Computation of Maxwell singular solution by nodal-continuous elements

In this paper, we propose and analyze a nodal-continuous and H1-conforming finite element method for the numerical computation of Maxwell's equations, with singular solution in a fractional order Sobolev space Hr(Ω), where r may take any value in the most interesting interval (0,1). The key feature of the method is that mass-lumping linear finite element L2 projections act on the curl and divergence partial differential operators so that the singular solution can be sought in a setting of L2(Ω) space. We shall use the nodal-continuous linear finite elements, enriched with one element bubble in each element, to approximate the singular and non-H1 solution. Discontinuous and nonhomogeneous media are allowed in the method. Some error estimates are given and a number of numerical experiments for source problems as well as eigenvalue problems are presented to illustrate the superior performance of the proposed method.