Stream function -- pressure virtual element methods for the Stokes--Darcy interface problem

Abstract

This paper introduces a novel Virtual Element Method (VEM) for the coupled Stokes--Darcy system in primal-primal form. In the free-flow Stokes domain, we implement a stream function formulation that inherently satisfies the incompressibility constraint and reduces computational cost. Across the interface, mass conservation, normal stress balance, and the Beavers--Joseph--Saffman slip condition are enforced to couple the biharmonic stream function equation with the Darcy's pressure equation. Leveraging VEM's ability to handle general polygonal meshes, the proposed method naturally accommodates irregular interface geometries without requiring remeshing or adaptive refinement. The accuracy of the method is validated through several numerical simulations that include applications to dead-end filtration, and network flow in bioartificial organs.