a numerical method for solving elliptic interface problems using...

Research ArticleA Numerical Method for Solving Elliptic Interface ProblemsUsing Petrov-Galerkin Formulation with Adaptive Refinement

LiqunWang 1 Songming Hou 2 and Liwei Shi 3

1Department of Mathematics College of Science China University of Petroleum (Beijing) 102249 China2Department of Mathematics and Statistics and Center of Applied Physics Louisiana Tech University Ruston LA 71272 USA3Department of Science and Technology Teaching China University of Political Science and Law Beijing China

Correspondence should be addressed to Liwei Shi sliweihmilygmailcom

Received 29 May 2018 Accepted 30 July 2018 Published 10 September 2018

Academic Editor Nunzio Salerno

Copyright copy 2018 Liqun Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Elliptic interface problems have wide applications in engineering and science Non-body-fitted grid has the advantage of saving thecost of mesh generation In this paper we propose a Petrov-Galerkin formulation using non-body-fitted grid for solving ellipticinterface problems In this method adaptive mesh refinement is employed for cells with large errors The new mesh still has alltriangles being right triangles of the same shape Numerical experiments show side-by-side comparison that to obtain the sameaccuracy our new method has much less overall CPU time compared with the previous method even with some cost on meshgeneration

1 Introduction

Elliptic interface problems have wide applications in a varietyof disciplines such as electromagnetism fluid dynamicsand material science There are two types of grids used forsolving such problems the body-fitted grid and non-body-fitted grid For the body-fitted gridmesh generationhasmorecomputational cost than the non-body-fitted gridThequalityof the triangles is also an issue that needs special care Wefocus on the discussion using non-body-fitted grid

In the past three decades much attention has been paidto the numerical solution of elliptic interface problems onnon-body-fitted regular Cartesian grids since the pioneeringwork of Peskin [1] on the first-order accurate immersedboundary method Motivated by the immersed boundarymethod to improve accuracy in [2] the immersed interfacemethod (IIM) was presented The method achieves second-order accuracy by incorporating the interface conditionsinto the finite difference stencil in a way that preserves theinterface conditions in both solution and its flux in thenormal direction [119906] = 0 and [120573119906119899] = 0 The correspondinglinear system is sparse but may not be symmetric or positivedefinite if there is a jump in the coefficient

Naturally the standard finite element method has theproperty of symmetricity and positive definiteness Howeverit used body-fitted grid Can finite elementmethod be appliedusing non-body-fitted grid In [3] the immersed finiteelement (IFE) method is introduced for interface problemswith homogeneous jump conditions The idea is that the testand trial function basis are the same and are continuous butnot smooth across the interface in an interface triangle In [4]the method is generalized to deal with nonhomogeneous fluxjump condition In [5] the partially penalized IFE method isintroduced to penalize the discontinuity of the IFE functionon neighboring interface triangles There are some work inthe IFE formulation in three dimensions aswell such as [6 7]

As mentioned above special treatment is required to usethe IFE on non-homogeneous jump conditions Is it possiblefor a finite element method to handle non-homogeneousjump conditions the same way as homogeneous jump con-ditions In [8] a non-traditional finite element formulationfor solving elliptic equations with smooth or sharp-edgedinterfaces was proposed with non-body-fitting grids for [119906] =0 and [120573119906119899] = 0 It achieved second-order accuracy in the119871infin norm for smooth interfaces and about 08th order forsharp-edged interfaces In [9 10] the method is analyzed and

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 3721258 12 pageshttpsdoiorg10115520183721258

2 Mathematical Problems in Engineering

implemented in three dimensionsThe resulting linear systemis non-symmetric but positive definite

In [11] the matched interface and boundary (MIB)method was proposed to solve elliptic equations with smoothinterfaces In [12] the MIB method was generalized to treatsharp-edged interfaces With an elegant treatment second-order accuracy was achieved in the 119871infin norm In [13] theMIB method was extended to three-dimensional interfaceproblems Also there has been a large body of work fromthe finite volume perspective for developing high ordermethods for elliptic equations in complex domains such as[14 15] for two-dimensional problems and [16] for three-dimensional problems Another class of methods is theBoundary Condition Capturing Method [17ndash19] In [20 21]gradient recovery techniques were developed to improve theaccuracy of gradient computation

In this paper based on our early work on non-traditionalfinite element method using a non-symmetric weak formu-lation with uniform Cartesian grid we improve the perfor-mance by using adaptive mesh refinement on the cells wherethe numerical error is large The main idea is as follows firstwe use two different uniform Cartesian coarse grids so thatwe could compare their results to find at which cells the erroris larger Then we do adaptive mesh refinement for thesecells Finally we use our non-symmetric weak formulationon this non-uniform grid For most problems the large errorsoccur only around the interface and therefore an alternativemethod refining around the interface is also proposedWe dopresent an example in which large errors occur away from theinterface though All triangles are right triangles of the sameshape in the new grid This grid is just a minor modificationfrom the uniform Cartesian grid and the mesh generationcost is very low However since smaller triangles are usedwhere the error is large in the end the 119871infin error is reducedNumerical results show that to obtain the same L-infinityerror this new method needs much less overall CPU timecompared with the previous method

2 Equations and Weak Formulation

In this section we briefly go through the equations andweak formulation in our previous work [8] This is thefoundation of our adaptive refinement method in this paperBefore adaptive refinement the only difference between ourprevious work and the work in this paper is that the uniformtriangular mesh is different

Consider an open bounded domain Ω sub 1198772 Let Γ be aninterface of co-dimension 1 which dividesΩ into twodisjointopen subdomains Ωminus and Ω+ hence Ω = Ωminus cup Ω+ cup Γ

We seek solutions of the variable coefficient ellipticequation away from the interface Γ given by

minus998810 sdot (120573 (x) 998810119906 (x)) = 119891 (x) x isin Ω Γ (1)

in which x = (1199091 1199092) denotes the spatial variables Thecoefficient 120573(x) is assumed to be a 2 times 2 matrix that isuniformly elliptic on each disjoint subdomain Ωminus and Ω+and its components are continuously differentiable on eachdisjoint subdomain but they may be discontinuous across

the interface Γ The right-hand side 119891(x) is assumed to lie in1198712(Ω)Given functions 119886 and 119887 along the interface Γ we

prescribe the jump conditions

[119906]Γ (x) equiv 119906+ (x) minus 119906minus (119909) = 119886 (x)[(120573998810119906) sdot n]Γ (x) equiv n sdot (120573+ (x) 998810119906+ (x)) minus n

sdot (120573minus (x) 998810119906minus (x)) = 119887 (x) (2)

The ldquoplusmnrdquo superscripts refer to limits taken from within thesubdomains Ωplusmn

Finally the boundary conditions are given by

119906 (x) = 119892 (x) x isin 120597Ω (3)

The jump conditions are enforced strongly in the localsystem Also the flux jump condition appears in the weakformulation

We introduce the weak solution by the standard proce-dure of multiplying (1) by a test function 120595 and integration byparts

intΩ+

120573998810119906 sdot 998810120595 + intΩminus

120573998810119906 sdot 998810120595 = intΩ

119891120595 minus intΓ

119887120595 (4)

where 120595 is in 11986710 (Ω) Note that although the test function120595 is the same as in the standard finite element method thefunction 119906 is not a linear combination of such basis functionsInstead the jump conditions are enforced strongly

