Hyperthermia has been shown to improve local tumour control of superficial and deep seated lesions when combined with radiotherapy. There remains difficulty in heating larger tumours with conventional applicators, but this is being addressed by several new applicator designs. This paper presents a new numerical model of microwave heating which is designed to aid in the development of new applicators for superficial heating. The model is based on a finite element method which utilises vector valued basis functions instead of the more conventional scalar valued basis functions. These basis functions were chosen since they are inherently suited for the solution of Maxwell's equations due to their vector nature. The model was successfully verified against an analytic solution to the Mie scattering problem as well as against previously published measurements of heating from a modified water bolus attached to a conventional waveguide applicator. An accompanying paper describes an application of this model to the design optimization of this modified bolus.