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A FINITE ELEMENT METHOD FOR HIGH-CONTRAST INTERFACE

PROBLEMS WITH ERROR ESTIMATES INDEPENDENT OF CONTRAST

JOHNNY GUZMAN†, MANUEL A. SANCHEZ†, AND MARCUS SARKIS‡

Abstract. We define a new finite element method for a steady state elliptic problem withdiscontinuous diffusion coefficients where the meshes are not aligned with the interface. Weprove optimal error estimates in the L2 norm and H1 weighted semi-norm independent of thecontrast between the coefficients. Numerical experiments validating our theoretical findings areprovided.

1. Introduction

In this article we develop a finite element method for a steady state interface problem. We payparticular attention to high contrast problems and will prove optimal error estimates independentof contrast for the numerical method.

Let Ω ⊂ R2 be a polygonal domain with an immersed smooth interface Γ such that; Ω =

Ω− ∪ Ω

+with Ω− ∩ Ω+ = ∅, and Γ = Ω

− ∩ Ω+

. We assume that Γ does not intersect ∂Ωand that Γ encloses either Ω− or Ω+. Our numerical method will approximate a solution of theproblem below.

−ρ±∆u± = f± in Ω±,(1.1a)

u = 0 on ∂Ω,(1.1b)

[u] = 0 on Γ,(1.1c)

[ρDnu] = 0 on Γ.(1.1d)

The jumps across the interface Γ are defined as

[ρDnu] = ρ−Dn−u− + ρ+Dn+u+ = ρ−∇u− · n− + ρ+∇u+ · n+,

[u] = u+ − u−,

where u± ≡ u|Ω± and n± is the unit outward pointing normal to Ω±. We furthermore assumethat ρ+ ≥ ρ− > 0 are constant and that the interface Γ is C2 curve.

There has been a recent surge in the development of finite element methods for interfaceproblems; see for instance [17, 20, 3, 7, 6, 21, 5, 14, 13, 16, 11, 18, 2, 19, 1, 8], to name afew. Although many of the methods focus on low contrast problems, there are several methodsaddressing high contrast problems; see for example [7, 8]. In particular, in the work of Chu etal. [8] they develop a multi-scale method and prove error estimates independent of contrast.Their method was designed to deal with interfaces with radius of curvature of the same orderas the size of mesh, however, their results do not hold for simple interfaces such as a straightline. For our method, we prove error estimates independent of contrast for any interface that issufficiently smooth.

2000 Mathematics Subject Classification. 65N30, 65N15.Key words and phrases. Interface problems, finite elements, pointwise estimates.

1

2 JOHNNY GUZMAN†, MANUEL A. SANCHEZ†, AND MARCUS SARKIS‡

In order to put our contribution in context, let us explain more carefully previous finiteelement methods for the above problem. There are two popular approaches for the problemabove. The first approach is to double the degrees of freedom on triangles that intersect theinterface and then add penalty terms to weakly enforce the continuity across the interface; seefor example [7]. Burman and Zunino [7] demonstrated that in addition to penalizing the jumpsacross Γ it is necessary to add a flux stabilization term. This method is the so-called stabilizedunfitted Nitsche method. The stabilization term penalizes the jumps of the gradient on edgesthat belong to triangles that intersect the triangulation. As Burman and Zunino [7] showed,in order to get a method that is robust with respect to diffusion contrast and robust withrespect to the way Γ cuts triangles, this type of penalization is necessary. The second commonapproach, and the one we focus in this paper, is to define local piecewise polynomial finiteelement spaces on triangles that intersect the interface Γ (see for instance [1, 12, 11, 8, 16, 15]).The basis functions are constructed by having them satisfy the continuity of the function and thecontinuity of the flux at certain points across Γ. Unlike the unfitted Nitsche method, at certainpoints on Γ the flux conservation and the continuity of the solution are enforced strongly withoutrequiring stabilization terms on Γ and this an important feature for the Immersed Interface andImmersed Boundary Methods. These basis function are defined locally on each triangle, so theyare naturally not continuous across edges of the triangulation. Namely, Adjerid et al. in [1]proposed to penalize jumps of the trial functions across the edges. Similarly, Tao Lin et al. in[17] add similar penalty terms and prove optimal error estimates. However, in that paper theydo not consider high contrast problems.

In this paper we follow this approach, defining local basis functions that are piecewise poly-nomials on triangles that are cut by Γ. However, an additional stabilization term is added ascompared to the methods of Adjerid et al. [1] and Tao Lin et al. [17], allowing us to proveestimates that are independent of the contrast. The stabilization term is penalizing the jumpsof the normal derivatives of the approximation across the edges that belong to triangles inter-secting Γ. This is the same idea used by Burman and Zunino [7], however here we use differentstabilization parameters (and of course different basis functions). Roughly speaking, the reasonthis flux stabilization is important for high contrast problems, is that one does not want to moveestimates from Ω+ to Ω− because ρ+ could be much larger than ρ−. However, triangles thatare cut by Γ might have a thin part in Ω+ and therefore inverse estimates might be affected.By adding the jumps of derivatives we can transfer the estimates to a neighboring triangle thatwill have a larger portion in Ω+.

In addition to proving error estimate independent of the contrast, we also prove optimalestimates only using H2 regularity on both Ω+ and Ω−. This is in contrast to the error estimatesby Chu et al. [8] and Tao Lin et al. [17] that use more regularity. To be more precise, the errorestimate we prove is

‖u− uh‖V ≤ C h√

ρ−(‖Du‖L2(Ω−) + ‖D2u‖L2(Ω−))(1.2)

+√ρ+(‖Du‖L2(Ω+) + ‖D2u‖L2(Ω+))

.

Assuming that Ω is convex and using regularity estimates (see [8]) we will have the result

(1.3) ‖u− uh‖V ≤ Ch√ρ−‖f‖L2(Ω).

Using a duality argument we can also prove the estimate ‖u− uh‖L2(Ω) ≤ C h2

ρ− ‖f‖L2(Ω).

An outline of this paper is as follows. We formulate the discrete problem as finding uh ∈ Vhsuch that ah(uh, vh) = (f, vh), ∀vh ∈ Vh. The discrete space Vh is introduced in section 2.1

FEM FOR INTERFACE 3

and the bilinear form ah(·, ·) in section 2.2. In section 3, fundamental results on element-wiseweighted L2 and H1 norm approximation for the space Vh are established. The coercivityc‖vh‖2V ≤ ah(vh, vh) for vh ∈ Vh is established in section 4.1, and the continuity ah(u, vh) ≤C‖u‖W ‖vh‖V for u ∈ H2

h(Ω±) + Vh and v ∈ Vh is studied in Section 4.2. We note that the useof the augmented norm W is necessary for the analysis due to the presence of the penalty termsinvolving flux jumps. In Section 5 the bound (1.2) is established by estimating the approximationerror in the W -norm and the consistency errors across Γ and across elements near Γ. In Section5.2 we prove error estimate in the L2 norm. In section 6 we present extensions to three dimensionsand discuss related methods. Finally, in Section 7 we provide numerical experiments.

2. The finite element method

2.1. Notation and local finite element space. In this section we present a finite elementmethod for problem (1.1) using piecewise linear polynomials. To prove optimal energy errorestimates if u± ∈ H2(Ω±) and f± ≡ f |Ω± ∈ L2(Ω±) and Γ is C2.

We next develop notation. Let Th, 0 < h < 1 be a sequence of triangulations of Ω, withΩ = ∪T∈ThT and the elements T are mutually disjoint. Let hT denote the diameter of theelement T and h = maxT hT . We let Eh be the set of all edges of the triangulation. We assumethat the mesh is shape-regular, see [4]. We adopt the convention that edges e, elements T ,sub-edges e± := e ∩ Ω±, sub-elements T± := T ∩ Ω± and sub-regions Ω± are open sets, andwe use the over-line symbol to refer to their closure.

Let T Γh denotes the set of triangles T ∈ Th such that T intersects Γ. We let EhΓ be the set

of all the three edges of triangles in T Γh . We define an element patch TE of a triangle T , its

restriction to Ω± and its intersection with Γ as

TE := int

⋃K∈Th

K : K ∩ T 6= ∅

, TE,± := TE ∩ Ω±, TEΓ := TE ∩ Γ.

The introduction of notation for the patch will be relevant in the proof of the interpolationerror further forward. We first need to build our local finite element space (on each element).To this end, for a triangle T ∈ T Γ

h we parameterize the curve TEΓ by its arc length and we let x0

be the midpoint on the curve TΓ := T ∩Γ. We note here that the midpoint choice is a preferenceof the authors, the proofs below hold for any x0 ∈ TΓ. Let LT ⊂ T be the line segment inside Twhich is tangent to Γ at x0. We denote t± as the unit tangent vector to Γ, such that the crossproduct of n± × t± = 1. Figure 1 illustrates the definitions and notations introduced above.

Figure 1. Illustrationof our notation on anelement T ∈ T Γ

h .

In order to define our finite element space, we will need the following lemma.


Lemma 1. Given v ∈ P1(T+) there exists a unique E(v) ∈ P1(T−) satisfying

E(v)(x0) := v(x0)(2.1)

(Dt+0E(v))(x0) := (Dt+0

v)(x0)(2.2)

ρ−(Dn+0E(v))(x0) := ρ+(Dn+

0v)(x0),(2.3)

where n±0 = n±(x0) and t±0 = t±(x0).

Note that the exact solution u± satisfies these transmission conditions on all points over Γ.Next, given T ∈ T Γ

h and for each v ∈ P1(T+) we can consider the unique correspondingfunction

G(v) =

v on T+,

E(v) on T−.

Let span v1, v2, v3 be a basis for P1(T ) restricted to T+. Then we define the local finiteelement space

(2.4) S1(T ) =

span

G(v1), G(v2), G(v3)

if T ∈ T Γ

h ,

P1(T ) if T ∈ Th\T Γh .