3 Numerical Method

In this paper we restrict ourselves to a rectangular domainΩ = (119909min 119909max) times (119910min 119910max) in the plane Given positiveintegers 119868 and 119869 set Δ119909 = (119909max minus 119909min)119868 and Δ119910 =(119910max minus119910min)119869 We define a uniform Cartesian grid (119909119894 119910119895) =(119909min + 119894Δ119909 119910min + 119895Δ119910) for 119894 = 0 119868 and 119895 = 0 119869 Each(119909119894 119910119895) is called a grid point For the case 119894 = 0 119868 or 119895 = 0 119869 agrid point is called a boundary point otherwise it is called aninterior point The grid size is defined as ℎ = max(Δ119909 Δ119910) gt0

Two sets of grid functions are needed and they aredenoted by

1198671ℎ = 120596ℎ = (120596119894119895) 0 le 119894 le 119868 0 le 119895 le 119869 (5)

and

1198671ℎ0 = 120596ℎ = (120596119894119895) isin 1198671ℎ 120596119894119895 = 0 if 119894 = 0 119868 or 119895= 0 119869 (6)

Cut every rectangular region [119909119894 119909119894+1] times [119910119895 119910119895+1] intofour pieces of right triangular regions Collecting all thosetriangular regions we obtain a uniform triangulation 119879ℎ ⋃119870isin119879ℎ 119870 see Figure 1 This is slightly different from thetriangulation in our previous work [8] for convenience ofadaptive mesh refinement

We call a cell 119870 an interface cell if its vertices belong todifferent subdomains we write 119870 = 119870+ cup 119870minus 119870+ and 119870minus are

Mathematical Problems in Engineering 3

minus1 minus05 0 05 1minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Figure 1 A uniform triangulation

1

2 3(a) Regular cell

2 3

1

4 5

(b) Interface cell

Figure 2 Cells

separated by a straight interface segment Γℎ119870 We call a cell 119870a regular cell if all its vertices belong to the same subdomaineither Ω+ or Ωminus see Figure 2

In order to introduce our non-traditional finite elementmethod two finite element isomorphism mappings areneeded from coefficient vector to finite element functionsThe first one is 119879ℎ 1198671ℎ 997888rarr 11986710 (Ω) For any 120595ℎ isin 1198671ℎ0 119879ℎ(120595ℎ) is a standard continuous piecewise linear functionwhich is a linear function in every triangular cell and 119879ℎ(120595ℎ)matches 120595ℎ on grid points The second one 119880ℎ is constructedas follows For any119906ℎ isin 1198671ℎwith119906ℎ = 119892ℎ at boundary points119880ℎ(119906ℎ) is a piecewise linear function and matches 119906ℎ on gridpoints It is a linear function in each regular cell just like thefirst extension operator 119880ℎ(119906ℎ) = 119879ℎ(119906ℎ) in a regular cell Ineach interface cell it consists of two pieces of linear functionsone is on 119870+ and the other is on 119870minus

In each cell we shall construct an approximate solution119906ℎ to the interface problem taking into account the jumpconditions

Case 1 If 119870 is a regular cell (see Figure 2(a)) we can have thefollowing equation

int119870

120573nabla119880ℎ (119906ℎ) sdot nabla119879ℎ (120595ℎ) = int119870

119891119879ℎ (120595ℎ) (7)

Case 2 If 119870 is an interface cell (see Figure 2(b)) notice thatpoints 2 3 4 5 are coplanar and the value of point 5 can bedenoted as a linear combination of the values of points 2 3 41199065 = 11988811199062 + 11988821199063 + 11988831199064 Then a local system defined on theinterface cell 119870 can be constructed as

[119906]4 = 1198864[119906]5 = 1198865

11988811199062 + 11988821199063 + 11988831199064 = 1199065[(120573998810119906) sdot n]6 = 1198876

(8)

where point 6 is the midpoint of the line segment from 4 to 5Solve this local system and get the values of 119906plusmn4 119906plusmn5 which

are denoted by the linear combinations of 1199061 1199062 1199063 Then wehave the following equation

int119870+

120573nabla119880ℎ (119906ℎ) sdot nabla119879ℎ (120595ℎ) + int119870minus

120573nabla119880ℎ (119906ℎ)sdot nabla119879ℎ (120595ℎ)

= int119870+

119891119879ℎ (120595ℎ) + int119870minus

119891119879ℎ (120595ℎ) minus intΓℎ119870

119887119879ℎ (120595ℎ) (9)


Input Triangulation set 119879 interior point set 119875Output New triangulation set 119879 new interior point set 1198751 Let 119905119899 be the length of triangulation 1198792 for 119894 = 1 to 119905119899 do3 if 119879119894rsquos refine sign is equal to 1 then4 119904 larr997888 1 119894119905 larr997888 15 while s=1 do6 Let (119909 119910) be the right angle point of 1198791198947 Let (119909119898 119910119898) be the middle point of the hypotenuse of 1198791198948 Let 119879119895 be the triangle next to 119879119894 and share the hypotenuse with 1198791198949 (119909119899 119910119899) larr997888 2(119909119898 119910119898) minus (119909 119910)10 if (119909119898 119910119898) is on the boundary then11 119904 larr997888 0 (see cell 11987010813 in Figure 3)12 else if (119909119899 119910119899) isin 119875 then13 119904 larr997888 0 (see cell 119870967 in Figure 3)14 else15 119904 larr997888 1 119894119905 larr997888 119894119905 + 1 119879119894 larr997888 119879119895 add 119879119894 into a triangle set 119873119879 (see cell 1198701547 in Figure 3)16 end if17 end while18 for 119895 = 119894119905 down to 1 do19 119879119894 larr997888 11987311987911989520 Let (119909 119910) be the right angle point of 11987911989421 Let (119909119898 119910119898) be the middle point of the hypotenuse of 11987911989422 (119909119899 119910119899) larr997888 2(119909119898 119910119898) minus (119909 119910)23 if (119909119898 119910119898) is on the boundary then24 Connect (119909 119910) and (119909119898 119910119898) separate the original cell into two new cells

delete the original cell from 119879 and add two new cells into 11987925 else if (119909119899 119910119899) isin 119875 then26 Connect (119909 119910) and (119909119898 119910119898) separate the two cells into four new cells

delete the two cells from 119879 add four new cells into 119879and add the new point (119909119898 119910119898) into 119875

27 end if28 end for29 end if30 end for

Algorithm 1 Adaptive mesh refinement

Putting all the cells together we propose the followingmethod

Method 1 Find a discrete function 119906ℎ isin 1198671ℎ such that 119906ℎ =119892ℎ on boundary points so that for all 120595ℎ isin 1198671ℎ0 we have

sum119870isin119879ℎ

(int119870+


120573nabla119880ℎ (119906ℎ)

sdot nabla119879ℎ (120595ℎ)) = sum119870isin119879ℎ

(int119870+

119891119879ℎ (120595ℎ)

+ int119870minus

119891119879ℎ (120595ℎ) minus intΓℎ119870

119887119879ℎ (120595ℎ))

(10)

