In the paradigm of Data-Driven Computational Mechanics [1], one seeks the solution of a PDE-constrained optimization problem that determines continuous primal fields (e.g., gradients and fluxes) that are closest, in an appropriate metric, to discrete fields drawn from a material data set. A promising strategy for solving such problems is based on alternating direction methods. In this approach, the optimality conditions derived from a Lagrangian that incorporates proximity measures together with balance and compatibility constraints are solved in a first step for a prescribed distribution of discrete fields, while in a second step the data are redistributed over the computational domain. These two steps are repeated until convergence is achieved. A key challenge in this class of algorithms is therefore the efficient and accurate discretization of the resulting optimality conditions. In this work, we compare two families of stable finite element approximations for these conditions in the context of a diffusion-reaction problem. On the one hand, we consider a family of approximations based on the Galerkin least-squares (GLS) method, recently introduced in [2]. On the other hand, we investigate a family of variational multiscale (VMS) formulations, considering both primal and dual variants, also introduced recently in [3]. In all cases, continuous equal-order interpolations are employed. The different methods are described in detail as well as its implementation in the open-source finite element platform Firedrake. Their approximation properties are assessed through a set of challenging two-dimensional numerical examples. REFERENCES [1] Kirchdoerfer T. and Ortiz, Data-driven computational mechanics, Computer Methods in Applied Mechanics and Engineering, 304:81–101, 2016. [2] Bazon P. B., Gebhardt C. G., Buscaglia G. C. and Ausas R. F., Numerical approximation of a PDE–constrained Optimization problem that appears in Data-Driven Computational Mechanics. arXiv, 2025 [3] Codina R., Ausas R. F., Bazon P. B. and Gebhardt C. G., A variational multiscale approach to PDE-constrained optimization problems arising in Data-Driven Computational Mechanics Mechanics. arXiv, 2025