Consequently, the global finite element space is given by

Vh :=v : v|T ∈ S1(T ), ∀T ∈ Th, v is continuous across all edges in Eh\EhΓ

.

2.2. Finite element method. We begin this section introducing some standard discontinuousfinite element notation for jumps and averages.

For a piecewise smooth function v with support on Th, we define its average and jump acrossan interior edge e ∈ Eh\∂Ω, shared by elements T1 and T2, as

v

=v|T1 + v|T2

2, JvK = v|T1n1 + v|T2n2,

where n1 and n2 are the unit normal vectors to e pointing outwards T1 and T2, respectively.Similarly, if τ is a vector-valued function, piecewise smooth on Th, its average and jump across

an interior edge e are defined asτ

=τ |T1 + τ |T2

2, Jτ K = τ |T1 · n1 + τ |T2 · n2,

For a boundary edge e ⊆ ∂Ω we simply letv

= v|T1 , JvK = v|T1n1,τ

= τ |T1 , Jτ K = τ |T1 · n1.

Next we introduce the finite element approximation to problem (1.1). Find uh ∈ Vh such that

(2.5) ah(uh, v) = (f, v) for all v ∈ Vh,where

ah(w, v) :=

∫Ωρ∇hw · ∇hv −

∑e∈EhΓ

∫e

(ρ∇hv

· JwK +

ρ∇hw

· JvK

)(2.6)

+∑e∈EhΓ

(γ

|e−|

∫e−ρ−JwK · JvK +

γ

|e+|

∫e+ρ+JwK · JvK

)∑e∈EhΓ

(|e−|

∫e−ρ−J∇hvK J∇hwK + |e+|

∫e+ρ+J∇hvK J∇hwK

),

FEM FOR INTERFACE 5

Figure 2. Illustration of basis functions on triangle figure 1 with ρ+ = 10 andρ− = 1.

(f, v) :=∑T∈Th

(∫T−

f−v +

∫T+

f+v),

for γ > 0 and e± = e ∩ Ω±. Here we denote by ∇hv the functions whose restriction to eachT± with T ∈ Th is ∇v.

Note that we not only penalize the jumps of the function but also the normal jumps of thefirst derivatives across edges. This will allow us to prove that the bilinear form is coerciveindependent of the contrast of the coefficients and also independent of how small T+ or T−

might be. Finally, we like to stress that only the normal derivative jumps are penalized, not thetangential derivative jumps.

3. Local Approximation

In this section we show that our finite element space has optimal local approximation prop-erties for T ∈ T Γ

h .In order to define the interpolation operator onto Vh we first assume that u± ∈ H2(Ω±) have

extensions u±E ∈ H2(Ω) with the following properties

u±E = u± in Ω±,(3.1)

‖D`u±E‖L2(Ω) ≤ C‖D`u±‖L2(Ω±) for ` = 0, 1, 2.(3.2)

This is particularly true for simple smooth curves, Γ ∈ C2, that are away from the boundary∂Ω, see Lemma 6.37 [10].

Definition 3.1. Let u± ∈ H2(Ω±). For each T ∈ T Γh we define ITu ∈ S1(T ). In fact, we define

ITu on all TE

ITu =

I+T u on TE,+

I−T u on TE,−,

where I±T are defined satisfying the following conditions

(I−T u)(x0) := (JTu+E)(x0) =: (I+

T u)(x0)(3.3)

(Dt+0I−T u)(x0) := (Dt+0

(JTu+E))(x0) =: (Dt+0

I+T u)(x0)(3.4)

ρ−(Dn+0I−T u)(x0) := ρ−(Dn+

0JTu

−E)(x0) =: ρ+(Dn+

0I+T u)(x0).(3.5)

Here JT is the L2 projection operator onto P1(TE). If T does not belong to T Γh then we simply

define IT (u)|T to be the Scott-Zhang interpolation operator of u on T .


We define the interpolation operator Ih onto the finite element space Vh as the restriction ofthe local interpolation operator IT , i.e.

(3.6) Ihu|T = IT (u)|T for all T ∈ Th.

It is well known that the following approximation properties hold:

(3.7) hjT ‖Dj(w − JTw)‖L2(TE) ≤ C h2

T ‖D2w‖L2(TE) for j = 0, 1.

Lemma 2. For every v ∈ P1(TE) the following bound holds

hjT ‖Djv‖L2(TE) ≤ C

(hT |v(x0)|+ h2

T |Dn+0v(x0)|+ h2

T |Dt+0v(x0)|

).

Proof. Using the Cauchy-Schwarz inequality one has

hjT ‖Djv‖L2(TE) ≤ C hj+1

T ‖Djv‖L∞(TE).

Using the fact that v is a linear function, the lemma follows.

We next prove a fundamental result of this paper, a local approximation property on thespace S1(T ).

Lemma 3. Let u± ∈ H2(Ω±) satisfying the interface conditions [u] = 0 and [ρDnu] = 0 on a

C2 curve Γ = Ω− ∩ Ω

+. Then, for any T ∈ T Γ

h and j = 0, 1, the following bounds hold:

hjT ‖Dj(u− ITu)‖L2(TE,−) ≤ C h2

T

(‖Du−E‖L2(TE) + ‖D2u−E‖L2(TE)) + ‖D2u+

E‖L2(TE)

)(3.8)

and

hjT ‖Dj(u− ITu)‖L2(TE,+) ≤ C h2

T

(‖Du+

E‖L2(TE) + ‖D2u+E‖L2(TE)(3.9)

+ρ−

ρ+‖D2u−E‖L2(TE)

).

Proof. Adding and subtracting JTu±E to u±E− I

±T u, using the triangle inequality and the approx-

imation result (3.7), the proof of the lemma reduces to estimate

hjT ‖Djw±T ‖L2(TE,±), where w±T := JTu

±E − I

±T u, for w±T ∈ S

1(T ).

According to the definition of IT and JT , and denoting n±0 = n±(x0) and t±0 = t±(x0), wehave

w−T (x0) = JTu−E(x0)− JTu+

E(x0),

(Dt+0w−T )(x0) = (Dt+0

JTu−E)(x0)− (Dt+0

JTu+E)(x0),

(Dn+0w−T )(x0) = 0,

and

w+T (x0) = 0,

(Dt+0w+T )(x0) = 0,

ρ+(Dn+0w+T )(x0) = ρ+(Dn+

0JTu

+E)(x0)− ρ−(Dn+

0JTu

−E)(x0).

Then, using Lemma 2 we have

(3.10) hjT ‖Djw−T ‖L2(TE,−) ≤ C

(hT |w−T (x0)|+ h2

T |Dt+0w−T (x0)|

)

FEM FOR INTERFACE 7

and

(3.11) hjT ‖Djw+

T ‖L2(TE,+) ≤ Ch2T |Dn+

0w+T (x0)|.

We proceed by bounding the two terms in (3.10) and the term in (3.11), separately.For the first term in (3.10), we use that u is continuous on the interface (1.1c), in particular

on TEΓ , and we apply the triangle inequality

|w−T (x0)| = |(JTu−E − JTu+E)(x0)| ≤ ‖JTu−E − JTu

+E‖L∞(TEΓ )

≤ ‖JTu−E − u−E‖L∞(TEΓ ) + ‖JTu+

E − u+E‖L∞(TEΓ ).

By means of a Sobolev inequality in one dimension, we have

‖JTu−E − u−E‖L∞(TEΓ ) ≤ C

(1√hT‖JTu−E − u

−E‖L2(TEΓ ) +

√hT ‖D(JTu

−E − u

−E)‖L2(TEΓ )

),

consequently, using a trace inequality we obtain

‖JTu−E − u−E‖L∞(TEΓ ) ≤ C

(h−1T ‖JTu

−E − u

−E‖L2(TE) + ‖D(JTu

−E − u

−E)‖L2(TE)

)+ C hT ‖D2(u−E)‖L2(TE).

Hence, using the approximation property (3.7), we get

‖JTu−E − u−E‖L∞(TEΓ ) ≤ ChT ‖D

2(u−E)‖L2(TE).

Analogously, we can show that

‖JTu+E − u

+E‖L∞(TEΓ ) ≤ ChT ‖D

2(u+E)‖L2(TE).

Therefore, we have the bound for the first term in (3.10)

(3.12) hT |w−T (x0)| ≤ C h2T

(‖D2(u−E)‖L2(TE) + ‖D2(u+

E)‖L2(TE)

).

We now turn to the second term in (3.10). We use the fact that Dt−0w−T is constant on TEΓ to

obtain

|Dt+0w−T (x0)| = |TEΓ |−1/2‖Dt+0

w−T ‖L2(TEΓ ) ≤ Ch−1/2T ‖Dt+0

(JTu−E − JTu

+E)‖L2(TEΓ ).

Using the identities

Dt+0u±E = (t+0 · t

+)Dt+u±E + (t+0 · n

+)Dn+u±E ,(3.13)

Dt+u+E = Dt+u

−E ,(3.14)

Dn+u+E =

ρ−

ρ+Dn+u−E ,(3.15)

on TEΓ , we have

‖Dt+0w−T ‖L2(TEΓ ) = ‖Dt+0

(JTu−E − u

−E + u+

E − JTu+E) + (1− ρ−

ρ+)(t+0 · n

+)Dn+u−E‖L2(TEΓ )

≤ ‖D(JTu−E − u

−E)‖L2(TEΓ ) + ‖D(JTu

+E − u

+E)‖L2(TEΓ )

+ (1− ρ−

ρ+)‖(t+0 · n

+)Du−E‖L2(TEΓ ).

For the first two terms in the previous bound we use a trace inequality to obtain

‖D(JTu±E − u

±E)‖L2(TEΓ ) ≤ C

(1√hT‖D(JTu

±E − u

±E)|L2(TE) +

√hT ‖D2(JTu

±E − u

±E)|L2(TE)

),


and for the third term we use that (t−0 ·n−) = O(hT ) on TEΓ when Γ is C2, ρ− ≤ ρ+ and a traceinequality to obtain

(1− ρ−

ρ+)‖(t+0 · n

+)Du−E‖L2(TEΓ ) ≤ C(√

hT ‖Du−E‖L2(TE) + h3/2T ‖D

2u−E‖L2(TE)

).