In order to improve the efficiency of our method weintroduce the adaptive refinement method to get a refinedmesh The idea is to use the error scale of the coarse gridto get the refinement triangulation set First calculate thenumerical result of the uniform coarse grids using (119899119909 119899119910)and (2119899119909 2119899119910) grid points in both 119909 and 119910 directions then

use this numerical result to get the numerical error |119890(119899119909119899119910)|on the corresponding points Based on the error scale ondifferent grid points we can decide the refinement scale ofeach triangulation and then get the refined mesh with ouradaptive refinementmethodwe call thismesh adaptivemeshThe detailed algorithm is given in Algorithm 1

Remark When it comes to numerical computation of inter-face problems the major part of the error is from the regionaround the interface so the refined mesh around this regionneeds to be the smallest triangulation This is why we proposethe adaptive mesh with further refinement on the interfacecalled adaptive interface mesh

4 Numerical Experiments

In all numerical experiments below the level-set function120601(119909 119910) the coefficients 120573plusmn(119909 119910) and the solutions

119906 = 119906+ (119909 119910) in Ω+119906minus (119909 119910) in Ωminus (11)


Table 1 Numerical results of Example 2 with different mesh

Mesh 119873 10038171003817100381710038171198901199061003817100381710038171003817infin CPU TimeUniform Mesh 404101 135e-3 63925sAdaptive Mesh 147664 132e-3 22183sAdaptive Interface Mesh 18249 128e-3 1208s

31 2

4 16 5

9 10 14

11 12 13

6 715 8

17

Figure 3 Refined mesh

are given Hence

119891 = minusnabla sdot (120573nabla119906) 119886 = 119906+ minus 119906minus119887 = (120573+nabla119906+) sdot n minus (120573minusnabla119906minus) sdot n

(12)

on the whole domain Ω 119892 is obtained as a proper Dirichletboundary condition since the solutions are given

In Examples 2 4 and 5 we solve the problem in therectangular domain Ω = [minus1 1] times [minus1 1] In Example 3 wesolve the problem in the rectangular domain Ω = [0 1] times[0 1] 119873 is the number of interior points and 11987312 is thenumber of points in one dimension The 119871infin norm and theenergy norm in the whole domain Ω are measured in thefollowing ways

10038171003817100381710038171198901199061003817100381710038171003817infin = max119909isinΩ

10038161003816100381610038161003816119906 minus 119906ℎ10038161003816100381610038161003816 10038171003817100381710038171198901199061003817100381710038171003817120573 = radicint

Ω(998810 (119906 minus 119906ℎ))119879 120573 (998810 (119906 minus 119906ℎ)) = radicint

Ω+(998810 (119906 minus 119906ℎ))119879 120573 (998810 (119906 minus 119906ℎ)) + int

Ωminus(998810 (119906 minus 119906ℎ))119879 120573 (998810 (119906 minus 119906ℎ))

(13)

Example 2 The level-set function 120601 the coefficients 120573plusmn andthe solution 119906plusmn are given as follows

120601 (119909 119910) = 1199092 + 1199102 minus 015120573+ (119909 119910) = 1000 (1199092 + 3 cos (119909)

cos (119909) 1199102 + 2)

120573minus (119909 119910) = (1199092 + 3 cos (119909)cos (119909) 1199102 + 2)

119906+ (119909 119910) = sin (5119909119910) + 1119906minus (119909 119910) = sin (10119909119910)

(14)

Table 1 shows the error and CPU time on uniformtriangular mesh adaptive mesh and adaptive interface meshFrom this table we can see that the adaptive mesh methodhas a better result than the uniform mesh method and themethod with adaptive interface mesh is significantly fasterand more accurate than the other two methods The left andright of Figure 4 are a comparison of the adaptive meshthe numerical solution and the numerical error using theadaptive mesh method with 3663 interior points (left) andthe adaptive interface meshmethod with 4167 interior points(right) The error plots do not include the outside boundarypoints where the solution is given with no error FromFigure 4(e) we can see that the maximum error comes fromthe area around the interface Therefore we refine this place


0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

0

02

04

06

08

1minus2

024

002

0406

081

002

0406

081

minus2024

0 608

10 2

0406

08

0 02 04 06 08 1

002

0406

081

minus004

minus002

0

002

068

0 02 04 06 08 1

002

0406

081

minus10

minus5

0

5x 10 minus3

08

10

(a) Grids of Adaptive Mesh (b) Grids of Adaptive Interface Mesh

(c) Numerical Solution of Adaptive Mesh (d) Numerical Solution of Adaptive Interface Mesh

(e) Numerical Error of Adaptive Mesh (f) Numerical Error of Adaptive Interface Mesh

Figure 4 Numerical results of Example 2

with the smallest triangulation Figure 4(f) is the numericalerror after the refinement it is clear in this figure that the erroraround the interface has been decreased to the same scale asthe error away from the interface


120601 (119909 119910) = 119909 + 05120573+ (119909 119910) = (1199092 + 1 0

0 1199102 + 1)

120573minus (119909 119910) = 1000 (1199092 + 1 00 1199102 + 1)

119906+ (119909 119910) = 1199094 + 1199093 minus 31199092 + 2119910 119909 gt 05119906+ (119909 119910) = minus74 (119909 minus 05) + 2119910 minus 916 119909 le 05119906minus (119909 119910) = 2119909 + 3119910 + 3

(15)

The left and right of Table 2 show the error on uniformtriangular meshes and adaptive meshes with different gridsrespectively From this table we can see that the methodwith adaptive mesh is much faster than the method withuniform mesh We plot the adaptive mesh using the adaptivemesh method with 5004 interior points in Figure 5(a) Thenumerical solution and numerical error are shown in (b) and



UniformMesh Adaptive Mesh119873 10038171003817100381710038171198901199061003817100381710038171003817infin CPU Time 119873 10038171003817100381710038171198901199061003817100381710038171003817infin CPU Time12641 302e-4 270s 4109 302e-4 213s30505 127e-4 844s 8377 130e-4 374s50881 769e-5 1795s 16171 769e-5 730s120541 327e-5 7238s 30096 327e-5 1557s204161 194e-5 17894s 57391 194e-5 4368s479221 829e-6 181197s 106387 822e-6 12068s

Order 198 Order 217


minus08minus06minus04minus02

002040608

1

(a) Grids

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1 minus1minus05

005

1

00


012345

(b) Numerical solution

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1

minus05

0

05

1x 10minus4

minus2024

(c) Numerical error

Uniform MeshAdaptive Mesh

minus12

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

Erro

r)

45 5 55 6 65 74log(N12)

(d) 119871infin error for different grids

Figure 5 Numerical results of adaptive mesh for Example 3

(c) of Figure 5 After calculating the coarse grid error |119890(119899119909119899119910)|we noticed that the error around the interface is smaller thanother areas away from the interface so in this example wedo not need to use the adaptive interface method to get afurther refinement mesh around the interface as we did inExample 2 In Figure 5(c) it is clear that the maximum errorlies on the line119909 = 1 and fromFigure 5(a)we can see that ourmethod refines the grid around this line automatically and the

smallest triangulation lies exactly on 119909 = 1 Figure 5(d) is acomparison of the 119871infin error on uniform meshes and adaptivemeshes with different grid which shows that the convergencyrate of the adaptivemesh is higher than the uniformmesh andthat the adaptive mesh can get to higher than second-orderaccuracy in 119871infin norm even when the coefficients are matrixand the ratio 120573minus120573+ = 1000 which is more efficient than theuniform mesh



