TRITA-­NA 2011:1 | ISRN KTH/NA-­-­11/1-­-­SE | ISSN 0348-­2952

Stockholm 2011

NA – Numerical Analysis

School of Computer Sciense and Communication

KTH Royal Institute of Technology

Sara Zahedi, Eddie Wadbro, Gunilla Kreiss, Martin Berggren

TECHNICAL REPORT

method for interface problems: Part I

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School of Computer Science and Communication

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Sara Zahedi, Eddie Wadbro, Gunilla Kreiss, Martin Berggren

NA – Numerical Analysis

Publicationsdate: June 2011

E-­mail to author: sara7@csc.kth.se

TRITA-­NA 2011:1

ISRN KTH/NA-­-­11/1-­-­SE

ISSN 0348-­2952

A uniformly well-conditioned, unfitted Nitschemethod for interface problems: Part I∗

Sara Zahedi† Eddie Wadbro‡ Gunilla Kreiss§ Martin Berggren‡

Abstract

A finite element method for elliptic partial differential equations thatallows for discontinuities along an interface not aligned with the mesh ispresented. The solution on each side of the interface is separately ex-panded in standard continuous, piecewise-linear functions, and a variantof Nitsche’s method enforces the jump conditions at the interface. Inthis method, the solutions on each side of the interface are extended tothe entire domain, which results in a fixed number of unknowns indepen-dent of the location of the interface. A stabilization procedure is includedto ensure well-defined extensions. Numerical experiments are presentedshowing optimal convergence order in the energy and L2 norms, and alsofor pointwise errors. The presented results also show that the conditionnumber of the system matrix is independent of the position of the interfacerelative to the grid.

1 Introduction

There is a growing interest in accurate and efficient numerical methods forproblems in which the solution exhibits strong or weak discontinuities—thatis, jumps in the solution or its derivatives—along a surface in the interior ofthe solution domain. The application we have in mind is two-phase flow wheredifferences in fluid viscosity lead to a weak discontinuity in the velocity fieldand the surface-tension force gives rise to a discontinuous pressure. The weakand strong discontinuities are for two-phase flows typically formulated as jumpconditions for the normal stress at the interface [12].

Weak discontinuities in the solution of a partial differential equation can ac-curately be approximated by globally continuous finite-element functions, pro-vided that the interface coincides with mesh lines. Such fitting of the mesh to

∗TRITA-NA 2011:1, ISRN KTH/NA–2011/01–SE, ISSN 0348-2952

†School of Computer Science and Communication, Royal Institute of Technology, SE–

100 44 Stockholm, Sweden

‡Department of Computing Science, Umea University, SE–901 87 Umea, Sweden

§Department of Information Technology, Uppsala University, Box 337, SE–751 05 Uppsala,

Sweden

1

2

the interface allows the construction of finite-element methods with optimal-order accuracy even in the presence of weak discontinuities [1, 5]. The mortarmethod introduces Lagrange multipliers supported on the interface to enforceweak discontinuities, allowing mesh vertices to be non-matching at the interfacewithout loss of accuracy [11, 13]. But the interface still needs to be aligned withmesh lines; otherwise, the approximation order will be ruined.

The requirement that the mesh should conform to the interface leads tosignificant complications for two-phase-flow simulations, where the interfaceevolves with time and may undergo topological changes. From this point ofview, it would be preferable to use a method in which the interface can bearbitrarily located with respect to a fixed background mesh.

One such approach involves regularization at the interface [16, 18]. In themultiphase flow case, the discontinuous density and viscosity are then approx-imated by smooth functions, while a volume force approximates the singularsurface tension. However, such regularization methods always lose accuracynear the interface. Methods that allow for true discontinuities internal to theelements have the prospect of being more accurate. Examples of such methodsare the extended finite element method XFEM [2, 6, 7, 8] and the unfitted fi-nite element method by Hansbo and Hansbo [9]. In the extended finite elementmethod, the finite element space is enriched with particular discontinuous basisfunctions. In the method by Hansbo and Hansbo [9], the discrete solution is con-structed from separate solutions defined on each subdomain, and the jump con-ditions at the interface are enforced by an extension of Nitsche’s method for theweak enforcement of essential boundary conditions [14]. Hansbo and Hansbo [9]show optimal convergence order without restrictions on the location of the in-terface. However, in both XFEM and the method of Hansbo and Hansbo, theconditioning of the problem is sensitive to the position of the interface. Thecondition number of the system matrix blows up for cases when the interfaceapproaches element boundaries. For unsteady problems, it is not unusual thatsuch situations occur, and some precaution is needed to prevent problems suchas breakdown of direct or iterative linear solvers. Reusken [15] addresses thisproblem by deleting basis functions in the XFEM space that have very smallsupport and may cause ill-conditioning. Burman and Hansbo [4] stabilize theclassical Nitsche’s method for the imposition of inhomogeneous Dirichlet con-ditions by adding a penalty term for normal-derivative jumps between thoseelements for which at least one of them crosses the boundary. Burman [3] con-siders the same problem, but stabilizes the method instead by a penalty termthat contains the difference between the solution and an L2-projection of thesolution on a patch of elements in the vicinity of the boundary.

In this paper we propose a simple method for elliptic PDEs that allows fordiscontinuities internal to the elements. Our method is similar to the method byHansbo and Hansbo [9], as it uses standard linear finite elements and a versionof Nitsche’s method to enforce the jump conditions at the interface. However,in our method, the solutions on the sub-domains separated by the interface areextended to the entire domain by a stabilization procedure that is very easyto implement. This approach yields a fixed problem size, which simplifies the

3

implementation. We can prove that the method is optimal-order accurate andthat the stabilization procedure yields a matrix condition number of O(h−2),independently of the position of the interface relative to the mesh. Moreover,we choose the coefficients in the Nitsche numerical fluxes in order to minimizethe penalty parameter. Numerical experiments indicate that this choice of coef-ficients has a positive impact on the pointwise errors at the interface as well ason the condition number of the matrix. The new method is applied to a steadydiffusion problem in two space dimensions. We allow for a discontinuous diffu-sion coefficient, and the solution may have both weak and strong discontinuitiesacross a smooth interface internal to the elements. We expect the method to beapplicable also to multiphase flow problems.