Hence, applying approximation property (3.7) we obtain

(3.16) h2T |Dt+0

w−T (x0)| ≤ Ch2T

(‖Du−E‖L2(TE) + ‖D2u−E‖L2(TE)) + ‖D2u+

E‖L2(TE)

).

If we combine the inequalities (3.12) and (3.16) with (3.10), we arrive at the first result of thelemma, inequality (3.8).

We now turn to prove inequality (3.11). We need to bound

|Dn+0w+T (x0)| = |TEΓ |−1/2‖Dn+

0w+T ‖L2(TEΓ ) ≤ Ch

−1/2T ‖Dn+

0(JTu

+E −

ρ−

ρ+JTu

−E)‖L2(TEΓ ).

Similarly to the previous bound, w note that

Dn+0u±E = (n+

0 · t+)Dt+u

±E + (n+

0 · n+)Dn+u±E ,

and then using that(3.14) and (3.15) we obtain

‖Dn+0w+T ‖L2(TEΓ )

= ‖Dn+0

((JTu

+E − u

+E)− ρ−

ρ+(JTu

−E − u

−E)

)+ (1− ρ−

ρ+)(n+

0 · t+)Dt+u

+E‖L2(TEΓ ).

The remaining of the proof is similar as above and we obtain

h2T |Dn+

0w+T (x0)| ≤ Ch2

T

(ρ−

ρ+‖D2u−E‖L2(TE)) + ‖Du+

E‖L2(TE) + ‖D2u+E‖L2(TE))

).

If we combine this inequality with (3.11), we arrive at our second result (3.9).

4. Coercivity and Continuity of Bilinear Form

The aim of this section is to prove coercivity and continuity of the bilinear form ah(·, ·) definedin (2.6).

4.1. Coercivity of Bilinear Form. We define the following energy norm ‖·‖V : H1(Ω)→ R+

‖v‖2V = ‖√ρ∇hv‖2L2(Ω) +∑e∈EhΓ

(ρ−

|e−|‖JvK‖2L2(e−) +

ρ+

|e+|‖JvK‖2L2(e+)

)(4.1)

+∑e∈EhΓ

(ρ−|e−|‖J∇vK‖2L2(e−) + ρ+|e+|‖J∇vK‖2L2(e+)

).

In order to prove coercivity we will need the following lemma.

Lemma 4. Let T1, T2 ∈ Th be such that e = T1 ∩ T2. There exists a constant θ > 0 such that

|e±|2 ≤ θmaxi=1,2

|T±i |.

The constant θ depends on the C2-norm of the parametrization of Γ and the shape regularity ofT1 and T2.

Proof. See Appendix A.

FEM FOR INTERFACE 9

Lemma 5. (Coercivity) If γ is large enough (depending on shape regularity of triangulation)there exists a constant c > 0 such that

(4.2) c‖v‖2V ≤ ah(v, v) for all v ∈ Vh.

Proof. Let v ∈ Vh, then

ah(v, v) = ‖√ρ∇hv‖2L2(Ω) − 2∑e∈EhΓ

∫eρ∇v · JvK

+∑e∈EhΓ

(γ

|e−|ρ−‖JvK‖2L2(e−) +

γ

|e+|ρ+‖J∇vK‖2L2(e+)

)+∑e∈EhΓ

(ρ−|e−|‖J∇vK‖2L2(e−) + ρ+|e+|‖J∇vK‖2L2(e+)

).

To prove the lemma, it is enough to bound the non-symmetric term∫eρ∇v · JvK =

∫e−ρ∇v · JvK +

∫e+ρ∇v · JvK.

Let T1, T2 ∈ Th be such that e = T1 ∩ T2 and set Te = T1 ∪ T2. Without loss of generalityassume T−1 = max

i=1,2|T−i |. Then, according to Lemma 4

(4.3) |e−|2 ≤ θ|T−1 |.

Then, we see that∫e−ρ∇v · JvK =

∫e−

ρ−

2((Dn1v) (v|T1 − v|T2) + (Dn2v) (v|T2 − v|T1))

=

∫e−ρ−(Dn1v) (v|T1 − v|T2) +

ρ−

2(Dn2v +Dn1v)(v|T2 − v|T1)

=

∫e−ρ−(Dn1v) (v|T1 − v|T2) +

ρ−

2(J∇vK) (v|T2 − v|T1).

Using the fact that v is a linear function on T−1 and using (4.3) we have

|2∫e−ρ−(Dn1v) (v|T1 − v|T2)| ≤ C ‖ρ−∇v‖L2(T−1 )

1

|e−|1/2‖JvK‖L2(e−),

where C depends on θ. Therefore, we have

|2∫e−ρ−(Dn1v) (v|T1 − v|T2)| ≤ ερ−‖∇v‖2

L2(T−e )+ρ−C2

ε|e−|‖JvK‖2L2(e−)

for any ε > 0. Furthermore, we have

(4.4) |∫e−ρ−(J∇vK) (v|T2 − v|T1)| ≤ ερ−|e−|‖J∇vK‖2L2(e−) +

ρ−

ε|e−|‖JvK‖2L2(e−)

Collecting the last two estimates gives

|2∫e−ρ∇v · JvK| ≤ ερ−(|e−|‖J∇vK‖2L2(e−) + ‖∇v‖2

L2(T−e )) +

(C2 + 1)ρ−

ε|e−|‖JvK‖2L2(e−).


Likewise, we can bound the integral over e+ to get a combined result

|2∫eρ∇v · JvK| ≤ε(‖√ρ∇hv‖2L2(Te)

+ ρ−|e−|‖J∇vK‖2L2(e−) + ρ+|e+|‖J∇vK‖2L2(e+))

+(C2 + 1)ρ−

ε|e−|‖JvK‖2L2(e−) +

(C2 + 1)ρ+

ε|e+|‖JvK‖2L2(e+).

Summing over all edges e we get

|2∑e∈EhΓ

∫eρ∇v · JvK| ≤

∑e∈EhΓ

ε(‖√ρ∇hv‖2L2(Te)+ ρ−|e−|‖J∇vK‖2L2(e−) + ρ+|e+|‖J∇vK‖2L2(e+))

+∑e∈EhΓ

(C2 + 1)ρ−

ε|e−|‖JvK‖2L2(e−) +

(C2 + 1)ρ+

ε|e+|‖JvK‖2L2(e+).

We now note that the result follows by choosing ε = 1/2 and then choosing γ = (C2+1)ε + 1

4.2. Continuity of bilinear form. Next we prove continuity of the bilinear form. To do thiswe define the augmented norm

(4.5) ‖v‖2W = ‖v‖2V +∑T∈Eh

(ρ−‖∇v · n‖2L2(e−)|e−|+ ρ+‖∇v · n‖2L2(e+)|e

+|),

and denote by H2h the broken Sobolev space

H2h(Ω±) = v : v|T± ∈ H2(T±), for all T ∈ Th.

Lemma 6. (Continuity) Suppose that v±, w± ∈ H2h(Ω±). There exists a constant independent

of v, w such that

ah(w, v) ≤ C ‖w‖W ‖v‖W .Additionally, if w± ∈ H2

h(Ω±) and v ∈ Vh we have

(4.6) ah(w, v) ≤ C ‖w‖W ‖v‖V .

Proof. We give a sketch of the proof by bounding each term of ah(·, ·) in (2.6) separately. Thefirst term can easily be bounded as follows∫

Ωρ∇hv · ∇hw ≤ ‖

√ρ∇hv‖L2(Ω) ‖

√ρ∇hw‖L2(Ω).

The second term can be written as∫eρ∇v · JwK =

∫e−

ρ−∇v

· JwK +

∫e+

ρ+∇v

· JwK.

Using the Cauchy-Schwarz inequality one has∫e±

ρ±∇v

JwK ≤

(√|e±|

√ρ±‖∇v · n‖L2(e±)

) √ρ±√|e±|‖JwK‖L2(e±).

Let e = T 1 ∩ T 2. If v ∈ Vh then ∇hv|T±i is constant for each i = 1, 2. Then, we can use

Lemma 4 to get√|e±|

√ρ±‖∇v ·n‖L2(e±) ≤

√|e±|

√ρ±‖J∇vK‖L2(e±) +C

√ρ±(‖∇hv‖L2(T±1 ) + ‖∇hv‖L2(T±2 ))

where C here depends on θ.

FEM FOR INTERFACE 11

The third term can be bounded by∫e±ρ±JwK · JvK ≤ ρ±‖JwK‖L2(e±)‖JvK‖L2(e±).

Finally, the fourth term can be bounded as∫e±ρ±JDwK JDvK ≤ ρ±‖JDvK‖L2(e±)‖JDwK‖L2(e±).

The proof is complete summing over the edges, using finite overlapping of the elements asso-ciated to the edges and using arithmetic-geometric mean inequality.

5. A priori error estimates

The purpose of this section is to prove a priori error estimates for the method defined in (2.5)in the energy and L2 norm.

5.1. Energy error estimates.

Theorem 1. (A priori energy error estimate) Let u be the solution to problem (1.1) and uh ∈ Vhbe its finite element approximation solution of (2.5) Then, there exists C > 0, independent ofh, ρ− and ρ+, such that

‖u− uh‖V ≤ C h(√

ρ−(||Du||L2(Ω−) + ‖D2u‖L2(Ω−)) +√ρ+(||Du||L2(Ω+) + ||D2u||L2(Ω+))

).

Proof. Note that ‖u − uh‖V ≤ ‖u − Ihu‖V + ‖Ih(u) − uh‖V . Since dh := Ih(u) − uh ∈ Vh,coercivity of ah(·, ·) (see (4.2)) gives

(5.1) c‖dh‖2V ≤ ah(dh, dh) = ah(Ihu− u, dh) + ah(u− uh, dh).