Mesh 119873 10038171003817100381710038171198901199061003817100381710038171003817120573 10038171003817100381710038171198901199061003817100381710038171003817infin CPU TimeAdaptive IFEMMesh in [22] 4021 411e-2

Adaptive Mesh 933 441e-2 132e-3 153s3417 132e-2 375e-4 220s

Adaptive Interface Mesh 2017 128e-2 384e-4 176s3453 886e-3 165e-4 260s


Mesh 119873 10038171003817100381710038171198901199061003817100381710038171003817infin CPU Time Order

Uniform Mesh

1105 665e-3 26s

1588065 159e-3 185s28561 561e-4 865s99905 188e-4 5613s400513 659e-5 64481s

Adaptive Mesh

567 613e-3 36s

1622385 156e-3 143s9711 539e-4 662s39038 182e-4 3466s155869 617e-5 33298s

Adaptive Interface Mesh

874 576e-3 58s

1863037 142e-3 184s11035 463e-4 720s41714 135e-4 3901s161208 425e-5 33434s

Example 4 This example is taken from [22] We use thisexample to compare our method with the adaptive immersedfinite element method The level-set function 120601 the coeffi-cients 120573plusmn and the solution 119906plusmn are given as follows

120601 (119909 119910) = 1199092 + 1199102 minus 025120573+ (119909 119910) = 1000120573minus (119909 119910) = 1

119903 (119909 119910) = radic1199092 + 1199102119906+ (119909 119910) = 11990331000 + (1 minus 11000) 053119906minus (119909 119910) = 1199033

(16)

Table 3 is a comparison of the numerical results andCPU time of the adaptive IFEM mesh in [22] the adaptivemesh and adaptive interface mesh in this paper From thistable we can see that our adaptive method with 933 interiorpoints gives a better result than the adaptive IFEM meshwith 4021 interior points and the result of our adaptiveinterface mesh is better than the adaptive mesh The left andright of Figure 6 are a comparison of the adaptive meshthe numerical solution and the numerical error using theadaptive mesh method with 585 interior points (left) andthe adaptive interface mesh method with 825 interior points

(right) From this figure we can see that after the refinementusing the adaptive method the maximum error comes fromarea inside the interface but not exactly lies on the interfaceso the difference of the results between the adaptive interfacemesh and the adaptive mesh is not as large as it is inExample 2

Example 5 This example is taken from [8] It is an interfaceproblem with a singular point The level-set function 120601 thecoefficients 120573plusmn and the solution 119906plusmn are given as follows Theinterface is Lipschitz continuous and it has a kink at (0 0)and 119906 is piecewise 1198672

120601 (119909 119910) = 119910 minus 2119909 119909 + 119910 gt 0120601 (119909 119910) = 119910 + 1199092 119909 + 119910 le 0

120573+ (119909 119910) = 1120573minus (119909 119910) = 2 + sin (119909 + 119910) 119906+ (119909 119910) = 8119906minus (119909 119910) = (1199092 + 1199102)56 + sin (119909 + 119910)

(17)

Figure 7 shows the jump of solution and flux in thenormal direction on the interface Table 4 is a comparison ofthe numerical results and CPU time of the uniform meshadaptive mesh and adaptive interface mesh with different




002040608

1



002040608

1

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1

x 10minus4

minus15

minus10

minus5

0

5

minus1minus05

005

1

minus1minus05

005

1

x 10minus4

minus10minus8minus6minus4minus2

02





minus1 0 1 2 3 minus1

minus05

0

05

1

0

2

4

6

8

10

(a) Jump of solution on the interface

minus10

1 23 minus1

minus05

0

05

1

005

115

225

3

(b) Jump of flux in the normal direction on the interface

Figure 7 Jumps on the interface of Example 5



002040608

1

minus05 0 05 1 15 2 25 3minus1minus1


002040608

1

minus05 0 05 1 15 2 25 3minus1

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

2005

x 10minus3

minus2minus1

01234

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

minus15minus1

minus050

051

15

Uniform MeshAdaptive MeshAdaptive Interface Mesh

minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

log(

Erro

r)

35 4 45 5 55 6 653log(N12)




(g) Linfin Error for Different Grids


grids All three methods can get to higher than 15th orderaccuracyThe result of the method using the adaptive mesh isobviously better than the method using the uniform meshand the result of the method using the adaptive interface

mesh is slightly better result than the method using theadaptive mesh and the difference of the results of threemeshes gets clearer as the triangulation becomes smaller Theleft and right of Figure 8 are a comparison of the adaptive


mesh the numerical solution and the numerical error usingthe adaptive meshmethodwith 2385 interior points (left) andthe adaptive interface meshmethod with 3037 interior points(right) In this example we can see that after the refinementusing the adaptive method the maximum error lies aroundthe interface so we further refined the mesh around theinterface and get a better result as shown in Figure 8(f)

5 Conclusion

In this paper we proposed the adaptive refinement method tosolve the elliptic interface problems using a non-symmetricweak formulation The key idea is to use two uniform coarsegrids to find the location of the maximum error Sometimesthis step can be omitted in practice because the cells withmaximum error often locates near the interface but weprovided examples in which the cells with maximum errorsare away from the interface After fixing the location ofthe maximum error we use the adaptive mesh refinementmethod to reduce the maximum error so that the error of thewhole area is of the same scale Numerical experiments showthat the adaptive refinement method has remarkably higherconvergency and accuracy than the method using uniformCartesian grid and for the same 119871infin error the overall CPUtime is highly reduced using our new method The study ofthe three-dimensional form of this new method will be ourfuture work so that we can solve problems that are morepractical

Data Availability

All data can be accessed in the Numerical Experimentssection of this article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

L Shirsquos research is supported by National Natural ScienceFoundation of China (No 11701569) S Hoursquos research issupported by Dr Walter Koss Professorship made avail-able through the State of Louisiana Board of Regents Sup-port Funds L Wangrsquos research is supported by ScienceFoundations of China University of PetroleumndashBeijing (No2462015BJB05)

References

[1] C S Peskin ldquoNumerical analysis of blood flow in the heartrdquoJournal of Computational Physics vol 25 no 3 pp 220ndash2521977

[2] R J LeVeque and Z L Li ldquoThe immersed interface method forelliptic equations with discontinuous coefficients and singularsourcesrdquo SIAM Journal on Numerical Analysis vol 31 no 4 pp1019ndash1044 1994

[3] Z Li ldquoThe immersed interface method using a finite elementformulationrdquoApplied Numerical Mathematics vol 27 no 3 pp253ndash267 1998

[4] X He T Lin and Y Lin ldquoImmersed finite element methodsfor elliptic interface problems with non-homogeneous jumpconditionsrdquo International Journal of Numerical Analysis ampModeling vol 8 no 2 pp 284ndash301 2011

[5] T Lin Y Lin and X Zhang ldquoPartially penalized immersedfinite element methods for elliptic interface problemsrdquo SIAMJournal on Numerical Analysis vol 53 no 2 pp 1121ndash1144 2015