The outline of this article is as follows. In Section 2 we formulate the modelinterface problem, and derive a variational formulation involving averaging op-erators and a penalty term to impose the interface conditions. In Section 3 theaveraging operators are defined so that the penalty parameter is minimized, andwe introduce the discrete solution space and the stabilization procedure. Theresulting finite dimensional variational problem is shown to be consistent. InSection 4 we prove continuity and coercivity. Moreover, we state an approxima-tion result and a convergence result establishing optimal order of convergencefor linear elements in the mesh dependent norm associated with the bilinearform as well as in the L2 norm. The section ends with a theorem boundingthe condition number of the corresponding system matrix independently of thelocation of the interface. Because of space concerns, the proofs of the approx-imation result, optimal order of convergence, and the bound on the conditionnumber are deferred to a second part of this article [17]. In Section 5 numericalexamples in one and two space dimensions are shown. The proposed methodis compared to existing numerical methods; the optimal convergence order andthe bound on the condition number are confirmed. We end the article with ashort discussion.

2 Problem formulation

2.1 The model interface problem

Let the domain Ω be an open, bounded, and connected point set in R2 with aconvex polygonal boundary ∂Ω. We assume that there is, as illustrated in fig-ure 1, a smooth internal boundary Γ that divides Ω into two open and connectedsets, Ω1 and Ω2, where Ω1 is strictly included in Ω, which means that ∂Ω1 = Γand ∂Ω∩ Γ = ∅. The outward-directed (with respect to Ω1) unit normal vectoron Γ and ∂Ω is denoted n, and directional derivatives in this normal directionare denoted ∂n .

Let u : Ω → R be a function whose restrictions to Ω1 and Ω2 are denoted u1

4

Ω2

Ω1

Γn

Figure 1: The internal smooth boundary Γ divides the computational domainΩ into two connected domains, an interior Ω1 and an exterior Ω2.

and u2, respectively. We consider the following boundary-value problem for u:

−∇ · (µ1∇u1) = f1 in Ω1, (1a)

−∇ · (µ2∇u2) = f2 in Ω2, (1b)

u2 = 0 on ∂Ω, (1c)

u = b on Γ, (1d)

µ ∂nu = g on Γ, (1e)

where µ1, µ2 > 0 are constants (but typically different), and where we use thenotation

u = u1 − u2, µ ∂nu = µ1∂nu1 − µ2∂nu2 (2)

for jumps of the function and its normal derivative on Γ. The functions f1, f2,b, and g are given functions.

2.2 A variational formulation of Nitsche type

We will derive a variational formulation of problem (1) in which jump condi-tions (1d) and (1e) are imposed through a version of Nitsche’s method. Thevariational form is based on an integration-by-parts formula that makes use ofthe averaging operators

f = κ1f1 + κ2f2, f∗ = κ2f1 + κ1f2, (3)

in which κ1 and κ2 are real numbers, to be specified in section 3.1, satisfyingκ1 + κ2 = 1.

Lemma 2.1. Let ψi ∈ H1(Ωi), ϕi ∈ H1(Ωi)2, i = 1, 2, and define ψ ∈ L2(Ω),ϕ ∈ L2(Ω)2 from

ψ =

ψ1 in Ω1,

ψ2 in Ω2,ϕ =

ϕ1 in Ω1,

ϕ2 in Ω2.(4)

5

Then

Ω1∪Ω2

(ψ∇ · ϕ+∇ψ · ϕ) =

∂Ωn · ϕψ +

Γn · ϕ ψ +

Γn · ϕ ψ∗ . (5)

Remark 2.1. Throughout this article, we surpress the symbols of measure(such as dx ) in all integrals. The kind of measure used in the integrals will beclear from the domain of integration.

Proof. Integration by parts yields

Ω1∪Ω2

(ψ∇ · ϕ+∇ψ · ϕ) =

∂Ωn · ϕψ +

Γn · (ϕ1ψ1 − ϕ2ψ2). (6)

The last integrand can be rewritten, using that κ1 + κ2 = 1, as follows:

ϕ1ψ1 − ϕ2ψ2 = (κ1 + κ2)ϕ1ψ1 − (κ1 + κ2)ϕ2ψ2

[add and subtract] = (κ1 + κ2)ϕ1ψ1 + κ1ϕ1ψ2 − κ1ϕ1ψ2

− (κ1 + κ2)ϕ2ψ2 + κ2ϕ2ψ1 − κ2ϕ2ψ1

[collecting terms] = (κ1ϕ1 + κ2ϕ2)(ψ1 − ψ2) + (ϕ1 − ϕ2)(κ1ψ2 + κ2ψ1)

= ϕ ψ + ϕ ψ∗ ,

(7)

from which the claim follows.

Now introduce the spaces of weak solutions to problem (1),

V1 = H1(Ω1), V2 =v ∈ H1(Ω2)

v|∂Ω = 0, (8)

where (·)|∂Ω denotes the trace operator from Ω2 to ∂Ω, the following bilinearand linear forms,

a(u1, u2, v1, v2) =

Ω1

µ1∇u1 ·∇v1 +

Ω2

µ2∇u2 ·∇v2

−

Γu µ ∂nv−

Γv µ ∂nu+

Γλ u v , (9a)

l(v1, v2) =

Ω1

f1v1 +

Ω2

f2v2 +

Γg v∗ −

Γb µ ∂nv+

Γλb v , (9b)

and the following variational characterization.

Theorem 2.2. Assume that u1 ∈ V1 ∩ H2(Ω1), u2 ∈ V2 ∩ H2(Ω2) solve theboundary-value problem (1). Then u1, u2 satisfies

a(u1, u2, v1, v2) = l(v1, v2) (10)

for each v1 ∈ V1 ∩H2(Ω1), v2 ∈ V2 ∩H2(Ω2) and for each λ ∈ L∞(Γ).

6

Proof. Let ϕ ∈ L2(Ω)2 be defined by ϕ|Ωi = µi∇ui and ψ ∈ L2(Ω) by ψ|Ωi = vi,for arbitrary vi ∈ Vi ∩H2(Ωi) i = 1, 2. Lemma 2.1 and equation (1) yields that

Ω1

µ1∇u1 ·∇v1 +

Ω2

µ2∇u2 ·∇v2 −

Γv µ ∂nu

=

Ω1

f1v1 +

Ω2

f2v2 +

Γg v∗ .

(11)

From boundary condition (1d) follows that

−

Γu µ ∂nv+

Γb µ ∂nv = 0 (12)

and

Γλ u v −

Γλb v = 0, (13)

for each λ ∈ L∞(Γ). Adding expressions (11), (12), and (13) yields the claim.