Using (4.6) we obtain

(5.2) ah(Ihu− u, dh) ≤ C ‖Ihu− u‖W ‖dh‖V ≤ C(ε

2‖dh‖2V +

1

2ε‖Ihu− u‖2W

).

Choosing ε sufficiently small we have

(5.3) ‖dh‖2V ≤ C ‖Ihu− u‖2W + ah(u− uh, dh).

The proof of the theorem follows by checking bellow Lemma 8 (approximation error in W -norm)and Lemma 10 (inconsistency error).

Now we would like to state a corollary to this result. First, we need to state regularityestimates which was proved in [8] appendix.

Proposition 1. In addition to the assumptions already made in this article, assume furtherthat Ω is convex. Let u solve (1.1) then the following regularity estimate holds

(5.4) ρ+‖D2u‖L2(Ω+) + ρ−‖D2u‖L2(Ω−) ≤ C ‖f‖L2(Ω)

where C is independent of ρ±.

We will also need a standard energy estimate.

Proposition 2. Let u solve (1.1), then the following energy estimate holds

(5.5)√ρ+‖Du‖L2(Ω+) +

√ρ−‖Du‖L2(Ω−) ≤

C√ρ−‖f‖L2(Ω)

where C is independent of ρ±.


Note that we are using here that ρ− ≤ ρ+.In fact, a better energy estimate holds as we proved in next proposition.

Proposition 3. Let u solve (1.1), then the following energy estimate holds

(5.6) ρ+‖Du‖L2(Ω+) + ρ−‖Du‖L2(Ω−) ≤ C ‖f‖L2(Ω).

Proof. We prove this estimate in the case that Ω+ is the inclusion (i.e. ∂Ω+ does not intersect∂Ω). The proof of the other case is similar. First, from the previous proposition we have

(5.7) ρ−‖Du‖L2(Ω−) ≤ C ‖f‖L2(Ω).

Let w = u+ − 1|Ω+|

∫Ω+ u

+dx. Therefore, using Poincare’s inequality we have

(5.8) ‖w‖L2(Ω+) ≤ C‖∇u‖L2(Ω+).

With the help of an extension theorem, there exists v ∈ H10 (Ω) such that v|Ω+ = w|Ω+ such

that

(5.9) ‖v‖H1(Ω) ≤ C‖w‖H1(Ω+).

We know that

(5.10)

∫Ω−

ρ−∇u · ∇v +

∫Ω+

ρ+∇u · ∇vdx =

∫Ωfv.

However,

ρ+‖∇u‖2L2(Ω+) = ρ+

∫Ω+

ρ+∇u · ∇vdx = ρ+

∫Ωfv − ρ+

∫Ω−

ρ−∇u · ∇v

Therefore, we have

‖ρ+∇u‖2L2(Ω+) ≤ ‖f‖L2(Ω)‖ρ+v‖L2(Ω) + ‖ρ−∇u‖L2(Ω−)‖ρ+∇v‖L2(Ω).

By (5.9) we have

(5.11) ‖ρ+∇u‖2L2(Ω+) ≤ C(‖f‖L2(Ω) + ‖ρ−∇u‖L2(Ω−))ρ+‖w‖H1(Ω+).

Hence, using (5.8) we get

(5.12) ‖ρ+∇u‖2L2(Ω+) ≤ C(‖f‖L2(Ω) + ‖ρ−∇u‖L2(Ω−))‖ρ+∇u‖L2(Ω+).

The proof is complete after applying (5.7).

In fact, we would like to point out that we can prove the improved energy estimateCombining theorem 1 and the previous propositions we have the following corollary.

Corollary 1. Let u be the solution to (1.1) and uh ∈ Vh be its finite element approximation.If Ω is convex, we have

(5.13) ‖u− uh‖V ≤C h√ρ−‖f‖L2(Ω).

Now we return to finishing the missing parts of Theorem 1.We need an estimate for interpolant Ih (see (3.6)) in the augmented norm W , and to do so,

we first need to prove a trace inequality that goes from a part of boundary to the interior of thedomain.

Lemma 7. Suppose that w ∈ H2(TE,±) then one has

1

|e±|‖w‖2L2(e±) ≤ C

(1

h2T

‖w‖2L2(TE,±) + ‖Dw‖2L2(TE,±) + h2T ‖D2w‖2L2(TE,±)

)


Proof. Let e−E ⊂ T−E be the straight prolongation of the line segment e− to all of T−E . It must

be that chT ≤ |e−E | for a constant c that only depends on the shape regularity of the mesh. Astandard result, see lemma 3 of [9],

1

|e−|‖w‖2L2(e−) ≤ C(

1

hT‖w‖2

L2(e−E)+ hT ‖∇w‖2L2(e−E)

)

and the trace inequality from e−E to T−E gives the result. The exact same argument would applyfor e+.

Lemma 8. (Best Approximation Error Estimate) It holds,

‖Ihu− u‖W ≤ C h(√

ρ−(‖Du‖L2(Ω−) + ‖D2u‖L2(Ω−)) +√ρ+(‖Du‖L2(Ω+) + ‖D2u‖L2(Ω+))

).

Proof. We bound each term of ‖u− Ihu‖W , see (4.5) and (4.1), separately.Using Lemma 3 we easily have

‖√ρ∇h(u− Ihu)‖2L2(Ω) ≤ C∑

T∈Th\T Γh

h2T

(ρ−‖D2(u−E)‖2L2(T ) + ρ+|D2u+

E |2L2(T )

)+ C

∑T∈T Γ

h

h2T

(ρ−(‖Du−E‖

2L2(TE) + ‖D2u−E‖

2L2(TE)) + ρ+(‖Du+

E‖2L2(TE) + ‖D2u+

E‖2L2(TE))

)≤ Ch2

(ρ−(‖Du‖2L2(Ω−) + ‖D2u‖2L2(Ω−)) + ρ+(‖Du‖2L2(Ω+) + ‖D2u‖2L2(Ω+))

).

In the last inequality we used (3.2). We also have used finite overlapping of the sets TE .We next estimate the term∑

e∈EhΓ

(ρ−

|e−|‖Ju− IhuK‖2L2(e−) +

ρ+

|e+|‖Ju− IhuK‖2L2(e+)

).

Suppose that e = ∂T1 ∩ ∂T2. Then using Lemma 7 we have

ρ−

|e−|‖Ju− IhuK‖2L2(e−) +

ρ+

|e+|‖Ju− IhuK‖2L2(e+)

≤ C ρ−

|e−|(‖u− IT1u‖2L2(e−) + ‖u− IT2u‖2L2(e−)) + C

ρ+

|e+|(‖u− IT1‖2L2(e+) + ‖u− IT2‖2L2(e+))

≤ Cρ−2∑i=1

(1

h2Ti

‖u− ITiu‖2L2(TE,−i )+ ‖D(u− ITiu)‖2

L2(TE,−i )+ h2

Ti‖D2(u− ITiu)‖2

L2(TE,−i )

)

+ Cρ+2∑i=1

(1

h2Ti

‖u− ITiu‖2L2(TE,+i )+ ‖D(u− ITiu)‖2

L2(TE,+i )+ h2


L2(TE,+i )

),

where we used lemma 7. If we now apply lemma 3 and use (3.2), we get∑e∈EhΓ

(ρ−

|e−|‖Ju− IhuK‖2L2(e−) +

ρ+

|e+|‖Ju− IhuK‖2L2(e+))

≤ Ch2(ρ−(‖Du‖2L2(Ω−) + ‖D2u‖2L2(Ω−)) + ρ+(‖Du‖2L2(Ω+) + ‖D2u‖2L2(Ω+))

).

In order to bound∑e∈EhΓ

(ρ−|e−|‖JD(u− Ihu)K‖2L2(e−) + ρ+|e+|‖JD(u− Ihu)K‖2L2(e+)

),


we denote e = ∂T1 ∩ ∂T2 and use a trace inequality to obtain

ρ−|e−|‖JD(u− Ihu)K‖2L2(e−) + ρ+|e+|‖JD(u− Ihu)K‖2L2(e+)

(5.14)

≤ ρ−|e|‖JD(u− Ihu)K‖2L2(e−) + ρ+|e|‖JD(u− Ihu)K‖2L2(e+)

≤ ρ−|e−|(‖D(u− IT1u)‖2L2(e−) + ‖D(u− IT2u)‖2L2(e−)

)+ ρ+|e+|

(‖D(u− IT1u)‖2L2(e+) + ‖D(u− IT2u)‖2L2(e+)

)≤ Cρ−

2∑i=1

(‖∇(u− ITiu)‖2

L2(TE,−i )+ h2


L2(TE,−i )

)+ Cρ+

2∑i=1

(‖∇(u− ITiu)‖2

L2(TE,+i )+ h2


L2(TE,+i )

).

Again, if we apply lemma 3 and (3.2) we obtain∑e∈EhΓ

(ρ−|e−|‖JD(u− Ihu)K‖2L2(e−) + ρ+|e+|‖JD(u− Ihu)K‖2L2(e+)

)≤ Ch2

(ρ−(‖Du‖2L2(Ω−) + ‖D2u‖2L2(Ω−)) + ρ+(‖Du‖2L2(Ω+) + ‖D2u‖2L2(Ω+))

).

Finally, the last term is estimated by∑T∈Eh

(|e−|ρ−‖∇(u− Ihu) · n‖2L2(e−) + |e+|ρ+‖∇(u− Ihu) · n‖2L2(e+))

≤ Ch2(ρ−(‖Du‖2L2(Ω−) + ‖D2u‖2L2(Ω−)) + ρ+(‖Du‖2L2(Ω+) + ‖D2u‖2L2(Ω+))

),

exactly in the same way as it was done above.

In order to bound the inconsistency term we need a technical lemma.

Lemma 9. Let T ∈ T Γh and enumerate the three edges of T by e1, e2, e3. There exists a constant

m such that

|TΓ| ≤ m maxi=1,2,3

|e−i | and |TΓ| ≤ m maxi=1,2,3

|e+i |

where e±i = ei ∩ Ω±. The constant m depends on the shape regularity of the mesh and on theC2-norm of the parametrization of Γ.