[6] R Kafafy T Lin Y Lin and J Wang ldquoThree-dimensionalimmersed finite element methods for electric field simulationin composite materialsrdquo International Journal for NumericalMethods in Engineering vol 64 no 7 pp 940ndash972 2005

[7] S Vallaghe andT Papadopoulo ldquoA trilinear immersed finite ele-ment method for solving the electroencephalography forwardproblemrdquo SIAM Journal on Scientific Computing vol 32 no 4pp 2379ndash2394 2010

[8] S Hou WWang and L Wang ldquoNumerical method for solvingmatrix coefficient elliptic equationwith sharp-edged interfacesrdquoJournal of Computational Physics vol 229 no 19 pp 7162ndash71792010

[9] L Wang S Hou and L Shi ldquoAn improved non-traditionalfinite element formulation for solving three-dimensional ellip-tic interface problemsrdquoComputers ampMathematics with Applica-tions An International Journal vol 73 no 3 pp 374ndash384 2017

[10] L Wang S Hou and L Shi ldquoA numerical method for solvingthree-dimensional elliptic interface problems with triple junc-tion pointsrdquo Advances in Computational Mathematics vol 44no 1 pp 175ndash193 2018

[11] Y C Zhou S Zhao M Feig and G W Wei ldquoHigh ordermatched interface and boundary method for elliptic equationswith discontinuous coefficients and singular sourcesrdquo Journal ofComputational Physics vol 213 no 1 pp 1ndash30 2006

[12] S Yu Y Zhou and G W Wei ldquoMatched interface andboundary (MIB)method for elliptic problemswith sharp-edgedinterfacesrdquo Journal of Computational Physics vol 224 no 2 pp729ndash756 2007

[13] S Yu andGWWei ldquoThree-dimensionalmatched interface andboundary (MIB) method for treating geometric singularitiesrdquoJournal of Computational Physics vol 227 no 1 pp 602ndash6322007

[14] H Johansen and P Colella ldquoA Cartesian grid embeddedboundarymethod for Poissonrsquos equation on irregular domainsrdquoJournal of Computational Physics vol 147 no 1 pp 60ndash85 1998

[15] M Oevermann and R Klein ldquoA Cartesian grid finite volumemethod for elliptic equations with variable coefficients andembedded interfacesrdquo Journal of Computational Physics vol219 no 2 pp 749ndash769 2006

[16] M Oevermann C Scharfenberg and R Klein ldquoA sharp inter-face finite volume method for elliptic equations on Cartesiangridsrdquo Journal of Computational Physics vol 228 no 14 pp5184ndash5206 2009

[17] X-D Liu R P Fedkiw and M Kang ldquoA boundary conditioncapturingmethod for Poissonrsquos equation on irregular domainsrdquoJournal of Computational Physics vol 160 no 1 pp 151ndash1782000

[18] X-D Liu and T C Sideris ldquoConvergence of the ghost fluidmethod for elliptic equations with interfacesrdquo Mathematics ofComputation vol 72 no 244 pp 1731ndash1746 2003


[19] P Macklin and J S Lowengrub ldquoA new ghost celllevel setmethod for moving boundary problems application to tumorgrowthrdquo Journal of Scientific Computing vol 35 no 2-3 pp266ndash299 2008

[20] H Guo and X Yang ldquoGradient recovery for elliptic interfaceproblem I body-fittedmeshrdquo Journal of Computational Physicsvol 338 pp 606ndash619 2017

[21] H Guo and X Yang ldquoGradient recovery for elliptic interfaceproblem II immersed finite element methodsrdquo Journal ofComputational Physics vol 338 pp 606ndash619 2017

[22] C-T Wu Z Li and M-C Lai ldquoAdaptive mesh refinement forelliptic interface problems using the non-conforming immersedfinite element methodrdquo International Journal of NumericalAnalysis amp Modeling vol 8 no 3 pp 466ndash483 2011

Hindawiwwwhindawicom Volume 2018

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Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


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Submit your manuscripts atwwwhindawicom


implemented in three dimensionsThe resulting linear systemis non-symmetric but positive definite

In [11] the matched interface and boundary (MIB)method was proposed to solve elliptic equations with smoothinterfaces In [12] the MIB method was generalized to treatsharp-edged interfaces With an elegant treatment second-order accuracy was achieved in the 119871infin norm In [13] theMIB method was extended to three-dimensional interfaceproblems Also there has been a large body of work fromthe finite volume perspective for developing high ordermethods for elliptic equations in complex domains such as[14 15] for two-dimensional problems and [16] for three-dimensional problems Another class of methods is theBoundary Condition Capturing Method [17ndash19] In [20 21]gradient recovery techniques were developed to improve theaccuracy of gradient computation

In this paper based on our early work on non-traditionalfinite element method using a non-symmetric weak formu-lation with uniform Cartesian grid we improve the perfor-mance by using adaptive mesh refinement on the cells wherethe numerical error is large The main idea is as follows firstwe use two different uniform Cartesian coarse grids so thatwe could compare their results to find at which cells the erroris larger Then we do adaptive mesh refinement for thesecells Finally we use our non-symmetric weak formulationon this non-uniform grid For most problems the large errorsoccur only around the interface and therefore an alternativemethod refining around the interface is also proposedWe dopresent an example in which large errors occur away from theinterface though All triangles are right triangles of the sameshape in the new grid This grid is just a minor modificationfrom the uniform Cartesian grid and the mesh generationcost is very low However since smaller triangles are usedwhere the error is large in the end the 119871infin error is reducedNumerical results show that to obtain the same L-infinityerror this new method needs much less overall CPU timecompared with the previous method

2 Equations and Weak Formulation

In this section we briefly go through the equations andweak formulation in our previous work [8] This is thefoundation of our adaptive refinement method in this paperBefore adaptive refinement the only difference between ourprevious work and the work in this paper is that the uniformtriangular mesh is different

Consider an open bounded domain Ω sub 1198772 Let Γ be aninterface of co-dimension 1 which dividesΩ into twodisjointopen subdomains Ωminus and Ω+ hence Ω = Ωminus cup Ω+ cup Γ

We seek solutions of the variable coefficient ellipticequation away from the interface Γ given by

minus998810 sdot (120573 (x) 998810119906 (x)) = 119891 (x) x isin Ω Γ (1)

in which x = (1199091 1199092) denotes the spatial variables Thecoefficient 120573(x) is assumed to be a 2 times 2 matrix that isuniformly elliptic on each disjoint subdomain Ωminus and Ω+and its components are continuously differentiable on eachdisjoint subdomain but they may be discontinuous across

the interface Γ The right-hand side 119891(x) is assumed to lie in1198712(Ω)Given functions 119886 and 119887 along the interface Γ we

prescribe the jump conditions

[119906]Γ (x) equiv 119906+ (x) minus 119906minus (119909) = 119886 (x)[(120573998810119906) sdot n]Γ (x) equiv n sdot (120573+ (x) 998810119906+ (x)) minus n

sdot (120573minus (x) 998810119906minus (x)) = 119887 (x) (2)

The ldquoplusmnrdquo superscripts refer to limits taken from within thesubdomains Ωplusmn

Finally the boundary conditions are given by

119906 (x) = 119892 (x) x isin 120597Ω (3)

The jump conditions are enforced strongly in the localsystem Also the flux jump condition appears in the weakformulation

