of unknowns and the effect of the numerically constrained bases on the conditioning of the converted matrix. Finally, a surface integral equation derived from Laplace’s equation is discretized using the locally-corrected Nyström method in order to calculate the electric fields created by impressed-current corrosion protection systems. An iterative technique is presented for handling nonlinear boundary conditions. In addition we examine different approaches for calculating the magnetic field radiated by the corrosion protection system. Numerical tests show the accuracy achievable by higher-order discretizations, validate the iterative technique presented. Various methods for magnetic field calculation are also applied to basic test cases.