Proof. See Appendix B.

Now we are able to establish the inconsistency error estimate. The constant will be indepen-dent of the contrast of the coefficients ρ− and ρ+ and also independent of how the mesh crossesΓ.

Lemma 10. (Inconsistency Error Estimate) Let u be the solution to (1.1) and uh ∈ Vh be itsfinite element approximation. Then for any dh ∈ Vh, it holds(5.15)

ah(u− uh, dh) ≤ C h(√

ρ−(||Du||L2(Ω−) + h‖D2u‖L2(Ω−)

)+√ρ−h‖D2u‖L2(Ω+)

)‖dh‖V .


Proof. First note that ah(u−uh, dh) = ah(u, dh)− (f, dh). The inconsistency term is then givenby

ah(u− uh, dh) =∑T∈T Γ

h

∫TΓ

(ρ−Dn−u−)[dh] ds,

where we have used that ρ−Dn−u− = ρ+Dn+u+ on Γ. We also have used the notation [dh] =

d−h − d+h .

Note that [dh](x) = 0 on the segment line LT ⊂ T tangent to Γ at x0. Since the distancebetween this line segment and the curve TΓ is O(r2), where we set r = |TΓ|, we have that|[dh](x)| ≤ C r2|D[dh](x)| on TΓ. Hence, we have∫

TΓ

(ρ−Dn−u−)[dh] ds ≤ C r2‖

√ρ−Dn−u

−E‖L2(TΓ)

√ρ−‖D[dh]‖L2(TΓ).

For the moment, let us assume that following inequality holds

(5.16)√ρ−√r‖D[dh]‖L2(TΓ) ≤ CMT ,

where

MT =∑e⊂∂T

(√|e+|‖

√ρ+JDdhK‖L2(e+) +

1√|e+|‖√ρ+ dh‖L2(e+) + ‖

√ρ+Ddh‖L2(TE,+)

)

+∑e⊂∂T

(√|e−|‖

√ρ−JDdhK‖L2(e−) +

1√|e−|‖√ρ− dh‖L2(e−) + ‖

√ρ−Ddh‖L2(TE,−)

).

Then, ∫TΓ

(ρ−Dn−u−)[dh] ds ≤ Cr3/2‖

√ρ−Du−E‖L2(TΓ)MT .

Letting Br be the ball of radius r centered at x0 we can use the trace inequality to get

‖√ρ−Du−E‖L2(TΓ) ≤ C(

1√r‖√ρ−Du−E‖L2(Br) +

√r‖√ρ−D2u−E‖L2(Br)).

Hence, ∫TΓ

(ρ−Dn−u−)[dh] ≤ Cr(‖

√ρ−Du−E‖L2(Br) + r‖

√ρ−D2u−E‖L2(Br))MT .

We see that (5.15) follows after using that∑T∈T Γ

h

M2T ≤ C ‖dh‖2V ,

the inequality (3.2) and the fact that r ≤ C hT .To complete the proof we need to prove (5.16). Firstly, by using the triangle inequality we

have √ρ−√r‖D[dh]‖L2(TΓ) ≤

√ρ−√r(‖Dd−h ‖L2(TΓ) + ‖Dd+

h ‖L2(TΓ)

).

Then by lemma 9, there exists a edge e of T such that

(5.17) r ≤ m|e+|.


Now let e = T ∩K where K ∈ Th. Using that Dd+h is constant we get

‖Dd+h ‖L2(TΓ) =

√r|Dd+

h | =

√r√|e+|‖Dd+

h ‖L2(e+)

≤√m‖Dd+

h ‖L2(e+).

According to Lemma 4

(5.18) |e+|2 ≤ θmax|T+|, |K+|.

First, suppose that |T+| = max|T+|, |K+|. Then,

‖Dd+h ‖L2(e+) =

√|e+|√|T+|

‖Dd+h ‖L2(T+)

Hence,

‖Dd+h ‖L2(e+) ≤

√θm√r‖Dd+

h ‖L2(T+).

On the other hand, if |K+| = max|T+|, |K+| then we have

‖Dd+h ‖L2(e+) ≤ ‖(D(d+

h |T − d+h |K)‖L2(e+) +

√θm√r‖Dd+

h ‖L2(K+).

If we write D(dh|T −dh|K) as its normal part and tangential part, then, use an inverse inequalityfor the tangential part on e+, and have

‖(D(d+h |T − d

+h |K)‖L2(e+) ≤ ‖JDdhK‖L2(e+) +

C

|e+|‖JdhK‖L2(e+),

therefore,

‖Dd+h ‖L2(e+) ≤ ‖JDdhK‖L2(e+) +

√θm√r‖Dd+

h ‖L2(K+) +C

|e+|‖JdhK‖L2(e+).

Combining the above inequalities we get

(ρ−r)1/2‖Dd+h ‖L2(TΓ) ≤

√m|e+|

√ρ+‖JDdhK‖L2(e+) +

C√ρ+√m√

|e+|‖JdhK‖L2(e+)

+ Cm√θ(‖√ρ+Ddh‖L2(T+) + ‖

√ρ+Ddh‖L2(K+)),

where we have used ρ− ≤ ρ+, so

(ρ−r)1/2‖Dd+h ‖L2(TΓ) ≤ C

∑e⊂∂T

(√m|e+|

√ρ+‖JDdhK‖L2(e+) +

√ρ+√m√

|e+|‖JdhK‖L2(e+))

+m√θ‖√ρ+∇hdh‖L2(TE,+).

We can identically prove

(ρ−r)1/2‖Dd−h ‖L2(TΓ) ≤ C∑e⊂∂T

(√m|e−|

√ρ−‖JDdhK‖L2(e−) +

√ρ−√m√

|e−|‖JdhK‖L2(e−))

+ Cm√θ‖√ρ−∇hdh‖L2(TE,−).

Combining the two last inequalities proves (5.3).


5.2. L2-Error Estimates. We now prove an L2 error estimate using a duality argument.

Theorem 2. Let u be the solution to problem (1.1) and uh ∈ Vh be its finite element approxi-mation. Assuming Ω is convex, then it holds

(5.19) ‖u− uh‖L2(Ω) ≤C h2

ρ−‖f‖L2(Ω).

Proof. Let φ be a solution of the problem

−ρ±∆φ± = (u− uh)± in Ω±,(5.20a)

φ = 0 on ∂Ω,(5.20b)

[φ] = 0 on Γ,(5.20c)

[ρDnφ] = 0 on Γ.(5.20d)

We have

‖u− uh‖2L2(Ω) = (u− uh, u)− (u− uh, uh) = ah(φ, u)− ah(φh, uh),

where φh is the finite element approximation of φ. Therefore, we see that

‖u− uh‖2L2(Ω) = ah(φ, u− uh) + ah(φ− φh, uh)

= ah(φ− φh, u− uh) + ah(u− uh, φh) + ah(φ− φh, uh)

= ah(φ− φh, u− uh) + ah(u− uh, φh − Ihφ) + ah(u− uh, Ihφ)

+ ah(φ− φh, uh − Ihu) + ah(φ− φh, Ihu)

Using continuity of the bilinear form we have

ah(φ− φh, u− uh) ≤ C‖φ− φh‖W ‖u− uh‖W .

Using the triangle inequality we have

‖u− uh‖W ≤ ‖u− Ihu‖W + ‖Ihu− uh‖W .

It is not difficult to show that

‖Ihu− uh‖W ≤ C‖Ihu− uh‖VHence,

‖u− uh‖W ≤ ‖u− Ihu‖W + ‖u− uh‖V .Now using theorem 8, theorem 1, (5.4) and (5.5) we get

‖u− uh‖W ≤ C h√ρ−‖f‖L2(Ω).

Similarly,

‖φ− φh‖W ≤ C h√ρ−‖u− uh‖L2(Ω).

Hence,

ah(φ− φh, u− uh) ≤ C h2

ρ−‖f‖L2(Ω) ‖u− uh‖L2(Ω).

Using lemma 10, (5.4) and (5.5) we have

ah(u− uh, φh − Ihφ) ≤ C h√ρ−‖f‖L2(Ω)‖Ihφ− φh‖V .


We therefore have

ah(u− uh, φh − Ihφ) ≤ C h2

ρ−‖f‖L2(Ω)‖u− uh‖L2(Ω).

Analogously, we have

ah(φ− φh, uh − Ihu) ≤ C h2

ρ−‖f‖L2(Ω)‖u− uh‖L2(Ω).

We write out the following term

ah(u− uh, Ihφ) =∑T∈T Γ

h

∫TΓ

(ρ−Dn−u−)[Ihφ] ds.

Using the Cauchy-Schwartz inequality we obtain

ah(u− uh, Ihφ) = ‖√ρ−Dn−u

−‖L2(Γ)

√ρ−

∑T∈T Γ

h

‖[Ihφ]‖2L2(TΓ)

1/2

Using a trace inequality we have

‖√ρ−Dnu

−‖L2(Γ) ≤ C√ρ−(‖D2u‖L2(Ω−) + ‖Du‖L2(Ω−)) ≤

C√ρ−‖f‖L2(Ω).

where we used (5.4), (5.5) and the fact that ρ− ≤ ρ+.We know that [Ihφ] = 0 on LT and we know that LT is at most distance Cr2 from TΓ where

r = |TΓ|. Therefore we can use, Taylor’s theorem to show that

[Ihφ](x) ≤ Cr2|D[Ihφ]|(x) ∀x ∈ TΓ.

Hence,‖[Ihφ]‖L2(TΓ) ≤ Cr2 ≤ ‖D[Ihφ]‖L2(TΓ) ≤ Ch2

T ≤ ‖D[Ihφ]‖L2(TΓ),

where we used that r ≤ C hT . Consequently, adding and subtracting D[φ] ∑T∈T Γ

h


1/2

≤ C

∑T∈T Γ

h

h4T ‖D[Ihφ− φ]‖2L2(TΓ)

1/2

+C

∑T∈T Γ

h

‖D[φ]‖2L2(TΓ)

1/2

.

