Abstract. The time-harmonic Maxwell equations are considered in the lowfrequency case. A finite element domain decomposition approach is proposed for the numerical approximation of the exact solution. This leads to an iteration-by-subdomain procedure, which is proven to converge. The rate of convergence turns out to be independent of the mesh size, showing that the preconditioner implicitly defined by the iterative procedure is optimal. For obtaining this convergence result it has been necessary to prove a regularity theorem for Dirichlet and Neumann harmonic fields.

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We present two a-posteriori error estimators for elliptic partial differential equations, when we use a mixed method. One is based on a adequate evaluation of the residual of the finite element solution and the other on the solution of a local problem. We prove their equivalence with the norm of the error, when the data is locally smooth.

This paper deals with the nonconforming spectral approximation of variationally posed eigenvalue problems. It is an extension to more general situations of known previous results about nonconforming methods. As an application of the present theory, convergence and optimal order error estimates are proved for the lowest order Crouzeix-Raviart approximation of the eigenpairs of two representative second-order elliptical operators.

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