We introduce the weak solution by the standard proce-dure of multiplying (1) by a test function 120595 and integration byparts

intΩ+

120573998810119906 sdot 998810120595 + intΩminus

120573998810119906 sdot 998810120595 = intΩ

119891120595 minus intΓ

119887120595 (4)

where 120595 is in 11986710 (Ω) Note that although the test function120595 is the same as in the standard finite element method thefunction 119906 is not a linear combination of such basis functionsInstead the jump conditions are enforced strongly

3 Numerical Method

In this paper we restrict ourselves to a rectangular domainΩ = (119909min 119909max) times (119910min 119910max) in the plane Given positiveintegers 119868 and 119869 set Δ119909 = (119909max minus 119909min)119868 and Δ119910 =(119910max minus119910min)119869 We define a uniform Cartesian grid (119909119894 119910119895) =(119909min + 119894Δ119909 119910min + 119895Δ119910) for 119894 = 0 119868 and 119895 = 0 119869 Each(119909119894 119910119895) is called a grid point For the case 119894 = 0 119868 or 119895 = 0 119869 agrid point is called a boundary point otherwise it is called aninterior point The grid size is defined as ℎ = max(Δ119909 Δ119910) gt0

Two sets of grid functions are needed and they aredenoted by

1198671ℎ = 120596ℎ = (120596119894119895) 0 le 119894 le 119868 0 le 119895 le 119869 (5)

and

1198671ℎ0 = 120596ℎ = (120596119894119895) isin 1198671ℎ 120596119894119895 = 0 if 119894 = 0 119868 or 119895= 0 119869 (6)

Cut every rectangular region [119909119894 119909119894+1] times [119910119895 119910119895+1] intofour pieces of right triangular regions Collecting all thosetriangular regions we obtain a uniform triangulation 119879ℎ ⋃119870isin119879ℎ 119870 see Figure 1 This is slightly different from thetriangulation in our previous work [8] for convenience ofadaptive mesh refinement

We call a cell 119870 an interface cell if its vertices belong todifferent subdomains we write 119870 = 119870+ cup 119870minus 119870+ and 119870minus are



minus08

minus06

minus04

minus02

0

02

04

06

08

1


1

2 3(a) Regular cell

2 3

1

4 5

(b) Interface cell

Figure 2 Cells





int119870


119891119879ℎ (120595ℎ) (7)


[119906]4 = 1198864[119906]5 = 1198865

11988811199062 + 11988821199063 + 11988831199064 = 1199065[(120573998810119906) sdot n]6 = 1198876

(8)



int119870+



= int119870+

119891119879ℎ (120595ℎ) + int119870minus

119891119879ℎ (120595ℎ) minus intΓℎ119870

119887119879ℎ (120595ℎ) (9)










(int119870+


120573nabla119880ℎ (119906ℎ)


(int119870+

119891119879ℎ (120595ℎ)

+ int119870minus

119891119879ℎ (120595ℎ) minus intΓℎ119870

119887119879ℎ (120595ℎ))

(10)










31 2

4 16 5

9 10 14

11 12 13

6 715 8

17


are given Hence


(12)



10038171003817100381710038171198901199061003817100381710038171003817infin = max119909isinΩ

10038161003816100381610038161003816119906 minus 119906ℎ10038161003816100381610038161003816 10038171003817100381710038171198901199061003817100381710038171003817120573 = radicint




(13)


120601 (119909 119910) = 1199092 + 1199102 minus 015120573+ (119909 119910) = 1000 (1199092 + 3 cos (119909)

cos (119909) 1199102 + 2)

120573minus (119909 119910) = (1199092 + 3 cos (119909)cos (119909) 1199102 + 2)

119906+ (119909 119910) = sin (5119909119910) + 1119906minus (119909 119910) = sin (10119909119910)

(14)



0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

0

02

04

06

08

1minus2

024

002

0406

081

002

0406

081

minus2024

0 608

10 2

0406

08

0 02 04 06 08 1

002

0406

081

minus004

minus002

0

002

068

0 02 04 06 08 1

002

0406

081

minus10

minus5

0

5x 10 minus3

08

10







120601 (119909 119910) = 119909 + 05120573+ (119909 119910) = (1199092 + 1 0

0 1199102 + 1)

120573minus (119909 119910) = 1000 (1199092 + 1 00 1199102 + 1)


(15)





Order 198 Order 217



002040608

1

(a) Grids


005

1

00


012345



minus05

0

05

1x 10minus4

minus2024

(c) Numerical error


minus12

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

Erro

r)

45 5 55 6 65 74log(N12)












Uniform Mesh

1105 665e-3 26s

1588065 159e-3 185s28561 561e-4 865s99905 188e-4 5613s400513 659e-5 64481s

Adaptive Mesh

567 613e-3 36s

1622385 156e-3 143s9711 539e-4 662s39038 182e-4 3466s155869 617e-5 33298s


874 576e-3 58s

1863037 142e-3 184s11035 463e-4 720s41714 135e-4 3901s161208 425e-5 33434s


120601 (119909 119910) = 1199092 + 1199102 minus 025120573+ (119909 119910) = 1000120573minus (119909 119910) = 1


(16)




120601 (119909 119910) = 119910 minus 2119909 119909 + 119910 gt 0120601 (119909 119910) = 119910 + 1199092 119909 + 119910 le 0


(17)





002040608

1



002040608

1

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1

x 10minus4

minus15

minus10

minus5

0

5

minus1minus05

005

1

minus1minus05

005

1

x 10minus4


02






minus05

0

05

1

0

2

4

6

8

10


minus10

1 23 minus1

minus05

0

05

1

005

115

225

3





002040608

1



002040608

1

minus05 0 05 1 15 2 25 3minus1

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

2005

x 10minus3

minus2minus1

01234

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

minus15minus1

minus050

051

15


minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

log(

Erro

r)

35 4 45 5 55 6 653log(N12)










5 Conclusion


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018










minus08

minus06

minus04

minus02

0

02

04

06

08

1


1

2 3(a) Regular cell

2 3

1

4 5

(b) Interface cell

Figure 2 Cells





int119870


119891119879ℎ (120595ℎ) (7)


[119906]4 = 1198864[119906]5 = 1198865

11988811199062 + 11988821199063 + 11988831199064 = 1199065[(120573998810119906) sdot n]6 = 1198876

(8)



int119870+



= int119870+

119891119879ℎ (120595ℎ) + int119870minus

119891119879ℎ (120595ℎ) minus intΓℎ119870

119887119879ℎ (120595ℎ) (9)










(int119870+


120573nabla119880ℎ (119906ℎ)


(int119870+

119891119879ℎ (120595ℎ)

+ int119870minus

119891119879ℎ (120595ℎ) minus intΓℎ119870

119887119879ℎ (120595ℎ))

(10)










31 2

4 16 5

9 10 14

11 12 13

6 715 8

17


are given Hence


(12)



10038171003817100381710038171198901199061003817100381710038171003817infin = max119909isinΩ

10038161003816100381610038161003816119906 minus 119906ℎ10038161003816100381610038161003816 10038171003817100381710038171198901199061003817100381710038171003817120573 = radicint




(13)


