STABILITY, CONVERGENCE AND ACCURACY OF STABILIZED FINITE ELEMENT METHODS FOR THE WAVE EQUATION IN MIXED FORM SANTIAGO BADIA †‡ , RAMON CODINA † , AND HECTOR ESPINOZA † Abstract. In this paper we propose two stabilized finite element methods for different functional frameworks of the wave equation in mixed form. These stabilized finite element methods are stable for any pair of interpolation spaces of the unknowns. The variational forms corresponding to dif- ferent functional settings are treated in an unified manner through the introduction of length scales related to the unknowns. Stability and convergence analysis is performed together with numerical experiments. It is shown that modifying the length scales allows one to mimic at the discrete level the different functional settings of the continuous problem and influence the stability and accuracy of the resulting methods. Key words. Wave equation, stabilized finite element methods, variational multiscale method, orthogonal subgrid scales, convergence, accuracy, stability AMS subject classifications. 65M60, 65M12, 65M15, 35M13, 35L04, 35F16 1. Introduction. When applied to approximate differential equations with sev- eral unknowns, and particularly saddle point problems, standard Galerkin mixed finite element (FE) formulations often require the use of inf-sup stable interpolations for the unknowns in order to be stable [10]. Inf-sup stable FE formulations have been formulated for several mixed problems, e.g. [2] for the Stokes problem, [11] for the Darcy problem, [26] for the Maxwell problem, [1, 25] for the Stokes-Darcy problem, [8] for the wave equation and [9, 24] for elastodynamics. On the contrary, stabilized FE methods [19] allow one to avoid inf-sup compat- ibility constraints. As a result, we can deal with different saddle-point problems by using the same equal interpolation for all the unknowns; see e.g. the unified frame- work for Stokes, Darcy and Maxwell problems in [7]. This way, we can certainly ease implementation issues, specially for multiphysics simulations. Stabilized FE methods can nicely be motivated in the Variational Multi-Scale (VMS) framework, as shown in [20]. This work is a follow-up of [15] for the wave equation in mixed form. The mixed wave equation is approximated in [15] using the Orthogonal Sub-scale Stabilization (OSS) method. In the present work, the OSS method is extended and the Algebraic Sub-Grid Scale Method (ASGS) is also considered. Additionally, length scales as- sociated to the unknowns are introduced, allowing one to treat different functional settings in a unified manner. A similar approach for the stationary Stokes-Darcy problem can be found in [5]. We focus on three variational forms of the mixed wave equation and the functional setting for each case. We obtain the different functional settings by transferring regularity from the scalar to the vector unknowns or vice-versa. More about functional settings for wave propagation problems of first and second order can be found in [22]. A priori error estimates for the mixed wave equation can be found in the literature. Some only bound L 2 norms of the error of the unknowns [18, 21], whereas others, such as [8], take into account the divergence of the vector unknown too. In this work we † Universitat Polit` ecnica de Catalunya, Jordi Girona 1-3, Edifici C1, E-08034 Barcelona ‡ Centre Internacional de M` etodes Num` erics en Enginyeria, Parc Mediterrani de la Tecnologia, Esteve Terradas 5, E-08860 Castelldefels, Spain 1