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This paper deals with a posteriori error estimators for the linear finite element approximation of second order elliptic eigenvalue problems in two or three dimensions. First, we give a simple proof of the equivalence, up to higher order terms, between the error and a residual type error estimator. Second, we prove that the volumetric part of the residual is dominated by a constant times the edge or face residuals, again up to higher order terms. This result was not known for eigenvalue problems.

In this work we use the Topological-Shape Sensitivity Method to obtain the topological derivative for three-dimensional linear elasticity problems, adopting the total potential energy as the cost function and the equilibrium equation as the constraint. This method, based on classical shape sensitivity analysis, leads to a systematic procedure to compute the topological derivative. In particular, firstly we present the mechanical model, later we perform the shape derivative of the corresponding cost function and, finally, we compute the final expression for the topological derivative using the Topological-Shape Sensitivity Method and results from classical asymptotic analysis around spherical cavities.

In this paper we introduce and analyze an a posteriori error estimator for the approximation of the eigenvalues and eigenvectors of a second-order elliptic problem obtained by the mixed finite element method of Raviart-Thomas of the lowest order. We define an error estimator of the residual type which can be computed locally from the approximate eigenvector and prove that the estimator is equivalent to the norm of the error in the approximation of the eigenvector up to higher order terms. The constants involved in this equivalence depend on the corresponding eigenvalue but are independent of the mesh size, provided the meshes satisfy the usual minimum angle condition. Moreover, the square root of the error in the approximation of the eigenvalue is also bounded by a constant times the estimator.