A stabilized mixed finite element method for the first-order form of advection-diffusion equation

This paper presents a stabilized mixed finite element method for the first-order form of advection-diffusion equation. The new method is based on an additive split of the flux-field into coarse- and fine-scale components that systematically lead to coarse and fine-scale variational formulations. Solution of the fine-scale variational problem is mathematically embedded in the coarse-scale problem and this yields the resulting method. A key feature of the method is that the characteristic length scale of the mesh does not appear explicitly in the definition of the stability parameter that emerges via the solution of the fine-scale problem. The new method yields a family of equal- and unequal-order elements that show stable response on structured and unstructured meshes for a variety of benchmark problems.