3 Discretization

3.1 Discretization in locally-defined spaces

Let T h be a triangulation of Ω, generated independently of the location of theinterface Γ; that is, the mesh is not assumed to be fitted to Ω1 and Ω2. Denotethe diameter of element K ∈ T h by hK and let

h = maxK∈T h

hK ≤ 1, (14)

which means that the problem is scaled so that diam(Ω) ≤ 1. We assume thatthe family of triangulations T h h>0 is non-degenerate; that is, there exists aCρ > 0 such that, uniformly with respect to h,

ρKhK

≥ Cρ, ∀K ∈ T h, (15)

where ρK is the diameter of the largest ball contained in element K. LetT h(ω) ⊂ T h denote the collection of elements that exhibit nonempty inter-section with ω ⊂ Ω, that is

T h(ω) =K ∈ T h | K ∩ ω = ∅

. (16)

Denote the set of elements for which the interface intersects the element with apositive surface measure by

Gh =K ∈ T h

|Γ ∩K| > 0, (17)

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and letGh =

K∈Gh

K. (18)

We also define the set

Gh = K ∈ Gh K ∩ Ω2 = ∅

, (19)

so that when a part of Γ coincides with an element edge, both elements sharingthat edge belong to Gh but only one of the two elements belongs to Gh.

For any x ∈ [0, 1] and i = 1, 2, denote the set of elements with a relativearea in Ωi less than x by

Ghi,x =

K ∈ Gh | |K ∩ Ωi|

|K| < x

, (20)

and letGh

i,x =

K∈Ghi,x

K. (21)

We make the following assumptions about the mesh.

Assumption 3.1. The curvature of the interface Γ is well resolved by the mesh.That is, for elements K ∈ Gh, the radius of curvature of interface portion Γ∩Kis greater than ChK , with the constant C 1.

Assumption 3.2. For each element K ∈ Gh we assume that either Γ intersectsthe element boundary ∂K exactly twice and each open edge at most once, or thatΓ ∩K coincides with an edge of the element.

Assumption 3.3. There exists a constant C < ∞ such that, uniformly withrespect to h, for any x > 0 and i = 1, 2,

Ghi,x

≤ C√xGh

. (22)

The third assumption is not very restrictive. For unstructed meshes and aninterface Γ that is well resolved by the mesh we expect that

cardGhi,x

≈ x cardGh. (23)

The fundamental approximation space consists of continuous, piecewise-linear functions on Ω,

V h =v ∈ C0(Ω)

v|K ∈ P1(K), ∀K ∈ T h, v|∂Ω = 0, (24)

where P1(K) is the space of linear polynomials on K. Now define the spaces ofrestrictions of V h on Ω1 and Ω2,

V hi = V h|Ωi , i = 1, 2. (25)

8

A finite-element approximation of boundary-value problem (1) is then the fol-lowing:

Find uh1 ∈ V h

1 and uh2 ∈ V h

2 such that

a(uh1 , u

h2 , v

h1 , v

h2 ) = l(vh1 , v

h2 ) ∀vh1 ∈ V h

1 , vh2 ∈ V h2 .

(26)

An immediate consequence of Theorem 2.2 is that problem (26) is a consistentapproximation of boundary-value problem (1) for each combination κ1, κ2 suchthat κ1 + κ2 = 1 and for each positive λ ∈ L∞(Γ).

Hansbo and Hansbo [9] introduced formulation (26) for the case of functionscontinuous at Γ (u = 0, b = 0). They chose, for i = 1, 2

κi|Γ∩K =|K ∩ Ωi|

|K| . ∀K ∈ Gh, (27)

and proved optimal order of convergence under the assumption that Γ intersectseach element in Gh exactly twice and each element edge only once. However,as we demonstrate in Section 5, this method is sensitive to the position of Γ inthe sense that the condition number of the discretized system may blow up asΓ ∩ K approaches ∂K, and it does not allow Γ ∩ K to coincide with elementedges.

We propose an optimal-order accurate method that differs from the methodof Hansbo and Hansbo in the following ways. First, the condition numberof the equation system will be bounded independently of the location of theinterface. Second we allow the interface to coincide with mesh boundaries.Third, the choices of κ1 and κ2 are different than in expression (27) and aremade to minimize the size of a locally defined penalty parameter λ. Finally,the method will utilize a fixed number of unknowns independent of the positionof the interface, which means that the method can be viewed as a “fictitiousdomain method”.

Regarding the choices of κ1, κ2, and λ, there are two cases, in accordancewith the intersection options of Assumption 3.2.

• For each element K ∈ Gh such that Γ intersects the element boundaryexactly twice, we have |K ∩ Ωi| = αi,Kh2

K and |Γ ∩K| = γKhK for someαi,K , γK > 0, and we define

κ1|Γ∩K =µ2α1,K

µ1α2,K + µ2α1,K, κ2|Γ∩K =

µ1α2,K

µ1α2,K + µ2α1,K, (28)

andλK = λ|Γ∩K =

ηKhK

, (29)

where

ηK = D +(1 +B)γKµ1µ2

A(µ1α2,K + µ2α1,K), A ∈ (0, 1); B,D > 0. (30)

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• If Γ ∩K coincides with an element edge, then Γ ∩K will also be an edgeof another triangle T ∈ Gh and Γ ∩K = Γ ∩ T . We may without loss ofgenerality assume that T ⊂ Ω1 and K ⊂ Ω2. We may write |T | = αTh

2T

and |K| = αKh2K for some αT , αK > 0, and |Γ ∩K| = |Γ ∩ T | = γKhK =

γThT for some γK , γT > 0. We define

κ1|Γ∩K =µ2αT

µ1αK (γT /γK)2 + µ2αT

,

κ2|Γ∩K =µ1αK

µ1αK + µ2αT (γK/γT )2 ,

(31)

andλK = λ|Γ∩K =

ηK|Γ ∩K|

, (32)

where

ηK = D +(1 +B)µ1µ2

A ((µ1αK)/γ2K + (µ2αT )/γ2

T ), A ∈ (0, 1); B,D > 0. (33)

Note that 0 ≤ κi ≤ 1 and κ1 + κ2 = 1 in both cases above. In Lemma 4.3,we prove that aε is coercive for all choices of the constants A ∈ (0, 1) andB,D > 0. (The coercivity constant depends on these constants, however.) Inthe numerical experiments, we use A = 0.5 and B = D = 1.

