C 0 elements for generalized indefinite Maxwell equations

Abstract

In this paper we develop the C 0 finite element method for a generalized curlcurl-grad div indefinite Maxwell problem in a Lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the H r regularity for some r < 1. The ingredients of our method are that two ‘mass-lumping’ L 2 projectors are applied to curl and div operators in the problem and that C 0 linear element or isoparametric bilinear element enriched with one element-bubble on every triangle element or with two-element-bubbles on every quadrilateral element, respectively, is employed for each component of the nonsmooth solution. Due to the fact that the element-bubbles can be statically eliminated at element levels, our method is essentially three-nodes or four-nodes C 0 Lagrange element method. With two Fortin-type interpolations established, a very technical duality argument is elaborated to estimate the error for the indefinite problem. For the nonsmooth solution having the H r regularity where r may vary in the interval [0, 1), we obtain the error bound $${{\mathcal O}(h^r)}$$ in an energy norm. Some numerical experiments are performed to confirm the theoretical error bounds.