Observe that ∑T∈T Γ

h

‖D[φ]‖2L2(TΓ)

1/2

≤ Ch2‖D[φ]‖L2(Γ) ≤ Ch2(‖Dφ+‖L2(Γ) + ‖Dφ−‖L2(Γ)).

Application of trace inequality gives ∑T∈T Γ

h

‖D[φ]‖2L2(TΓ)

1/2

≤ Ch2(‖D2φ‖L2(Ω−) + ‖Dφ‖L2(Ω−) + ‖D2φ‖L2(Ω+) + ‖Dφ‖L2(Ω+)

).

For the next term we use that Ihφ|T = ITφ and use a trace inequality to bound

‖D[Ihφ− φ]‖L2(TΓ) = ‖D[ITφ− φ]‖L2(TΓ)

≤ C√hT

(‖D(ITφ− φ)‖L2(TE,−) + ‖D(ITφ− φ)‖L2(TE,+))

+ C√hT (‖D2φ‖L2(TE ,−) + ‖D2φ‖L2(TE ,+)).


From (3.8) and (3.9) (using that ρ− ≤ ρ+) we obtain

‖D(ITφ− φ)‖L2(TE,−) + ‖D(ITφ− φ)‖L2(TE,+) ≤ ChT (‖D2φ‖L2(TE,−) + ‖D2φ‖L2(TE,+)).

Therefore, ∑T∈T Γ

h

h4T ‖D[Ihφ− φ]‖2L2(TΓ)

1/2

≤ C h2(‖D2φ‖L2(Ω−) + ‖Dφ‖L2(Ω−))

+ C h2(‖D2φ‖L2(Ω+) + ‖Dφ‖L2(Ω+)).

Hence, using the regularity result (5.4) and (5.5) we have

√ρ−

∑T∈T Γ

h


1/2

≤ Ch2√ρ−‖u− uh‖L2(Ω).

Thus, we obtain

ah(u− uh, Ihφ) ≤ Ch2

ρ−‖f‖L2(Ω)‖u− uh‖L2(Ω).

In a similar fashion we can prove

ah(φ− φh, Ihu) ≤ Ch2

ρ−‖f‖L2(Ω)‖u− uh‖L2(Ω).

Combining the above inequalities proves our result.

6. Extensions and related methods

6.1. Extension to three dimensions. We now show that the space S1(T ), introduced in(2.4), can easily be extended in three-dimensions. We consider the problem (1.1) where Ω is athree-dimensional domain and Γ is a C2 surface. Let now Th be a simplicial triangulation of Ω.Let Eh be all the faces of the mesh Th. Finally, let EhΓ be the set of all faces that belong to atetrahedron that intersects Γ.

We now define the local finite element space. Let T ∈ Th be a tetrahedron. Let x0 be a fixedpoint on Γ ∩ T . Let n+

0 be the unit vector normal to Γ at x0 pointing out of Ω+. Let t+0 , s+0 be

such that n+0 , t

+0 , s

+0 forms an orthonormal system.

Given v ∈ P1(T+) there exists a unique E(v) ∈ P1(T−) satisfying

E(v)(x0) = v(x0)(Dt+0

E(v))

(x0) =(Dt+0

v)

(x0)(Ds+

0E(v)

)(x0) =

(Ds+

0v)

(x0)

ρ−(Dn+

0E(v)

)(x0) = ρ+

(Dn+

0v)

(x0).

Given T ∈ T Γh and for each v ∈ P1(T+) we can consider the unique corresponding function

G(v) =

v on T+,

E(v) on T−.


Let span v1, v2, v3, v4 be a basis for P1(T ) restricted to T+. Then we define the local finiteelement space

S1(T ) =

span

G(v1), G(v2), G(v3), G(v4)

, if T ∈ T Γ

h

P1(T ) , if T ∈ Th\T Γh .

Consequently, the global finite element space is given by

Vh :=v : v|T ∈ S1(T ), ∀T ∈ Th, v is continuous across all faces in Eh\EhΓ

.

The numerical method is now given by (2.5), where now e± = e∩Ω± are pieces of e a face ofa tetrahedra.

6.2. Alternative Local Spaces. An alternative approach to enforce weak continuity of ourlocal spaces S1(T ) is normally used in the literature (see [8]). Instead of enforcing continuityof function and tangential derivative at x0 one imposes continuity at two distinct points x1, x2.More precisely, define x1, x2 be the two points in Γ that intersect T . Then, one can define E(v)(in contrast to the definition in Lemma 1) by

E(v)(xi) = v(xi) for i = 1, 2

ρ−(Dn+0E(v))(x0) = ρ+(Dn+

0v)(x0).

Subsequently, ITu is now defined by

(I−T u)(xi) := (JTu+E)(xi) =: (I+

T u)(xi) for i = 1, 2

ρ−(Dn+0I−T u)(x0) := ρ+(Dn+

0JTu

+E)(x0) =: ρ+(Dn+

0I+T u)(x0).

Defining the finite element method using these local spaces, we can prove all the a-priori esti-mates above.

Another alternative is to enforce the matching conditions by averaging. More precisely, onecan replace E(v) in Lemma 1 by ∫

TΓ

E(v) ds :=

∫TΓ

v ds∫TΓ

E(v) s ds :=

∫TΓ

v s ds∫TΓ

ρ+Dn+E(v) ds :=

∫TΓ

ρ−Dn+v ds

or alternatively, one can replace the second equation by∫TΓ

Dt+E(v) ds :=

∫TΓ

Dt+ vds.

Even though their numerical implementation is more complicated and require numerical inte-grations, these alternatives have the advantage that the analysis is shorter specially for theconsistency error, and also can be formulated using Lagrange multipliers.

6.3. Extension to cartesian grids. We note that the analysis and methods developed herecan be easily extended to cartesian grids and Q1 elements. One possibility would be to usesame local spaces S1(T ) defined before for quadrangular elements T ∈ T Γ

h , that is, one has

three degrees of freedom for T ∈ T Γh and four degrees otherwise. In case one wants to have four


degrees of freedom for T ∈ T Γh , let the space S1(T ) then be a piecewise bilinear function under

the coordinates n0, t0 and impose

E(v)(x0) := v(x0)

(Dt+0E(v))(x0) := (Dt+0

v)(x0)

ρ−(Dn+0E(v))(x0) := ρ+(Dn+

0v)(x0)

ρ−(Dn+0Dt+0

E(v))(x0) := ρ+(Dn+0Dt+0

v)(x0)

6.4. Alternative Bilinear forms. Here we will describe alternative bilinear forms that seemnatural. In some cases, we can prove all the results as the method above, but in some cases wecannot. We will point out what are the difficulties. When we provide numerical experiments inthe following section we will also see that, although we cannot prove stability for some methods,the methods seem to do well experimentally.

Firstly, formulation without flux stabilization has been used for example in [17]. Methods(6.1) and (6.2) experiment this direction.

ah(w, v) :=


∑e∈EhΓ

∫e

(ρ∇hv

· JwK +

ρ∇hw

· JvK

)(6.1)

+ γ∑e∈EhΓ

1

|e|

(∫e−ρ−JwK · JvK +

∫e+ρ+JwK · JvK

).

The problem with this method is that we cannot prove coercivity independent of contrast. Infact, we cannot prove coercivity in the method

ah(w, v) :=


∑e∈EhΓ

∫e

(ρ∇hv

· JwK +

ρ∇hw

· JvK

)(6.2)

+ γ∑e∈EhΓ

(1

|e−|


1

|e+|

∫e+ρ+JwK · JvK

).

However, both of these methods seem to do quite well numerically.The method that seems to do the best numerically, slightly better than the proposed method,

is the following one:

ah(w, v) :=


∑e∈EhΓ

∫e

(ρ∇hv

· JwK +

ρ∇hw

· JvK

)(6.3)

+ γ∑e∈EhΓ

(1

|e−|


1

|e+|

∫e+ρ+JwK · JvK

)

γF∑e∈EhΓ

|e|(∫

e−ρ−J∇hvK J∇hwK +


).

The difference between this method and the one analyzed in this paper is that we are usinga stronger flux stabilization (i.e. we replace |e±| by |e| and we introduce a flux stabilizationparameter γF ). It is not difficult to see, that we can prove all the error estimates contained inthis paper for this method. In particular, the coercivity of the bilinear form is obvious since we


are adding even more stabilization. Clearly, now we would have to redefine our V and W normsbut the approximation properties will still hold. In summary, Theorem 1 and Corollaries 1 and2 hold for this formulation.

One would be tempted to use the next formulation which seems natural:

ah(w, v) :=


∑e∈EhΓ

∫e

(ρ∇hv

· JwK +

ρ∇hw

· JvK

)(6.4)

+ γ∑e∈EhΓ

1

|e|

(∫e−ρ−JwK · JvK +

∫e+ρ+JwK · JvK

)

γF∑e∈EhΓ

|e|(∫

e−ρ−J∇hvK J∇hwK +


),

However, at the present moment we cannot prove coercivity of the bilinear form. This methodalso does well numerically.

7. Numerical examples

In this section we explore the properties of the methods presented in sections above applied tothe two dimensional interface problem (1.1). In particular, we are interested in the computationof the following errors and their respective ratio of convergence

e0h := ‖uh − u‖L2(Ω), e∞h := ‖uh − u‖L∞(Ω),

e1h := ‖√ρ(∇uh −∇u)‖L2(Ω), e1,∞

h := ‖√ρ(∇uh −∇u)‖L∞(Ω),

e1h := ‖ρ(∇uh −∇u)‖L2(Ω), e1,∞

h := ‖ρ(∇uh −∇u)‖L∞(Ω),

en,∞h := ‖ρ(Dnuh −Dnu)‖L∞(Γ), e1h := ‖ρ(∇uh −∇u)‖L2(Ω)\T Γ

h,

r(e) :=log(ehl+1

/ehl)

log(hl+1/hl).