3.2 Extending and stabilizing the solution spaces

The dimensions of the solution spaces introduced above, V h1 and V h

2 , dependon the location of the interface. A simple way to fix the number of unknownsindependently of Γ is to replace the local spaces V h

i with the globally definedV h, both for uh

1 and uh2 . Then u1 will be defined also inside Ω2 and vice versa

for u2. The number of unknowns will then always equal twice the number ofunknowns associated with V h. Unfortunately, problem (26) does not provideequations for (most of) the “ghost functions” uh

1 |Ω2 and uh2 |Ω1 . We will therefore

introduce a stabilization procedure that makes uh1 |Ω2 and uh

2 |Ω1 well defined.The challenge is to introduce a stabilization that yields a well-conditioned linearsystem without destroying the accuracy of the proper solutions uh

1 |Ω1 and uh2 |Ω2 .

Given a triangulation T h with N vertices, denote by ϕj the standard nodalbasis function, associated with mesh vertex j, for continuous, piecewise-linearfunctions. We have

V h = span ϕj Nj=1 . (34)

We propose to replace problem (26) with the following stabilized problem.

Find uh1 uh

2 ∈ V h such that

aε(uh1 , u

h2 , v

h1 , v

h2 ) = l(vh1 , v

h2 ) ∀vh1 , vh2 ∈ V h,

(35)

10

Ω1

Ω2

Γ

Figure 2: Illustration of the effects of projector Ph2,τ in definition (37); Ph

2,τϕj =

ϕj for all j marked with • and Ph2,τϕj = 0 for all j marked with .

where

aε(uh1 , u

h2 , v

h1 , v

h2 ) = a(uh

1 , uh2 , v

h1 , v

h2 )

+ ε

Ω2

Ph2,τu

h1P

h2,τv

h1 + ε

Ω1

Ph1,τu

h2P

h1,τv

h2 ,

(36)

and where the projection operators Phi,τ on V h|Ωi are defined through their

action on the basis functions, as detailed below. Letting τK > 0 denote a localtolerance parameter, we define the global tolerance parameter τ = maxK∈T h τKand projection operators

Ph1,τϕj =

ϕj when

|K ∩ Ω2||K| < τK ∀K ∈ T h(suppϕj),

0 otherwise;

Ph2,τϕj =

ϕj when

|K ∩ Ω1||K| < τK ∀K ∈ T h(suppϕj),

0 otherwise.

(37)

The stabilization terms in the bilinear form (36) add an ε fraction of the massmatrix to the rows and columns associated with those basis functions that areundetermined—or almost undetermined, as measured by parameters τK—bythe bilinear form a. Figure 2 exemplifies the action of the stabilization. Nodalvalues of function uh

1 at mesh vertices marked with the symbol can all bestably determined by the original unstabilized equation (26). However, thenodal values in the middle of the colored region would be almost undeterminedby equation (26), since the support of the corresponding basis function onlybarely intersects Ω1, and nodal values of uh

1 at the rest of the vertices markedwith • are completely undetermined by equation (26). The inclusion of thestabilization term involving Ph

2,τ in the form (36) will make solution uh1 well

determined also at the vertices marked with •.The proof of consistency of problem (35) follows almost immediately after a

formal extension by zero of the ui’s to the whole of Ω:

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Theorem 3.1. For i = 1, 2, let ui ∈ L2(Ω) such that ui|Ωi ∈ Vi ∩H2(Ωi) solvethe boundary-value problem (1) and such that ui|Ω\Ωi

≡ 0. Then

aε(u1, u2, vh1 , v

h2 ) = l(vh1 , v

h2 ) ∀vh1 , vh2 ∈ V h. (38)

Proof. Since u1|Ω2 and u2|Ω1 vanish identically, they are (trivial) objects inV h|Ω2 and V h|Ω1 , respectively. Thus, Ph

2,τu1|Ω2 and Ph1,τu2|Ω1 are well defined

and both vanishing. Hence,

aε(u1, u2, vh1 , v

h2 ) = a(u1, u2, v

h1 , v

h2 ) = l(vh1 , v

h2 ) ∀vh1 , vh2 ∈ V h, (39)

where the second equality follows from Theorem 2.2.

4 Analysis

In this section we show that aε with parameters chosen according to expres-sions (28)–(33) is continuous and coercive. We will also state an approximationresult, a convergence result, and an estimate of the condition number of thesystem matrix corresponding to problem (35).

In Section 3.2, we extended the numerical solutions uh1 , u

h2 in order to make

them well defined throughout Ω. Moreover, in Theorem 3.1, we introduceda formal extension by zero of the exact solutions u1, u2 so that also both ofthese are defined throughout Ω. This section therefore uses u = (u1, u2) anduh = (uh

1 , uh2 ) to denote the pair of extended solutions.

Remark 4.1. To retain the previous meanings of jumps and averages of thesolution along the interface as defined in expressions (2) and (3), the notationui on Γ should be interpreted as the trace of ui|Ωi (as opposed to ui|Ω\Ωi

, whichis zero!) on Γ.

The analysis will be carried out in the following mesh dependent norm:

|||v|||2 = ∇v120,Ω1+ ∇v220,Ω2

+ v21/2,h,Γ+ ∂nv2−1/2,h,Γ +

Ph2,τv1

20,Ω2

+Ph

1,τv220,Ω1

,(40)

where, for functions w ∈ L2(Γ),

w21/2,h,Γ =

K∈Gh

h−1K w20,Γ∩K , (41)

andw2−1/2,h,Γ =

K∈Gh

hK w20,Γ∩K . (42)

The Cauchy–Schwarz inequality and definitions (41) and (42) yield the inequal-ity

Γwz ≤ w1/2,h,Γ z−1/2,h,Γ . (43)

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In order to provide an error estimate in the norm (40), we need to prove continu-ity and coercivity of the form aε with respect to this norm and an interpolationestimate in the same norm.

4.1 Continuity and coercivity

In this section we show that aε, with parameters chosen according to expres-sions (28)–(33), is continuous and coercive with respect to the triple-norm |||·|||.

Lemma 4.1. Given any A ∈ (0, 1), B, D > 0 in expressions (30) and (33),there exists a constant C such that

aε(u1, u2, v1, v2) ≤ (C + 2ε) |||u||| |||v||| (44)

for ui, vi ∈ L2(Ω), ui|Ωi , vi|Ωi ∈ Vi ∩H2(Ωi), and ui|Ω\Ωi, vi|Ω\Ωi

∈ V h|Ω\Ωi,

i = 1, 2.

