An a Posteriori Error Estimator for a New Stabilized Formulation of the Brinkman Problem Tomás Barrios, Rommel Bustinza, Galina C. García, and María González Abstract We present in this work an a posteriori error estimator for a porous media flow problem that follows the Brinkman model. First, we introduce the pseudostress as an auxiliary unknown, which let us to eliminate the pressure and thus derive a dual-mixed formulation in velocity-pseudostress. Next, in order to circumvent an inf-sup condition for the unique solvability, we stabilize the scheme by adding some appropriate least squares terms. The existence and uniqueness of solution are guaranteed and we derive an a posteriori error estimator based on the Ritz projection of the error, which is reliable and efficient up to high order terms. Finally, we report one numerical example confirming the good properties of the estimator. 1 Introduction This note deals with the numerical approximation of the velocity and pressure of a porous media flow problem defined on a bounded and simply connected domain ˝ in R 2 , with polygonal boundary  WD @˝. Indeed, this boundary value problem corresponds to the well-known Brinkman model and reads as follows: Given the T. Barrios Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile e-mail: tomas@ucsc.cl R. Bustinza () CI 2 MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile e-mail: rbustinz@ing-mat.udec.cl G.C. García Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile e-mail: galina.garcia@usach.cl M. González Universidade da Coruña, Campus de Elviña s/n, 15071, A Coruña, Spain e-mail: mgtaboad@udc.es © Springer International Publishing Switzerland 2015 A. Abdulle et al. (eds.), Numerical Mathematics and Advanced Applications - ENUMATH 2013, Lecture Notes in Computational Science and Engineering 103, DOI 10.1007/978-3-319-10705-9__25 253