Tables for the example display the computation of the errors using different methods. The errorsin the, L2 norm and W 1,2 weighted with

√ρ semi-norm correspond to the only results proved

in this paper, Theorem 2 and 1 respectively. We expect optimal convergence, second order inthe L2 norm and first order in the weighted W 1,2 semi-norm. We also compute the analogousin the L∞ norm and W 1,∞ weighted with

√ρ semi-norm. In addition, we compute the errors in

the W 1,2 and W 1,∞ semi-norm weighted with full ρ. The computation of these errors is guidedby the correct flux ρ∇u. Note that the estimate for the interpolation error is also achieved inthis semi-norm weighted with ρ. Indeed, is a consequence of Lemma 3 and Proposition 1 and 3

‖ρ∇h(u− Ihu)‖L2(Ω) ≤ C‖f‖L2(Ω).

A revealing result of our experiments is that the ratio of convergence of the error is optimalwhen we compute using the triangles non intersecting the interface. Error e1

h illustrates thisobservation. Finally, error en,∞h is a standard error for interface problems, illustrating theapproximation of the normal derivative on the interface.

The experiment presented below shows that method (2.5)-(6.3) produces the best results.This method (and all the others) does not seem optimal for the W 1,∞ error weighted with fullρ. However, it is optimal when we do not consider the elements in T Γ

h . We highlight that thisresult was not remotely addressed in this paper, and this kind of estimates appear to be moredifficult. For the method that we analyzed (2.5)-(2.6) we observe a interesting behavior in the


error e1,∞h for the last mesh, not achieving the optimal convergence as in the previous method.

This possibly due to conditioning of the system. We also present tables for one of the methodwithout flux stabilization, method (2.5)-(6.2). We observe non convergence so far for the error

e1,∞h and in the last mesh and a deterioration in the L∞ norm. We also do not have the optimal

convergence for the normal flux error en,∞h .In our numerical experiment we consider the two dimensional domain Ω = [−1, 1]2 with the

immersed interface Γ = x ∈ Ω : x21 + x2

2 = R2. We define Ω− := x ∈ Ω : x21 + x2

2 < R2 andΩ+ = Ω\(Ω− ∪ Γ). Example (1) considers the following exact solution

u(x) =

rα

ρ− , if x ∈ Ω−,rα

ρ+ +Rα( 1ρ− −

1ρ+ ) , if x ∈ Ω+,

where r =√x2

1 + x22, R = 1/3 and α = 2. For example (1) we have Γ = ∂Ω−. Similar results

were obtained for the case Γ = ∂Ω+. We provide plots of the approximate solution for bothcases.

Finite element uniform triangular meshes Th non matching the interface were used. In thetables we compute with h = 2−(l+3/2), for l = 1, ..., 7.

All the computations were done in MATLAB, including the linear system (we used “\”).

(1) Case Γ = ∂Ω−, ρ+ = 104 and ρ− = 1. Tables 1 and 2 show the results obtained bymethod (2.5)-(6.3) with stabilization parameters γ = 10 and γF = 10. Note that thismethod has a stronger flux stabilization that the method analyzed in the paper.

l e0h r e∞h r e1h r e1,∞h r

1 8.2e-3 2.5e-2 1.1e-1 3.7e-1

2 1.7e-3 2.28 5.7e-3 2.12 4.4e-2 1.30 2.1e-1 0.86

3 2.7e-4 2.63 1.3e-3 2.19 1.8e-2 1.29 9.7e-2 1.08

4 4.6e-5 2.57 3.2e-4 1.97 8.3e-3 1.12 5.2e-2 0.90

5 9.0e-6 2.34 7.2e-5 2.15 3.9e-3 1.07 2.5e-2 1.08

6 2.0e-6 2.19 1.8e-5 2.01 1.9e-3 1.03 1.3e-2 0.92

7 4.7e-7 2.08 4.6e-6 1.94 9.5e-4 1.02 6.8e-3 0.94

l e1h r e1,∞h r e1,∞h r en,∞h r

1 3.9e-1 7.0e-1 7.0e-1 3.7e-1

2 1.6e-1 1.32 6.1e-1 0.19 4.0e-1 0.81 2.1e-1 0.86

3 6.4e-2 1.29 1.9e-1 1.68 1.9e-1 1.07 9.7e-2 1.08

4 2.9e-2 1.15 2.2e-1 -0.20 1.0e-1 0.91 4.9e-2 1.00

5 1.4e-2 1.09 1.4e-1 0.65 5.0e-2 1.01 2.5e-2 0.98

6 6.6e-3 1.04 6.6e-2 1.09 2.5e-2 0.99 1.2e-2 1.00

7 3.2e-3 1.02 5.1e-1 -2.95 1.3e-2 0.98 6.2e-3 1.00

Table 1. Example (1); errors and convergence orders with ρ− = 1 and ρ+ = 104,using method (2.5)-(6.3) with stabilization parameters γ = 10 and γF = 10.

The results in Table 1 show optimal convergence for the errors e1h and e0

h validatingthe theoretical results Theorem 1 and 2. In addition we observe optimal convergencefor the error e∞h and e1,∞

h . The error e1h, weighted with ρ instead of

√ρ, converges


ρ+ e0h e1,∞h e1h

101 2.3e-6 6.5e-3 2.8e-3

102 2.0e-6 6.6e-3 2.0e-3

103 2.0e-6 6.6e-3 1.9e-3

104 2.0e-6 6.6e-3 1.9e-3

105 2.0e-6 6.6e-3 1.9e-3

106 2.0e-6 6.6e-3 1.9e-3

Table 2. Example (1); errors with ρ− = 1 and h = 2−(6+3/2), using method(2.5)-(6.3) with stabilization parameters γ = 10 and γF = 10.

optimally. However, the rate of convergence of the error e∞h is not optimal. An interestingobservation is that the error e∞h converges optimally, indicating that is only a couple ofelements where the error does not converge. Another appealing feature of the methodis the optimal order of convergence for the error en,∞h .

Table 2 shows that the errors are independent of the contrast. We can observe thatalthough we increase ρ+ the errors remain constant, showing the contrast independencyof our estimates.

We present in Table 3 the errors and convergence orders for the method analyzed inthe paper (2.5)-(2.6).


1 8.6e-3 2.6e-2 1.1e-1 3.7e-1

2 1.7e-3 2.31 6.0e-3 2.14 4.5e-2 1.30 2.0e-1 0.90

3 2.8e-4 2.63 1.3e-3 2.18 1.8e-2 1.31 9.3e-2 1.09

4 4.7e-5 2.57 3.4e-4 1.97 8.4e-3 1.12 5.5e-2 0.76

5 9.2e-6 2.35 7.6e-5 2.16 4.0e-3 1.08 2.6e-2 1.09

6 2.0e-6 2.20 1.9e-5 2.01 1.9e-3 1.03 1.4e-2 0.91

7 4.7e-7 2.09 5.1e-6 1.88 9.5e-4 1.02 7.1e-2 -2.37


1 4.0e-1 7.4e-1 7.4e-1 3.7e-1

2 1.6e-1 1.33 2.5e+0 -1.79 4.0e-1 0.90 2.0e-1 0.90

3 6.4e-2 1.31 2.8e-1 3.20 1.9e-1 1.07 9.2e-2 1.11

4 2.9e-2 1.15 1.6e+0 -2.56 9.7e-2 0.96 4.5e-2 1.01

5 1.4e-2 1.09 4.8e-1 1.76 4.7e-2 1.04 2.3e-2 0.96

6 6.6e-3 1.05 4.8e-1 0.01 2.4e-2 1.00 1.1e-2 1.05

7 3.2e-3 1.02 7.1e+0 -3.88 1.8e-2 0.42 5.9e-3 0.93

Table 3. Example (1); errors and convergence orders with ρ− = 1 and ρ+ = 104,using method (2.5)-(2.6) with stabilization parameters γ = 10.

The results in Table 3 show optimal convergence for the errors e1h and e0

h validatingthe theoretical results Theorem 1 and 2. In addition we observe optimal convergence


for the error e∞h . The error e1,∞h seems to converge optimal up to mesh l = 6, and then

for the last mesh the rate can possibly be affected by the choice of the flux stabilizationparameter. As in the previous test the error e1

h converges optimally, however for the restof the errors the convergence is not as clear as in the previous method. We suspect thatthis phenomena is related to the weights |e±| in the flux stabilization.

Finally, Table 4 displays error and convergence orders for the method without fluxstabilization (2.5)-(6.2).


1 1.7e-3 7.0e-3 5.8e-2 2.2e-1

2 4.5e-3 1.88 2.9e-3 1.29 3.1e-2 0.89 1.6e-1 0.51

3 3.6e-4 0.31 6.1e-3 -1.08 1.2e-1 -1.89 1.3e+1 -6.37

4 2.8e-5 3.69 2.1e-4 4.87 7.7e-3 3.92 2.0e-1 5.98

5 7.3e-6 1.96 5.0e-5 2.05 3.8e-3 1.02 1.5e-1 0.40

6 1.7e-6 2.06 1.2e-5 2.05 1.9e-3 1.01 2.5e-2 -0.70

7 4.5e-7 1.95 4.0e-6 1.60 9.5e-4 0.99 1.1e-1 0.23


1 2.1e-1 5.8e-1 2.2e-1 1.5e-1

2 1.3e-1 0.69 1.4e+1 -4.58 1.5e-1 0.49 1.1e-1 0.39

3 7.3e-1 -2.46 1.3e+2 0.10 8.9e-1 -2.5 1.3e+1 -6.83

4 2.8e-2 4.70 7.2e+0 0.84 5.3e-2 4.07 2.0e-1 6.00

5 1.3e-2 1.07 2.6e+0 1.50 2.9e-2 0.86 1.5e-1 0.39

6 7.0e-3 0.93 2.1e+0 0.27 3.7e-2 -0.35 2.5e-1 -0.71

7 3.3e-3 0.99 8.6e+0 -2.10 2.1e-2 0.86 1.1e-3 0.23

Table 4. Example (1); errors and convergence orders with ρ− = 1 and ρ+ = 104,using method (2.5)-(6.2) with stabilization parameters γ = 10.