Proof. First we recall that aε is given by

aε(u1, u2, v1, v2) =

Ω1

µ1∇u1 ·∇v1 +

Ω2

µ2∇u2 ·∇v2 −

Γu µ ∂nv

−

Γv µ ∂nu+

Γλ u v + ε

Ω2

Ph2,τu1P

h2,τv1 + ε

Ω1

Ph1,τu2P

h1,τv2.

(45)The two first terms can be bounded using the Cauchy–Schwarz inequality,

Ωi

µi∇ui ·∇vi ≤ µi ∇ui0,Ωi∇vi0,Ωi

≤ µi |||u||| |||v||| . (46)

The following two terms in aε can be bounded using inequality (43),

Γu µ ∂nv ≤ max

i=1,2µi u1/2,h,Γ ∂nv−1/2,h,Γ ≤ max

i=1,2µi |||u||| |||v||| ,

(47)and similarly,

Γv µ ∂nu ≤ max

i=1,2µi |||u||| |||v||| . (48)

For the fifth term in aε, we first recall from defintions (29) and (32) that λK =ηK/hK if Γ intersectsK twice and λK = ηK/|Γ∩K| ≤ ηK/ρK , if Γ∩K coincideswith the element edge. Thus in both cases, we have that λK ≤ ηK/(CρhK),where Cρ is the constant of condition (15). We define

Cλ =1

CρmaxK∈Gh

ηK , (49)

so that λ|Γ∩K = λK ≤ Cλ/hK . By using the Cauchy–Schwarz inequality, we

13

bound the fifth term of aε as

Γλ u v ≤

Γλ u2

1/2

Γλ v2

1/2

≤

K∈Gh

Cλ

hKu20,Γ∩K

1/2

K∈Gh

Cλ

hKv20,Γ∩K

1/2

= Cλ u1/2,h,Γ v1/2,h,Γ ≤ Cλ |||u||| |||v||| .

(50)

Finally, applying the Cauchy–Schwarz inequality on the last two terms of aεgives

ε

Ω2

Ph2,τu1P

h2,τv1 ≤ ε

Ph2,τu1

0,Ω2

Ph2,τv1

0,Ω2

≤ ε |||u||| |||v||| (51)

and similarly,

ε

Ω1

Ph1,τu2P

h1,τv2 ≤ ε |||u||| |||v||| . (52)

Hence, by inequalities (46), (47), (48), (50), and (51),

aε(u1, u2, v1, v2) ≤µ1 + µ2 + 2max

i=1,2µi + Cλ + 2ε

|||u||| |||v||| . (53)

The coercivity proof will use the following inverse inequality.

Lemma 4.2. Assume that K is an element in Gh such that, for i = 1 or 2,|K ∩ Ωi| = αi,Kh2

K , where αi,K > 0 and let |Γ ∩K| = γKhK . For any functionvh ∈ V h, we have,

hK

κi ∂nvh2

0,Γ∩K≤ κ2

i γKαi,K

∇vh20,K∩Ωi

. (54)

Proof. Since ∇vh|K is constant for vh ∈ V h and by the Cauchy-Schwarz in-equality, we find that

hK

κi ∂nvh2

0,Γ∩K≤ hKκ2

i |Γ ∩K|∇vh|K

2

= κ2ihK |Γ ∩K||K ∩ Ωi|

∇vh20,K∩Ωi

=κ2i γKαi,K

∇vh20,K∩Ωi

.

(55)

Now we are ready to prove coerciveness of aε for functions in V h.

14

Lemma 4.3. Given any A ∈ (0, 1), B, D > 0 in expressions (30) and (33),there exists a positive constant C independent of the location of the interfacesuch that

aε(v1, v2, v1, v2) ≥ min(C, ε) |||v|||2 ∀v1, v2 ∈ V h. (56)

Proof. By the definition of aε, for each v1, v2 ∈ V h,

aε(v1, v2, v1, v2) = µ1 ∇v120,Ω1+ µ2 ∇v220,Ω2

+ εPh

2,τv120,Ω2

+ εPh

1,τv220,Ω1

+

Γλ v2 − 2

Γv µ ∂nv .

(57)

By the Cauchy–Schwarz and arithmetic–geometric mean inequalities, the lastterm in expression (57) can be bounded as follows:

2

Γv µ ∂nv = 2

2

i=1

K∈Gh

Γ∩Kv (µiκi ∂nvi)

≤

K∈Gh

2

i=1

δi,K|Γ ∩K|

Γ∩Kv2

+

K∈Gh

2

i=1

|Γ ∩K|δi,K

Γ∩K(µiκi ∂nvi)

2,

(58)where δi,K are positive constants defined below. Substituting inequality (58)into expression (57) yields

aε(v1, v2, v1, v2) ≥ µ1 ∇v120,Ω1+ µ2 ∇v220,Ω2

+ εPh

2,τv120,Ω2

+ εPh

1,τv220,Ω1

+

K∈Gh

λK − δ1,K + δ2,K

|Γ ∩K|

v 2

0,Γ∩K

−2

i=1

K∈Gh

|Γ ∩K|µ2i

δi,Kκi ∂nvi20,Γ∩K

.

(59)

Since for each K ∈ Gh,Γ ∩K

= γKhK for some γK > 0, and by definition (41)of ·1/2,h,Γ, the second-to-last term in (59) can be bounded as

K∈Gh

λK − δ1,K + δ2,K

|Γ ∩K|

v 2

0,Γ∩K

≥ minK∈Gh

λKhK − δ1,K + δ2,K

γK

v 21/2,h,Γ.

(60)

Moreover, from the definition (42) of ·−1/2,h,Γ, the last term in (59) satisfies

2

i=1

K∈Gh

|Γ ∩K|µ2i

δi,Kκi ∂nvi20,Γ∩K

≥ mini=1,2K∈Gh

µ2i γKδi,K

∂nv−1/2,h,Γ . (61)

15

Substituting expressions (60) and (61) into inequality (59), and writing −1 =B − (1 +B) for the given B > 0 implies that

aε(v1, v2, v1, v2) ≥ µ1 ∇v120,Ω1+ µ2 ∇v220,Ω2

+ εPh

2,τv120,Ω2

+ εPh

1,τv220,Ω1

+ minK∈Gh

λKhK − δ1,K + δ2,K

γK

v 21/2,h,Γ

+B mini=1,2K∈Gh

µ2i γKδi,K

∂nv−1/2,h,Γ

− (1 +B)2

i=1

K∈Gh

|Γ ∩K|µ2i

δi,Kκi ∂nvi20,Γ∩K

.

