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This work presents a priori and a posteriori error analyses of a new multiscale hybridmixed method (MHM) for an elliptic model. Specially designed to incorporate multiple scales into the construction of basis functions, this finite element method relaxes the continuity of the primal variable through the action of Lagrange multipliers, while assuring the strong continuity of the normal component of the flux (dual variable). As a result, the dual variable, which stems from a simple postprocessing of the primal variable, preserves local conservation. We prove existence and uniqueness of a solution for the MHM method as well as optimal convergence estimates of any order in the natural norms. Also, we propose a face-residual a posteriori error estimator, and prove that it controls the error of both variables in the natural norms. Several numerical tests assess the theoretical results.

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Abstract. In this paper we propose a novel way, via finite elements to treat problems that can be singular perturbed, a reaction-diffusion equation in our case. We enrich the usual piecewise linear or bilinear finite element trial spaces with local solutions of the original problem, as in the Residual Free Bubble (RFB) setting, but do not require these functions to vanish on each element edge, a departure from the RFB paradigm. Such multiscale functions have an analytic expression, for triangles and rectangles. Bubbles are the choice for the test functions allowing static condensation, thus our method is of Petrov-Galerkin type. We perform several numerical validations which confirm the good performance of the method.

This work concerns the development of stabilized finite element methods for the Stokes problem considering nonstable different (or equal) order of velocity and pressure interpolations. The approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary. On the other hand, since the test function space is enriched with bubble-like functions, a Petrov-Galerkin approach is employed. We use such a strategy to propose stable variational formulations for continuous piecewise linear in velocity and pressure and for piecewise linear/piecewise constant interpolation pairs. Optimal order convergence results are derived and numerical tests validate the proposed methods. Introduction.Finite element solution of the Stokes problem poses the basic problem of satisfying the discrete Babuska-Brezzi (or inf-sup) condition (see [24] and the references therein). This is indeed a restriction from the point of view of implementation since equal order velocity and pressure spaces do not satisfy this condition. On the other hand, the minimal space to imagine, namely continuous piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure, does not satisfy this condition either.Several solutions have been proposed to overcome this restriction, starting with that in [11] and the first consistent method in [28]. Moreover,in [23,27,29,34] the possibility of considering discontinuous spaces for the pressure was considered and justified. On the other hand, in [14,13], the idea from [16] has been used to propose a new kind of stabilized finite element methods, with stabilizing terms now containing only jump terms across the interelement boundaries. For an overview of stabilized finite element methods for the Stokes problem, see [19] and [5].On the other hand, the theoretical justification of stabilized methods has become a subject of interest in the last decade. In [2,3,4,31], the connection between stabilized finite element methods and Galerkin methods enriched with bubble functions has been used to propose new stabilized finite element methods for Stokes-like and linearized Navier-Stokes problems. Also, in [22] macro bubbles were used to derive a method analogous to the locally stabilized method from [29] containing jump terms across the interelement boundaries of the macroelements. In the resulting method, the stabilizing

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This work presents a family of stable finite element methods for two-and threedimensional linear elasticity models. The weak form posed on the skeleton of the partition is a byproduct of the primal hybridization of the elasticity problem. The unknowns are the piecewise rigid body modes and the Lagrange multipliers used to relax the continuity of displacements. They characterize the exact displacement through a direct sum of rigid body modes and solutions to local elasticity problems with Neumann boundary conditions driven by the multipliers. The local problems define basis functions which are in a one-to-one correspondence with the basis of the subspace of Lagrange multipliers used to discretize the problem. Under the assumption that such a basis is available exactly, we prove that the underlying method is well posed, and the stress and the displacement are super-convergent in natural norms driven by (high-order) interpolating multipliers. Also, a local post-processing computation yields strongly symmetric stress which is in local equilibrium and possesses continuous traction on faces. A face-based a posteriori estimator is shown to be locally efficient and reliable with respect to the natural norms of the error.Next, we propose a second level of discretization to approximate the basis functions. A two-level numerical analysis establishes sufficient conditions under which the well-posedness and super-convergent properties of the one-level method is preserved.