Approximation of singular solutions and singular data for Maxwell’s equations by Lagrange elements

Abstract

Abstract A Lagrange finite element method is proposed for Maxwell’s equations in Lipschitz domains. The method is suitable for the approximation of singular solutions lying outside $(H^1(\varOmega ))^3$, with nonhomogeneous singular boundary data in the tangential trace space of $H({\mathbf{curl}}; \varOmega )$ and a singular right-hand-side source term in $(H_0({\mathbf{curl}}; \varOmega ))^{\prime}$ (the dual space of $H_0({\mathbf{curl}}; \varOmega ))$. Numerical results are presented to illustrate performance and the theoretical results.