(62)

According to the intersection options of Assumption 3.2, for each elementK ∈ Gh, either Γ∩K coincides with an edge of the element, or Γ intersects theelement boundary exactly twice. Denote by Eh

i the set of elements K ∈ T h(Ωi)for which ΓK coincides with an edge of the element. Applying Lemma 4.2 toelements K ∈ Gh\Eh

2 with |Γ ∩K| = γKhK and summing, we get

K∈Gh\Eh2

|Γ ∩K|µ2i

δi,Kκi ∂nvi20,Γ∩K

≤

K∈T h(Ωi)\Ehi

µ2iκ

2i γ

2K

αi,Kδi,K∇vi20,K∩Ωi

. (63)

The elements K ∈ Eh2 are such that Γ ∩K coincides with an edge that is also

an edge of another triangle T ⊂ Ω1, and Γ∩K = Γ∩T with |Γ∩K| = γKhK =γThT . Apply Lemma 4.2 with the element T when i = 1 and with the elementK when i = 2. This yields

K∈Eh2

|Γ ∩K|µ2i

δi,Kκi ∂nvi20,Γ∩K

≤

T∈Eh

1

µ21κ

21γ

2T

αT δ1,K∇v120,T , i = 1,

K∈Eh

2

µ22κ

22γ

2K

αKδ2,K∇v220,K , i = 2.

(64)

For the given A ∈ (0, 1), let

δ1,K =(1 +B)µ1κ2

1γ2K

Aα1,K, δ2,K =

(1 +B)µ2κ22γ

2K

Aα2,K, (65)

when K is such that Γ intersects the element boundary exactly twice, and

δ1,K =(1 +B)µ1κ2

1γ2T

AαT, δ2,K =

(1 +B)µ2κ22γ

2K

AαK, (66)

when Γ ∩ K coincides with an edge shared by element T ⊂ Ω1 and K ⊂ Ω2.Substituting inequalities (63) and (64) with the choices (65) and (66) for δi,K

16

into expression (62) yields the bound

aε(v1, v2, v1, v2) ≥ (1−A)µ1 ∇v120,Ω1+ (1−A)µ2 ∇v220,Ω2

+

+ εPh

2,τv120,Ω2

+ εPh

1,τv220,Ω1

+B mini=1,2K∈Gh

µ2i γKδi,K

∂nv2−1/2,h,Γ

+ minK∈Gh

λKhK − δ1,K + δ2,K

γK

v 21/2,h,Γ.

(67)To show coerciveness, we need to show that the coefficients in front of the two

last terms in expression (67) are uniformly positive for each interface location.ForK ∈ Gh\Eh

2 , expressions (65) and (28) imply that for each interface location,there is a constant c such that δi,K ≤ cαi,Kγ2

K . Thus,

Bµ2i γKδi,K

≥ mini=1,2

K∈Gh\Eh2

B

cαi,KγK

= Cf > 0. (68)

An analogous lower bound holds for K ∈ Eh2 ; we redefine Cf to denote the

smallest of these bounds. The particular values of κ1 and κ2 in expressions (28)and (31) are the solution to the problem of minimizing δ1,K + δ2,K , which isquadratic in κ1 and κ2, subject to κ1 + κ2 = 1. Moreover, the values of λK

according to expressions (29) and (32) satisfies

λKhK =

D + (δ1,K + δ2,K)/γK for K ∈ Gh \ Eh

2 ,

(D + δ1,K + δ2,K)/γK for K ∈ Eh2 ,

(69)

from which follows that

minK∈Gh

λKhK − δ1,K + δ2,K

γK

≥ D, (70)

where we have used that 1/γK ≥ 1 for K ∈ Eh2 . Hence, from expressions (67),

(68), and (70) follow that

aε(v1, v2, v1, v2) ≥ min (1−A)µ1, (1−A)µ2, Cf , D, ε |||v|||2 . (71)

4.2 Error estimates

Theorem 3.1 shows consistency of the stabilized problem (35) for functions u1,u2 that exhibit jump discontinuities at Γ: u1 vanishes on Ω2 and vice versa foru2. However, the solutions uh

1 , uh2 to the discrete, stabilized problem (35) will be

continuous throughout Ω. This regularity difference complicates the construc-tion of a suitable nodal interpolant of the exact solution. Another complicatingfactor for the approximation theory is the presence of the projection operatorsPhi,τ in the norm (40) that is used in the analysis.

17

In order to circumvent these complications, we construct in Part II of thisarticle a more complicated-than-usual nodal interpolant Ih,∗τ and show that itis of optimal order. A consequence of consistency, continuity, coercivity, andthe approximation result is the following convergence result which is proved inPart II of this article.

Theorem 4.4. Let u = (u1, u2) ∈ L2(Ω)× L2(Ω), where, for i = 1, 2, ui|Ωi ∈Vi∩H2(Ωi) solve boundary-value problem (1) and where ui|Ω\Ωi

≡ 0. Moreover,let uh = (uh

1 , uh2 ) ∈ V h × V h be the solution to problem (35), where, for some

c > 0, τK = ch2K in the projection operators (37). Then there is a constant C

such that|||u− uh||| ≤ Ch (u12,Ω1 + u22,Ω2) (72)

and

i

ui − uhi 0,Ωi

1/2

≤ Ch2 (u12,Ω1 + u22,Ω2) . (73)

4.3 Condition number

Under a stronger assumption on the mesh family (compared to assumption (15)),the spectral condition number of the system matrix, that is, the largest-to-smallest eigenvalue ratio, is O(h−2), independently of the location of the inter-face. Here, we make the standard assumption that the mesh family is quasi-uniform; that is, there is a C∗ > 0 such that, uniformly for all h,

ρK ≥ C∗h ∀K ∈ T h, (74)

where ρK is the diameter of the largest disk inscribed in element K. Let Adenote the stiffness matrix associated with the form aε(uh

1 , uh2 , v

h1 , v

h2 ). The

matrix A has dimension 2N × 2N , is symmetric and positive definite.

Theorem 4.5. Let T hh>0 be a quasi-uniform family of triangulations of Ωand let uh = (uh

1 , uh2 ) ∈ V h × V h be the solution to problem (35), where, for

some c > 0, ch2 ≤ τK ≤ 1/4 in the projection operators (37). Then there exista constant C such that the spectral condition number of matrix A associatedwith problem (35) satisfies

κ(A) ≤ C(Cc + 2ε)

min Ccoer, εh2, (75)

where Cc and Ccoer, are the constants from Lemma 4.1, 4.3, respectively.

