Enriched P 1-Conforming Methods for Elliptic Interface Problems with Implicit Jump Conditions

We develop a numerical method for elliptic interface problems with implicit jumps. To handle the discontinuity, we enrich usual P1-conforming finite element space by adding extra degrees of freedom on one side of the interface. Next, we define a new bilinear form,which incorporates the implicit jump conditions.We show that the bilinear form is coercive and bounded if the penalty term is sufficiently large.We prove the optimal error estimates in both energy-like norm and L2-norm.We provide numerical experiments. We observe that our scheme converges with optimal rates, which coincides with our error analysis.

Firstly, partial differential equations may have different coefficients along the interface due to the change of material properties.When the geometry of interface is complex, one needs to generate grids that align with the interfaces.Once a fitted grid is generated, one uses finite element method (FEM) or finite volume method (FVM) based on this grid.Secondly, the problem may have nonhomogeneous jump conditions along the interface.When the jumps along the interface are known explicitly, (say [] =  1 , [/n] =  2 , with known  1 and  2 ), these jumps may be handled effectively by discontinuous Galerkin (DG) [10,11] by incorporating jumps into the bilinear form with proper penalty terms.For example, an effective DG scheme was developed to describe discontinuous phenomena arising from porous media with discontinuous capillary pressure [12].The interface problems with known jumps can be solved with immersed interface methods [13][14][15] or discontinuous bubble-immersed finite element methods [16].
However, when the jumps are implicit along the interface problems, numerically solving the governing equations becomes more challenging.Let us consider some problems with interface conditions, where the jumps of primary variables are related to the normal fluxes.Firstly, these problems arise in the medical imaging of cancer cells using MREIT [3,4] or electrochemotherapy [17], where the jumps of an electric voltage across the cell membrane appear.Next, an elastic body has spring-type jumps that are related to stress [18,19].The heat in the material interface may have implicit jump conditions along the interface [20,21].Also, a generalized jump condition for Laplace equation or Helmholtz equation has been considered in [22][23][24].
The first attempt to solve the elliptic interface problems having implicit jump conditions seems to be introduced in [25], where the iterative method was used.Recently, some XFEM-based nonfitted methods were developed in [26,27] for the elliptic problems and elasticity problems, respectively, where the extra degrees of freedom are introduced on elements cut by the interface.On the other hand, an immersed finite element type method was developed in [28].
In this work, we introduce a new numerical method to solve elliptic interface problems, where the jumps are related to the normal fluxes and some known functions.A main idea of our work is to include the jump conditions implicitly on the bilinear form so that the numerical solutions for the weak problems satisfy the implicit jump conditions.We enrich the usual  1 FEM space near the interface.We show that our 2 Advances in Mathematical Physics bilinear form is coercive and bounded and prove the optimal error estimates.In numerical section, we provide several numerical examples supporting our analysis.
Let Ω be a convex domain in R  ( = 2,3), which is divided into Ω + and Ω − by a  2 closed interface Γ.The governing equations on Ω are given by where  ∈  2 (Ω) and  0 ∈  3/2 (Ω),  1 ∈  3/2 (Γ),  2 ∈  1/2 (Γ), and  is a positive piecewise constant; that is,  =  + in Ω + and  =  − in Ω − , where  + and  − are some positive constants.Here, n  is the outer unit normal vector to Ω  ( = −, +) and [⋅] Γ is the jump along the interface; that is, [] Γ = | Ω − − | Ω + .Also, we define n Γ to be an outer normal vector to Ω − .The jump of normal derivatives of  is defined as We assume that  is a positive constant.We introduce some notations.Let  be any domain and let   (),  = 1, 2, be a usual Sobolev space with norm ‖‖ , .We define  1 0 () as the set of functions in  1 () with vanishing trace on .We define the subspaces of   (), equipped with the (semi)norms: Finally, we define subspace of H1 (Ω): We state a theorem regarding the existence and regularity of the problem [29,30].
The rest of the paper is organized as follows.In Section 2, we derive the variational forms for the problems with implicit jump conditions.We introduce new numerical methods in Section 3 and in Section 4 we prove the error estimates.In Section 5, we give numerical results that support our analysis.The conclusion follows in Section 6.

