The problem of absorption and scattering of electromagnetic radiation by particles can be solved analytically for only the simplest cases, but established numerical methods can allow a straightforward extension to particles with arbitrary inhomogeneities, shapes, and nonlinear response. In this paper a recently developed finite element technique for the scattering and absorption by a homogeneous sphere is extended to the problem of a coated dielectric sphere of arbitrary size parameter. Numerical results showing good agreement with analytical solutions for the size parameter of 5.93 and various core radii are presented. Results obtained suggest that finite element methods have promise for analytically intractable scattering/absorption problems and show that the Debye amplitude formulation of the problem offers some advantages in a numerical scheme.