For a proof, see Part II [17]. The theorem implies that if the stabilizationparameter ε is chosen as a positive constant, the condition number is O(h−2).

18

5 Numerical Experiments

In this section we present results for numerical experiments in one and two spacedimensions. We study the impact of the introduced stabilization procedure fordifferent stabilization tolerances and confirm the optimal convergence order.The domain Ω is triangulated using a uniform mesh, and a direct solver is appliedin the solution of the linear system. The system-matrix assembly routine usesa piecewise-linear approximation of the interface–element intersections.

Recall that in our method, parameters κ1 and κ2 are chosen according toexpressions (28) and (31), and that penalty parameter λ is chosen locally ac-cording to expressions (29) and (32). We use τK = h2, where h is the legsize of the (right) triangles in the mesh, as the stabilization tolerance in equa-tion (37). The error is not sensitive to the size of the stabilization parameterε; it is the choice τK = O(h2) that limits the impact of the stabilization on theerror. Therefore, ε is chosen so that it does not affect the condition number.In the one-dimensional examples ε = 10, and in the two-dimensional examplesε = 100.

When setting τ = 0 in expression (37), we refer to the method as the un-stabilized method. The method of Hansbo and Hansbo [9] is equivalent tosetting τ = 0, choosing κ1 and κ2 according to expression (27), and usingλ = η/h. In our implementation of the method of Hansbo and Hansbo we setη = 1 + 16max(µ1, µ2).

5.1 Numerical examples in one dimension

As a first experiment we study the behavior of the proposed stabilized methodas well as the unstabilized method for a one-dimensional interface problem. Theinterface Γ is located at the point xΓ, dividing the computational domain Ω inthe two subdomains Ω1 = (0, xΓ) and Ω2 = (xΓ, 1). The parameters in thegoverning equation (1) are set as µ1 = 1, µ2 = 500, f = sinx, g = −50, andb = 0. The solution contains only a week discontinuity at the interface point, sowhen xΓ agrees with a grid point, the standard finite element method (withoutthe Nitsche treatment of the interface) using continuous functions and a pointsource at xΓ works fine and shows optimal convergence.

First, we consider three interface positions: (i) the interface is located at agrid point xΓ = 0.5, (ii) the interface is located close to a point xΓ = 0.5 + εM ,where εM ≈ 2 · 10−16 is machine epsilon (for IEEE binary64), and (iii) theinterface is located at the point xΓ = e−0.5. We use three methods, (a) theunstabilized method (τK = 0), (b) our proposed stabilized method with τK =h2 in expression (37), and (c) stabilization for all interface positions, that is,choosing τK = 1 in expression (37). However, in the experiments for case (i),that is when the interface is located at a grid point, we replace the unstabilizedmethod with the standard finite element method with a point source at xΓ.Fig. 3 shows the spectral condition number (left) and the L2-error (right) asfunctions of grid size h for the three choices of τ described above, and for (fromtop to bottom) the interface positions (i), (ii), and (iii). The dashed lines in

19

10−5

10−3

10−1

105

1010

1015

1020

Co

nd

itio

n n

um

be

r

Element size, h10

−510

−310

−110

−10

10−8

10−6

10−4

Err

or

Element size, h

Interface position = 0.5

10−5

10−3

10−1

105

1010

1015

1020

Co

nd

itio

n n

um

be

r

Element size, h10

−510

−310

−110

−10

10−8

10−6

10−4

Err

or

Element size, h

Interface position = 0.5+eps

10−5

10−3

10−1

105

1010

1015

1020

Co

nd

itio

n n

um

be

r

Element size, h10

−510

−310

−110

−10

10−8

10−6

10−4

Err

or

Element size, h

Interface position = exp(−0.5)

Figure 3: Condition number (left) and the L2-error (right) from 1D computa-tions as functions of grid size for stabilization parameters τ = 0 (stars), τ = h2

(circles), and τ = 1 (diamonds). The interface is positioned at a grid point(top), close to a grid point (middle), at the point xΓ = e−0.5(bottom). Thedashed lines are µ2h−2 for the condition number and h2/20 for the error.

20

10−15

10−10

10−5

100

105

1010

1015

Conditi

on n

um

ber

Relative distance between interface and gridline

Figure 4: Condition number as function of interface position in 1D computa-tions. The interface is located at xΓ = 0.5 − dh, where h = 2−7 is the elementsize and d the relative distance from the interface to the mesh point 0.5. Di-amonds illustrate the unstabilized method (τ = 0) and circles the proposedstabilized method with parameter τK = h2. The dotted line is placed at d = h2.For the proposed method the stabilization is active to the left of the dotted line.

Fig. 3 are µ2h−2 for the condition number and h2/20 for the error. The resultsindicate that the two first methods have optimal second-order convergence inthe L2-norm but that stabilizing independently of the interface position ruinsthe optimal convergence order. This result demonstrates that it is essentialto limit the influence of the stabilization. We have performed several othercomputations that all suggest that when the third method is used the errorbehaves as O(εh). In Fig. 3 we also see that for all three interface positions(i), (ii), and (iii), the condition number is O(h−2) when stabilization is active.When the unstabilized method is used, the condition number reaches extremelylarge values when the interface is located close to a grid point (case (ii)).

In Fig. 4 we show the spectral condition number of the system matrix asa function of interface position using the proposed stabilized method (usingτK = h2) and the unstabilized method. The interface is located at xΓ = 0.5−dh,with h = 2−7 and d = 2−i, for i = 1, 2, . . . , 46. The dotted line in Fig. 4 is placedat d = h2. Since τK = h2 in the proposed method the stabilization is activewhen d < h2 (see definition (37)). For the unstabilized method, the conditionnumber grows inversely proportional to the distance between the interface andthe grid point. On the other hand, due to the stabilization, the condition numberof the system matrix for the proposed method stays almost constant and doesnot grow as the interface approaches the grid point.

5.2 Numerical examples in two dimensions

We consider two examples in two space dimensions with different types of singu-larities. We compare the condition numbers of the discretized problem obtainedusing the proposed method with the condition number obtained using the unsta-

21

bilized method. The computed condition number is an estimate of the 1-normcondition number of the stiffness matrix. We also measure the convergence ofthe proposed method to confirm the optimal convergence order in the L2(Ω)and H1(Ω1 ∪ Ω2) norm. The convergence of the maximum error over all nodepoints is also studied.