Numerical Methods
In this section, we develop a numerical method for (1)-( 4).Our method is obtained by adding extra degrees of freedom to  1 -conforming space on one side of the interface.For simplicity, we assume that Ω ⊂ R 2 .However, similar constructions are possible for the case of Ω ⊂ R 3 as well.
Let T ℎ be a given regular triangulation of Ω fitted with the interface.We let T + ℎ and T − ℎ be set of elements in T ℎ which belong to Ω + and Ω − , respectively.We let  ℎ (Ω) be the usual  1 -conforming space; that is, any function in  ℎ (Ω) is continuous and piecewise linear and is vanishing on the boundary.We use notation  ℎ () for the set of linear functions on .
We let I ℎ be the set of all neighboring elements of interface Γ in T + ℎ ; that is,  ∈ T ℎ belongs to I ℎ if and only if  ∈ T + ℎ and at least one node of  is located on Γ.We let  ℎ () be the space of functions in  ℎ () vanishing on nodes not lying at the interface.For example, suppose that  has three nodes  1 ,  2 , and  3 , where  1 and  2 are located on Γ.Then, a function in  ℎ () is linear on  vanishing at  3 .In this case,  ℎ () has dimension two.On the other hand, if  have only one node located on Γ, the dimension of  ℎ () is one.A function in  ℎ () is extended to Ω as follows: For example, suppose that there are seven elements aligning with interface (see Figure 1).Then function  ℎ in  ℎ (Ω) has a support on grey region.Moreover,  ℎ has vanishing values on outside nodes on Ω + .Thus,  ℎ has seven degrees of freedom, that is, , , , , , , , ℎ.
(i) E  ℎ is the set of edges of E ℎ whose two endpoints are located on Γ.
(ii) E  ℎ is the set of edges of E ℎ whose one endpoint is located on Γ and the other is located in the interior We note that For all edges  in E ℎ , we fix a normal vector n  once and for all.We define jumps and averages across the edges: where  1  and  2  are two neighboring elements of .We multiply V ℎ ∈ Sℎ (Ω) to (1) and use integration by parts to obtain the following: where Let us classify   into three categories.Firstly, if  ∈ E  ℎ , then using the similar method used in deriving ( 12), we have Secondly, if  ∈ E  ℎ , then by using the identity and the fact that we have Finally, if  ∈ E  ℎ ,   vanishes, since both ∇ and V ℎ are continuous across .Thus, (25) becomes Now we propose our method based on enriched  1conforming space: find  ℎ ∈ Sℎ (Ω) satisfying for all V ℎ ∈ Sℎ (Ω), where In (33), the parameter  is positive and the parameter  is 1, 0, or −1, which is motivated by NIPG, IIPG, and SIPG of DG scheme [11].We show that our scheme is consistent.

Lemma 3 (trace theorem).
There exists a constant  such that for all V ∈  1 ().

Corollary 4.
There exists a constant   such that, for all  ∈ H2 (Ω), Proof.Since  −  ℎ is continuous across  ∈ E  ℎ , we have It suffices to show that for some constant  > 0. Let  =  −  ℎ .By the trace inequality (41), we have We have the following coercivity property.
By a similar technique, we can show that  ℎ is bounded.Theorem 6.There exists a constant   > 0 such that following holds: for all  ℎ , V ℎ ∈ Sℎ (Ω).
Finally, we prove the error estimate in the energy-like norm.
Proof.We define an auxiliary problem.Let  ∈ H2 (Ω) be solution of [] Γ = −  + n + , on Γ, (60) where  ℎ fl  −  ℎ ∈  2 (Ω).We multiply  ℎ to (59) and we use integration by parts to have We use similar techniques in the classification of   of (25) to derive Combining with (63) and (64) and the fact that  is continuous on each subdomain Ω − and Ω + , we have By definition of  ℎ and the fact that   ℎ (⋅, ⋅) is symmetric and by (36), (54), (42), and (9), we have Thus, we have the conclusion.

Numerical Results
In this section, we provide some numerical experiments of elliptic interface problems with implicit jump conditions.We consider circle-and ellipse-type interface shapes.
We let the domain Ω = [−1, 1] 2 and we let T ℎ be a triangulation of Ω by regular triangles, which aligns with interfaces.We set  = −1 in bilinear form (33).We take  in (33) as a multiple of .We report the number of elements, degrees of freedom,  2 -errors, and  1 -errors for ℎ  = 2 − ,  = 1, 2, . .., in Tables 1 and 2. For both examples, we observe optimal error convergence, which supports our analysis in Section 4.

Conclusion
In this work, we introduce a numerical method for elliptic interface problems, where the jumps are related to the normal fluxes.We enrich usual  1 space by extra degrees of freedom on one side of the interface.We define bilinear form that includes the jump conditions implicitly.We prove that the bilinear form is coercive and bounded.Using Cea's Lemma, we prove the error estimates in energy-like norm.Next, we prove  2 error estimate using the duality arguments.We provide numerical experiments that support our analysis.

Figure 1 :
Figure 1: The support and the degrees of freedom of a function  ℎ in  ℎ (Ω).