The results in Table 4 show optimal asymptotical convergence for the errors e1h and e0

h.In addition we observe a sub-optimal convergence for the error e∞h , approximately 1.8.

The error e1,∞h does not seem to converge. As in the previous test the error e1

h convergesasymptotically to 1, however for the rest of the errors we do not observe convergence.

(2)


Figure 3. Approximate solution Example (1) (left) case Γ = ∂Ω− and for caseΓ = ∂Ω+ (right).


Appendix A. Proof of Lemma 4.

Let T1, T2 ∈ T Γh be such that e = T1 ∩ T2 and e± = e ∩ Ω±. We analyze the case in Ω− but

the same analysis is valid in Ω+. Note that in the case either T−1 or T−2 contains an entire edgethe result is trivial. In consequence, let us analyze the remaining case, see Figure 4.

Let αi for i = 1, 2 be the angle of the triangle Ti associated to e−, and let li be the part ofthe edge of Ti contained in Ω− such that this edge with e generate the angle αi. Let L denotethe segment connecting l1 and l2. These definitions are illustrated in Figure 4.

Figure 4. Proof of Lemma 4, illustration of definitions.

Due to our assumptions on the mesh (shape regularity) we have a lower bound α for the anglesof the triangles of the triangulation. Let TΓ

1 and TΓ2 be the porting of the interface contained

in the triangles T1 and T2, respectively. Consider the common vertex p of T1 and T2 in Ω− andthe points x1 = X(t1) and x2 = X(t2) defined as

xi ∈ TΓi : ri := ‖xi − p‖ = min

x∈TΓi

‖x− p‖

Computing the semi circular areas we obtain

|T−i | ≥1

2αir

2i ≥

1

2αr2

i , i = 1, 2.

We divide our analysis in two cases. Firstly, assume that |e−| is one of these distances, withoutloss of generality say |e−| = r1, then since the semi circular area is contained in |T−1 | the proofis complete as follows

maxi|T−i | ≥

1

2α1|e−|2.

On the other hand, suppose that r1, r2 < |e−|. Consider the distance function to the point p

dp(t) = ‖p−X(t)‖2.

Since the curve Γ is smooth and r1, r2 < |e−|, imply that there exists an interior maximum fordp(t), a point on the curve X(tmax) between t1 and t2 such that d′p(tmax) = 0 and d′′p(tmax) < 0.Then, Taylor expansion around tmax gives, for t∗ ∈ (t, tmax)

dp(t) = dp(tmax) +d′′p(t

∗)

2(t− tmax)2.


Thus, we have for t1

dp(t1) = r21 = dp(tmax) +

d′′p(t∗1)

2(t1 − tmax)2.

Observe that d′′p(t∗1) < 0, otherwise we have a contradiction with the assumption r2

1 < dp(tmax).Now, using the arc-length parameterization of Γ and similarly to the proof of Lemma 9, thereexists a constant m1, depending only on ‖X ′′‖L∞ , such that

(t1 − tmax)2 ≤ m1dp(tmax).

Thus, we have

r21 ≥ dp(tmax) +m1d

′′p(tmax)dp(tmax) = dp(tmax)(1 +m1d

′′p(tmax)).

Observe that

d′′p(t∗1) = 2(X ′21 (t∗1) +X ′22 (t∗1)) + 2(X1(t∗1)− p1)X ′′1 (t∗1) + 2(X2(t∗1)− p2)X ′′2 (t∗1)

= 2 + 2(X1(t∗1)− p1)X ′′1 (t∗1) + 2(X2(t∗1)− p2)X ′′2 (t∗1)

=⇒ d′′p(t∗1) ≥ 2− 2h‖X ′′‖L∞

where, X = (X1, X2) and p = (p1, p2). The last inequality shows that for h small enough,h < (3/2 +m1)/(m1‖X ′′‖L∞), we have the result

r21 ≥ dp(tmax)(1 +m1(2− 2h‖X ′′‖L∞)) ≥ dp(tmax)

2≥ |e

−|2

2,

and hence we can conclude that

maxi|T−i | ≥ |T

−1 | ≥

1

4α|e−|2.

Appendix B. Proof of Lemma 9.

We analyze the case in Ω−. Consider the piece of interface TΓ = T ∩Γ and the line segmentLT that interpolates TΓ and its endpoints. We first observe that

|LT | ≤3∑i=1

|e−i | ≤ 3 maxi=1,2,3

|e−|.

Assuming that Γ is smooth enough with arc-length parametrization X (C2 for example), and|LT | < 1 we have that

|TΓ| ≤ |LT |+|LT |3

8‖X ′′‖L∞ ≤ (1 +

1

8‖X ′′‖L∞)|LT |.

Therefore, we conclude that there exists a constant m, only depending on the smoothness ofΓ, such that

|TΓ| ≤ 3(1 +1

8‖X ′′‖L∞) max

i=1,2,3|e−i | = m max

i=1,2,3|e−i |.

Same analysis is valid to prove the statement in Ω+.


References

[1] Slimane Adjerid, Mohamed Ben-Romdhane, and Tao Lin. Higher degree immersed finite element methodsfor second-order elliptic interface problems. Int. J. Numer. Anal. Model., 11(3):541–566, 2014.

[2] Nelly Barrau, Roland Becker, Eric Dubach, and Robert Luce. A robust variant of NXFEM for the interfaceproblem. C. R. Math. Acad. Sci. Paris, 350(15-16):789–792, 2012.

[3] Roland Becker, Erik Burman, and Peter Hansbo. A Nitsche extended finite element method for incompressibleelasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Engrg., 198(41-44):3352–3360, 2009.

[4] Susanne C. Brenner and L. Ridgway Scott. The mathematical theory of finite element methods, volume 15 ofTexts in Applied Mathematics. Springer-Verlag, New York, 1994.

[5] Erik Burman. Ghost penalty. C. R. Math. Acad. Sci. Paris, 348(21-22):1217–1220, 2010.[6] Erik Burman and Peter Hansbo. Fictitious domain methods using cut elements: III. A stabilized Nitsche

method for Stokes’ problem. ESAIM Math. Model. Numer. Anal., 48(3):859–874, 2014.[7] Erik Burman and Paolo Zunino. Numerical approximation of large contrast problems with the unfitted

Nitsche method. In Frontiers in numerical analysis—Durham 2010, volume 85 of Lect. Notes Comput. Sci.Eng., pages 227–282. Springer, Heidelberg, 2012.

[8] C.-C. Chu, I. G. Graham, and T.-Y. Hou. A new multiscale finite element method for high-contrast ellipticinterface problems. Math. Comp., 79(272):1915–1955, 2010.

[9] Maksymilian Dryja and Olof B. Widlund. Domain decomposition algorithms with small overlap. SIAM J.Sci. Comput., 15(3):604–620, 1994. Iterative methods in numerical linear algebra (Copper Mountain Resort,CO, 1992).

[10] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order, volume 224of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].Springer-Verlag, Berlin, second edition, 1983.

[11] Yan Gong, Bo Li, and Zhilin Li. Immersed-interface finite-element methods for elliptic interface problemswith nonhomogeneous jump conditions. SIAM J. Numer. Anal., 46(1):472–495, 2007/08.

[12] Yan Gong and Zhilin Li. Immersed interface finite element methods for elasticity interface problems withnon-homogeneous jump conditions. Numer. Math. Theory Methods Appl., 3(1):23–39, 2010.

[13] Anita Hansbo and Peter Hansbo. An unfitted finite element method, based on Nitsche’s method, for ellipticinterface problems. Comput. Methods Appl. Mech. Engrg., 191(47-48):5537–5552, 2002.

[14] Peter Hansbo. Nitsche’s method for interface problems in computational mechanics. GAMM-Mitt., 28(2):183–206, 2005.

[15] Xiaoming He, Tao Lin, and Yanping Lin. Immersed finite element methods for elliptic interface problemswith non-homogeneous jump conditions. Int. J. Numer. Anal. Model., 8(2):284–301, 2011.

[16] Xiaoming He, Tao Lin, and Yanping Lin. The convergence of the bilinear and linear immersed finite elementsolutions to interface problems. Numer. Methods Partial Differential Equations, 28(1):312–330, 2012.

[17] Tao Lin, Yanping Lin, and Xu Zhang. Partially penalized immersed finite element methods for ellipticinterface problems. SIAM J. Numer. Anal., 53(2):1121–1144, 2015.

[18] Andre Massing, Mats G. Larson, Anders Logg, and Marie E. Rognes. A stabilized Nitsche fictitious domainmethod for the Stokes problem. J. Sci. Comput., 61(3):604–628, 2014.

[19] Andre Massing, Mats G. Larson, Anders Logg, and Marie E. Rognes. A stabilized Nitsche overlapping meshmethod for the Stokes problem. Numer. Math., 128(1):73–101, 2014.

[20] Paolo Zunino. Analysis of backward Euler/extended finite element discretization of parabolic problems withmoving interfaces. Comput. Methods Appl. Mech. Engrg., 258:152–165, 2013.

[21] Paolo Zunino, Laura Cattaneo, and Claudia Maria Colciago. An unfitted interface penalty method for thenumerical approximation of contrast problems. Appl. Numer. Math., 61(10):1059–1076, 2011.


† Division of Applied Mathematics, Brown University, Providence, RI 02912, USAE-mail address: johnny [email protected]

E-mail address: manuel sanchez [email protected]

‡Department of Mathematical Sciences at Worcester Polytechnic Institute, 100 Institute Road,Worcester, MA 01609, USA

E-mail address: [email protected]

introduction - brown university · fem for interface 3 and the bilinear form a h(;) in section...

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