Nitsche's Type Stabilized Finite Element Method For The Fully Mixed Stokes-darcy System

Multi-domain,multi-physics problems exist widely in practical engineering applications,and are one of the current research hotspots in computational mathematics.Stokes-Darcy model can simulate the coupling system of surface water and groundwater,which is a typical multi-domain,multi-physics problem.This paper studies the fully-mixed Stokes-Darcy problem with more physical significance Beavers-Joseph(BJ)interface conditions.Both the Stokes and Darcy equations contain two unknown quantities of velocity and pressure(or hydraulic head),so that we can use different mixed finite element methods for simulation.Since it is difficult to prove the well-posedness of the fully-mixed Stokes-Darcy problem with BJ interface conditions,we introduce new Sobolev spaces as well as an adjusted weak formulation to demonstrate the equivalency between the adjusted and the classical weak formulation.Moreover,this paper employs a velocity subspace subtly,which verifies the well-posedness of the adjusted weak form.Based on the improved weak formulation,we construct a Nitsche's type stabilized mixed finite element method for the Stokes-Darcy problem by introducing two stability terms with strongly consistent interfaces.We don't require any additional unknown variables because none of the Lagrange multiplier terms are incorporated in our method.Due to its continuity and weak coercivity,our proposal ensures the stability of the full mixed format and has the optimal error estimates.In order to improve the computational efficiency of numerical simulations,we design a Nitsche's decoupling numerical algorithm.Ample numerical modelings of application scenarios,such as vascular flow and petroleum extraction,further validate the effectiveness of the Nitsche's type stabilized mixed finite element method.