The Weak Galerkin Finite Element Method For Partial Differential Equations With Interface

In this paper,we consider two kinds of partial differential equations with interfaces,namely Stokes-Darcy model and linear elasticity interface model.In the Stokes-Darcy model,the equations in subdomain divided by interface are different,the equations ex-change information on the interface.The research of this problem focuses on the coupling between different equations.Therefore,when we design the numerical method for this coupled model,two points will be involved.On the one hand,we concern with the appro-priate methods to separate equations,on the other hand,we also need to pay attention to the computational efficiency of the whole system.For the linear elasticity interface model,the equations in both sides of the interface are linear elasticity equations.However,the coefficients,displacement,tensor and stress may appear jumps while crossing the interface,which will bring some obstacles in designing an efficient numerical scheme.For example,the discontinuity results in low global regularity,then the numerical methods based on regularity assumption may not work.In addition,the geometric complexity of interface sometimes increases demands on the flexibility of the mesh.Finally,for the linear elastic-ity problem,when the material is nearly incompressible(one of Lam'e parameters tends to infinite),the numerical solution may show poor convergence rate,which is called locking phenomenon.In this paper,we will design the numerical methods for Stokes-Darcy model and linear elasticity interface model based on the weak Galerkin finite element methods.The features mentioned above of these two problems will be fully considered.The weak Galerkin(WG)method is a new numerical method developed in recent years,which is an extension of the standard finite element method.The key idea to the WG method is that the weak function is introduced as the approximation function,and the corresponding weak differential operators are defined for the weak function.The weak function has the form{v₀,v_b}.For each cell,v₀denotes the interior value in cell,and v_bdenotes the boundary value on the boundary.We emphasize that v_bis not necessarily related to the trace of v₀.Therefore,we can choose different polynomial combinations of v₀and v_bin practical calculations.Later on,in order to ensure the uniqueness of numerical solutions,a stabilizer is introduced to the numerical formulation.Since the WG elements are constructed element by element,this method allows arbitrary shape of polygon mesh in two dimension or polyhedra mesh in three dimension.As we can see,the WG method has high flexibility,which is convenient to deal with the models with interface.The Stokes-Darcy model describes the fluid flow coupled with porous media flow,the fluid flow is governed by Stokes equations and the flow in porous media is governed by the Darcy equations.Three interface conditions are imposed on the interface,namely mass conservation,balance of force and the Beavers-Joseph-Saffman condition.The Darcy equations have two different forms,i.e.the primal form and the mixed form.We first consider the standard Stokes equations coupled with the Darcy equations in primal form.The WG method shows some advantages for Stokes equation.First,it is straight forward to design the WG scheme for Stokes equations and and there is no need to choose a large parameter to ensure the stability of the scheme.Secondly,the low-order WG finite elements pairs is available to approximate the velocity and pressure.Thirdly,the mesh generation is flexible.As for the Darcy equation in primal form,which is a second order elliptic formulation,we use the finite element method to discrete such Darcy equations.It is not trival to achieved these advantages by simply combining different dis-crete formulations because the formulations should be consistent on the interface.Thanks to the flexibility of the WG method,we choose the local WG elements in the Stokes region matching the global continuous finite elements in the Darcy region.In addition,since the WG method allows the use of meshes with hanging points,the meshes on both sides of the interface do not have to be aligned.In the analysis of the numerical scheme,we prove the Korn inequality for weak finite element to deal with the stress tensor involved in S-tokes equations.Furthermore,we derive the error estimates and present some numerical experiments to demonstrate the theoretical analysis.Next,we consider the Stokes equations coupled with the Darcy equations in mixed form.This coupled model involved the velocity and pressure in fluid region and medi-um flow region simultaneously.Thus,it is natural to describe the interface conditions compared to the primal situation.The stable elements for Stokes equations and Darcy e-quations in mixed form are always different,although the WG elements are stable for both the Stokes equations and Darcy equations,we still employs different elements in different region to reduce the number of degree of freedom.We still use the WG method to discrete the Stokes equations and use the mixed finite element method to discrete Darcy equations,some classical mixed elements can be chosen,such as RT element,BDM element,BDFM element and so on.One of the difficulties is the proof of the inf-sup condition in the whole region.We construct two projection operators in Stokes region and Darcy region to overcome the difficulty.We also give the error estimates for velocity and pressure in two regions.Some numerical examples illustrate the efficiency of our numerical approach.The second model with interface investigated in this paper is the linear elasticity in-terface model.Linear elasticity equations often occur in modeling solid mechanics.Such equations describe the relationship between displacement,stress and strain of an object under external force.Usually,the elastic body is made of multiple materials,the material properties and physical properties change across the interface,then the coefficients,dis-placement,tensor and stress in equations may appear jumps on the interface,which leads to the linear elasticity interface problem.In this paper,we use the WG method to discrete the linear elasticity interface model.Since the solution may become discontinuous while crossing the interface,we first discretize the model by double-valued weak functions on the interface.Then,in order to facilitate theoretical analysis and algorithm implementation,we substitute interface conditions into the WG formulation and establish a WG scheme with single-valued functions on the interface.Furthermore,we give the the error estimates for displacement in energy norm and L²norm,which also implies the locking-free proper-ty of the WG method.Finally,we present some numerical examples to demonstrate the efficiency and locking-free of the WG method.