Two robust virtual element methods for the Brinkman equations

In this paper, we present two stable virtual element methods for the Brinkman equations robust in the Stokes and Darcy limits. We use a pair of stable virtual elements for the velocity and pressure. Firstly, we define an H ( div ) -conforming velocity reconstruction operator for the velocity test function. By employing it in the discretizations of the zero-order term and the right-hand-side source term, we propose the first virtual element method on convex polygonal meshes. Secondly, we construct the enhanced virtual element space in place of the original virtual element space such that one can exactly compute the L 2 -projection of virtual function onto linear polynomial space. We make full use of this projection to introduce the second virtual element method on general polygonal meshes. The well-posedness and uniform energy-error estimates of each method are strictly established. Finally, numerical experiments are provided to validate the theoretical results and illustrate the good performance of our methods.