120601 (119909 119910) = 1199092 + 1199102 minus 015120573+ (119909 119910) = 1000 (1199092 + 3 cos (119909)

cos (119909) 1199102 + 2)

120573minus (119909 119910) = (1199092 + 3 cos (119909)cos (119909) 1199102 + 2)

119906+ (119909 119910) = sin (5119909119910) + 1119906minus (119909 119910) = sin (10119909119910)

(14)



0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

0

02

04

06

08

1minus2

024

002

0406

081

002

0406

081

minus2024

0 608

10 2

0406

08

0 02 04 06 08 1

002

0406

081

minus004

minus002

0

002

068

0 02 04 06 08 1

002

0406

081

minus10

minus5

0

5x 10 minus3

08

10







120601 (119909 119910) = 119909 + 05120573+ (119909 119910) = (1199092 + 1 0

0 1199102 + 1)

120573minus (119909 119910) = 1000 (1199092 + 1 00 1199102 + 1)


(15)





Order 198 Order 217



002040608

1

(a) Grids


005

1

00


012345



minus05

0

05

1x 10minus4

minus2024

(c) Numerical error


minus12

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

Erro

r)

45 5 55 6 65 74log(N12)












Uniform Mesh

1105 665e-3 26s

1588065 159e-3 185s28561 561e-4 865s99905 188e-4 5613s400513 659e-5 64481s

Adaptive Mesh

567 613e-3 36s

1622385 156e-3 143s9711 539e-4 662s39038 182e-4 3466s155869 617e-5 33298s


874 576e-3 58s

1863037 142e-3 184s11035 463e-4 720s41714 135e-4 3901s161208 425e-5 33434s


120601 (119909 119910) = 1199092 + 1199102 minus 025120573+ (119909 119910) = 1000120573minus (119909 119910) = 1


(16)




120601 (119909 119910) = 119910 minus 2119909 119909 + 119910 gt 0120601 (119909 119910) = 119910 + 1199092 119909 + 119910 le 0


(17)





002040608

1



002040608

1

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1

x 10minus4

minus15

minus10

minus5

0

5

minus1minus05

005

1

minus1minus05

005

1

x 10minus4


02






minus05

0

05

1

0

2

4

6

8

10


minus10

1 23 minus1

minus05

0

05

1

005

115

225

3





002040608

1



002040608

1

minus05 0 05 1 15 2 25 3minus1

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

2005

x 10minus3

minus2minus1

01234

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

minus15minus1

minus050

051

15


minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

log(

Erro

r)

35 4 45 5 55 6 653log(N12)










5 Conclusion


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018

















(int119870+


120573nabla119880ℎ (119906ℎ)


(int119870+

119891119879ℎ (120595ℎ)

+ int119870minus

119891119879ℎ (120595ℎ) minus intΓℎ119870

119887119879ℎ (120595ℎ))

(10)










31 2

4 16 5

9 10 14

11 12 13

6 715 8

17


are given Hence


(12)



10038171003817100381710038171198901199061003817100381710038171003817infin = max119909isinΩ

10038161003816100381610038161003816119906 minus 119906ℎ10038161003816100381610038161003816 10038171003817100381710038171198901199061003817100381710038171003817120573 = radicint




(13)


120601 (119909 119910) = 1199092 + 1199102 minus 015120573+ (119909 119910) = 1000 (1199092 + 3 cos (119909)

cos (119909) 1199102 + 2)

120573minus (119909 119910) = (1199092 + 3 cos (119909)cos (119909) 1199102 + 2)

119906+ (119909 119910) = sin (5119909119910) + 1119906minus (119909 119910) = sin (10119909119910)

(14)



0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

0

02

04

06

08

1minus2

024

002

0406

081

002

0406

081

minus2024

0 608

10 2

0406

08

0 02 04 06 08 1

002

0406

081

minus004

minus002

0

002

068

0 02 04 06 08 1

002

0406

081

minus10

minus5

0

5x 10 minus3

08

10







120601 (119909 119910) = 119909 + 05120573+ (119909 119910) = (1199092 + 1 0

0 1199102 + 1)

120573minus (119909 119910) = 1000 (1199092 + 1 00 1199102 + 1)


(15)





Order 198 Order 217



002040608

1

(a) Grids


005

1

00


012345



minus05

0

05

1x 10minus4

minus2024

(c) Numerical error


minus12

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

Erro

r)

45 5 55 6 65 74log(N12)












Uniform Mesh

1105 665e-3 26s

1588065 159e-3 185s28561 561e-4 865s99905 188e-4 5613s400513 659e-5 64481s

Adaptive Mesh

567 613e-3 36s

1622385 156e-3 143s9711 539e-4 662s39038 182e-4 3466s155869 617e-5 33298s


874 576e-3 58s

1863037 142e-3 184s11035 463e-4 720s41714 135e-4 3901s161208 425e-5 33434s


120601 (119909 119910) = 1199092 + 1199102 minus 025120573+ (119909 119910) = 1000120573minus (119909 119910) = 1


(16)




120601 (119909 119910) = 119910 minus 2119909 119909 + 119910 gt 0120601 (119909 119910) = 119910 + 1199092 119909 + 119910 le 0


(17)





002040608

1



002040608

1

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1

x 10minus4

minus15

minus10

minus5

0

5

minus1minus05

005

1

minus1minus05

005

1

x 10minus4


02






minus05

0

05

1

0

2

4

6

8

10


minus10

1 23 minus1

minus05

0

05

1

005

115

225

3





002040608

1



002040608

1

minus05 0 05 1 15 2 25 3minus1

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

2005

x 10minus3

minus2minus1

01234

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

minus15minus1

minus050

051

15


minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

log(

Erro

r)

35 4 45 5 55 6 653log(N12)










5 Conclusion


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018











31 2

4 16 5

9 10 14

11 12 13

6 715 8

17


are given Hence


(12)



10038171003817100381710038171198901199061003817100381710038171003817infin = max119909isinΩ

10038161003816100381610038161003816119906 minus 119906ℎ10038161003816100381610038161003816 10038171003817100381710038171198901199061003817100381710038171003817120573 = radicint




(13)


120601 (119909 119910) = 1199092 + 1199102 minus 015120573+ (119909 119910) = 1000 (1199092 + 3 cos (119909)

cos (119909) 1199102 + 2)

120573minus (119909 119910) = (1199092 + 3 cos (119909)cos (119909) 1199102 + 2)

119906+ (119909 119910) = sin (5119909119910) + 1119906minus (119909 119910) = sin (10119909119910)

(14)



0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

0

02

04

06

08

1minus2

024

002

0406

081

002

0406

081

minus2024

0 608

10 2

0406

08

0 02 04 06 08 1

002

0406

081

minus004

minus002

0

002

068

0 02 04 06 08 1

002

0406

081

minus10

minus5

0

5x 10 minus3

08

10







120601 (119909 119910) = 119909 + 05120573+ (119909 119910) = (1199092 + 1 0

0 1199102 + 1)

120573minus (119909 119910) = 1000 (1199092 + 1 00 1199102 + 1)


(15)





Order 198 Order 217



002040608

1

(a) Grids


005

1

00


012345



minus05

0

05

1x 10minus4

minus2024

(c) Numerical error


minus12

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

Erro

r)

45 5 55 6 65 74log(N12)