5.2.1 2D example 1: discontinuous normal derivative

Here we consider the same two-dimensional boundary-value problem as studiedby Hansbo and Hansbo [9]. The problem is defined by equation (1) with µ1 = 1,µ2 = 1000, f1 = f2 = −4, b = 0, and g = 0. The interface Γ is a circle withradius r0 = 0.75 centered at (xc, yc) = (0, 0). The computational domain Ω is[−1, 1]×[−1, 1] and Ω1 is the domain inside the circle Γ. The Dirichlet boundaryconditions on ∂Ω are chosen such that the exact solution is given by

u(x, y) =

r2

µ1, if r ≤ r0,

r2

µ2− r20

µ2+ r20

µ1if r > r0,

(76)

where r =(x− xc)2 + (y − yc)2.

The approximate solution on a 20 × 20 uniform mesh is shown in Fig. 5.The condition number, the maximum error over all node points and the errormeasured in the H1(Ω1 ∪ Ω2), and L2(Ω)-norm are shown in Fig. 6. In thisexample the interface Γ is not close to any element boundaries and thereforethe stabilization does not influence u1|Ω1 and u2|Ω2 . Hence, in this case, the onlydifference between our proposed method and Hansbo and Hansbo’s method is inthe choice of the parameters κi and λ. The error measured in the H1(Ω1 ∪Ω2)and L2(Ω) norms using the proposed stabilized method is similar to the errorusing the unstabilized method of Hansbo and Hansbo. We see the expectedoptimal second order convergence in the L2(Ω) norm and first order convergencein the H1(Ω1 ∪ Ω2) norm. In Fig. 6 we also observe second order convergencefor the maximum error over all node points. Our choice of the parameters κi

and λ yield a smaller maximum error over all node points and smaller conditionnumbers compared to the parameter choices of [9].

5.2.2 2D example 2: discontinuous solution

Consider the two-dimensional boundary-value problem given by equation (1)with µ1 = µ2 = 1, f1 = f2 = 0, b = log(2r), and g = (1/r0 − xcx/(r0r2)),where r =

x2 + y2. The interface Γ is a circle with radius r0 = 0.5 centered

at (xc, yc) = (0.2, 0). The computational domain Ω is [−1, 1] × [−1, 1] and Ω1

is the domain inside the circle Γ. The Dirichlet boundary conditions on ∂Ω arechosen such that the exact solution is given by

u(x, y) =

1, in Ω1,

1− log(2r) in Ω2.(77)

22

−1

0

1

−1

0

10

0.2

0.4

0.6

0.8

xy

Figure 5: The approximate solution of the 2D example described in Section 5.2.1when h = 0.1 using the proposed method.

10−3

10−2

10−1

105

106

107

108

109

Element size, h

Co

nd

itio

n n

um

be

r

10−3

10−2

10−1

10−6

10−5

10−4

10−3

Element size, h

Err

or

10−3

10−2

10−1

10−6

10−4

10−2

100

Element size, h

Err

or

Figure 6: Condition number and errors for the 2D example described in Sec-tion 5.2.1. In all figures circles represent the proposed method and stars repre-sent our implementation of the method in [9]. Left panel: Condition number.The dashed line is µ1h−2. Middle panel: The maximum error over all nodepoints. Right panel: Convergence in the H1(Ω1∪Ω2) norm (symbols connectedby a dashed line) and the L2(Ω) norm.

23

−1−0.5

00.5

1 −1

0

1

−0.5

0

0.5

1

1.5

y

x

Figure 7: The approximate solution of the 2D example described in Section 5.2.2when h = 0.1 using the proposed method.

10−3

10−2

10−1

100

1010

1020

1030

1040

Co

nd

itio

n n

um

ber

Element size, h10

−310

−210

−110

−5

10−4

10−3

10−2

10−1

Err

or

Element size, h10

−310

−210

−110

−5

10−4

10−3

10−2

10−1

Err

or

Element size, h

Figure 8: Condition number and errors for the 2D example described in Sec-tion 5.2.2. In all three figures circles represent the proposed method and starsrepresent the unstabilized method, that is, τ = 0. Squares represent resultsextracted from Fig.5(a) in [10]. Left panel: Condition number. The dashed lineis h−2. Middle panel: The maximum error over all node points. Right panel:Convergence in the H1(Ω1 ∪ Ω2) norm (symbols connected by a dashed line)and the L2(Ω) norm.

The approximate solution on a 20 × 20 uniform mesh is shown in Fig. 7.The condition number, the maximum error over all node points, and the errormeasured in the H1(Ω1 ∪ Ω2) and L2(Ω) norms are shown in Fig. 8. In thisexample, stabilization is essential and we see in Fig. 8 that the condition num-ber for the unstabilized method reaches extremely large values. However, thecondition number of our proposed stabilized method is O(h−2). We also seethe optimal second order convergence in the L2(Ω)-norm and first-order con-vergence in the H1(Ω1 ∪ Ω2)-norm. We observe second order convergence forthe maximum error over all mesh points. This example was used by Hue andSethian [10] as a test case for their nonconforming finite element method. Theyreport convergence results in maximum norm that for coarse meshes are verysimilar to our results, but our method shows better asymptotic convergence.

24

6 Discussion and outlook

Our method constitutes a further development of the idea by Hansbo andHansbo [9], and offers a way to solve interface problems in which the inter-face can be located arbitrarily with respect to a fixed background mesh. Themain features of our method are

• optimal-order accuracy for linear elements; we can prove optimal ordersin the energy and L2 norms, and numerical experiments indicate alsooptimal-order pointwise errors;

• the condition number of the system of equations is O(h−2), independentof the interface location;

• the problem size is fixed, independent of the location of the interface;

• the stabilization procedure of Section 3.2 is easy to implement.

We expect the current method to be applicable also to three space dimensionswithout major changes. However, the present stabilization method is explicitlydesigned for linear elements, and a possible extension to higher-order elementsrequires further investigations.

The properties of the presented method are very desirable in, for example,multiphase flow simulations, where accurate simulations with surface tensioneffects on unfitted meshes remains a challenge. We plan therefore to extend themethod to Stokes equations and moving interface problems.

Acknowledgements

We are grateful to Maya Neytcheva for sharing a finite element code that fa-cilitated the implementation of the 2D finite element code. S. Z. acknowledgefinancial support from Linne FLOW center.

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NA – Numerical AnalysisSchool of Computer Science and Communication

KTH Royal Institute of Technology

www.csc.kth.se

NA – Numerical AnalysisTRITA-­NA 2011:1

ISRN KTH/NA-­-­11/1-­-­SE

ISSN 0348-­2952