Uniform Mesh

1105 665e-3 26s

1588065 159e-3 185s28561 561e-4 865s99905 188e-4 5613s400513 659e-5 64481s

Adaptive Mesh

567 613e-3 36s

1622385 156e-3 143s9711 539e-4 662s39038 182e-4 3466s155869 617e-5 33298s


874 576e-3 58s

1863037 142e-3 184s11035 463e-4 720s41714 135e-4 3901s161208 425e-5 33434s


120601 (119909 119910) = 1199092 + 1199102 minus 025120573+ (119909 119910) = 1000120573minus (119909 119910) = 1


(16)




120601 (119909 119910) = 119910 minus 2119909 119909 + 119910 gt 0120601 (119909 119910) = 119910 + 1199092 119909 + 119910 le 0


(17)





002040608

1



002040608

1

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1

x 10minus4

minus15

minus10

minus5

0

5

minus1minus05

005

1

minus1minus05

005

1

x 10minus4


02






minus05

0

05

1

0

2

4

6

8

10


minus10

1 23 minus1

minus05

0

05

1

005

115

225

3





002040608

1



002040608

1

minus05 0 05 1 15 2 25 3minus1

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

2005

x 10minus3

minus2minus1

01234

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

minus15minus1

minus050

051

15


minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

log(

Erro

r)

35 4 45 5 55 6 653log(N12)










5 Conclusion


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1

0

02

04

06

08

1minus2

024

002

0406

081

002

0406

081

minus2024

0 608

10 2

0406

08

0 02 04 06 08 1

002

0406

081

minus004

minus002

0

002

068

0 02 04 06 08 1

002

0406

081

minus10

minus5

0

5x 10 minus3

08

10







120601 (119909 119910) = 119909 + 05120573+ (119909 119910) = (1199092 + 1 0

0 1199102 + 1)

120573minus (119909 119910) = 1000 (1199092 + 1 00 1199102 + 1)


(15)





Order 198 Order 217



002040608

1

(a) Grids


005

1

00


012345



minus05

0

05

1x 10minus4

minus2024

(c) Numerical error


minus12

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

Erro

r)

45 5 55 6 65 74log(N12)












Uniform Mesh

1105 665e-3 26s

1588065 159e-3 185s28561 561e-4 865s99905 188e-4 5613s400513 659e-5 64481s

Adaptive Mesh

567 613e-3 36s

1622385 156e-3 143s9711 539e-4 662s39038 182e-4 3466s155869 617e-5 33298s


874 576e-3 58s

1863037 142e-3 184s11035 463e-4 720s41714 135e-4 3901s161208 425e-5 33434s


120601 (119909 119910) = 1199092 + 1199102 minus 025120573+ (119909 119910) = 1000120573minus (119909 119910) = 1


(16)




120601 (119909 119910) = 119910 minus 2119909 119909 + 119910 gt 0120601 (119909 119910) = 119910 + 1199092 119909 + 119910 le 0


(17)





002040608

1



002040608

1

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1

x 10minus4

minus15

minus10

minus5

0

5

minus1minus05

005

1

minus1minus05

005

1

x 10minus4


02






minus05

0

05

1

0

2

4

6

8

10


minus10

1 23 minus1

minus05

0

05

1

005

115

225

3





002040608

1



002040608

1

minus05 0 05 1 15 2 25 3minus1

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

2005

x 10minus3

minus2minus1

01234

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

minus15minus1

minus050

051

15


minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

log(

Erro

r)

35 4 45 5 55 6 653log(N12)










5 Conclusion


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018











Order 198 Order 217



002040608

1

(a) Grids


005

1

00


012345



minus05

0

05

1x 10minus4

minus2024

(c) Numerical error


minus12

minus115

minus11

minus105

minus10

minus95

minus9

minus85

minus8

log(

Erro

r)

45 5 55 6 65 74log(N12)












Uniform Mesh

1105 665e-3 26s

1588065 159e-3 185s28561 561e-4 865s99905 188e-4 5613s400513 659e-5 64481s

Adaptive Mesh

567 613e-3 36s

1622385 156e-3 143s9711 539e-4 662s39038 182e-4 3466s155869 617e-5 33298s


874 576e-3 58s

1863037 142e-3 184s11035 463e-4 720s41714 135e-4 3901s161208 425e-5 33434s


120601 (119909 119910) = 1199092 + 1199102 minus 025120573+ (119909 119910) = 1000120573minus (119909 119910) = 1


(16)




120601 (119909 119910) = 119910 minus 2119909 119909 + 119910 gt 0120601 (119909 119910) = 119910 + 1199092 119909 + 119910 le 0


(17)





002040608

1



002040608

1

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1

x 10minus4

minus15

minus10

minus5

0

5

minus1minus05

005

1

minus1minus05

005

1

x 10minus4


02






minus05

0

05

1

0

2

4

6

8

10


minus10

1 23 minus1

minus05

0

05

1

005

115

225

3





002040608

1



002040608

1

minus05 0 05 1 15 2 25 3minus1

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

2005

x 10minus3

minus2minus1

01234

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

minus15minus1

minus050

051

15


minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

log(

Erro

r)

35 4 45 5 55 6 653log(N12)










5 Conclusion


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018















Uniform Mesh

1105 665e-3 26s

1588065 159e-3 185s28561 561e-4 865s99905 188e-4 5613s400513 659e-5 64481s

Adaptive Mesh

567 613e-3 36s

1622385 156e-3 143s9711 539e-4 662s39038 182e-4 3466s155869 617e-5 33298s


874 576e-3 58s

1863037 142e-3 184s11035 463e-4 720s41714 135e-4 3901s161208 425e-5 33434s


120601 (119909 119910) = 1199092 + 1199102 minus 025120573+ (119909 119910) = 1000120573minus (119909 119910) = 1


(16)




120601 (119909 119910) = 119910 minus 2119909 119909 + 119910 gt 0120601 (119909 119910) = 119910 + 1199092 119909 + 119910 le 0


(17)





002040608

1



002040608

1

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1

x 10minus4

minus15

minus10

minus5

0

5

minus1minus05

005

1

minus1minus05

005

1

x 10minus4


02






minus05

0

05

1

0

2

4

6

8

10


minus10

1 23 minus1

minus05

0

05

1

005

115

225

3





002040608

1



002040608

1

minus05 0 05 1 15 2 25 3minus1

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

2005

x 10minus3

minus2minus1

01234

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

minus15minus1

minus050

051

15


minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

log(

Erro

r)

35 4 45 5 55 6 653log(N12)










5 Conclusion


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018











002040608

1



002040608

1

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1minus005

0

005

01

015

minus1minus05

005

1

minus1minus05

005

1

x 10minus4

minus15

minus10

minus5

0

5

minus1minus05

005

1

minus1minus05

005

1

x 10minus4


02






minus05

0

05

1

0

2

4

6

8

10


minus10

1 23 minus1

minus05

0

05

1

005

115

225

3





002040608

1



002040608

1

minus05 0 05 1 15 2 25 3minus1

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

2005

x 10minus3

minus2minus1

01234

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

minus15minus1

minus050

051

15


minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

log(

Erro

r)

35 4 45 5 55 6 653log(N12)










5 Conclusion


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018










002040608

1



002040608

1

minus05 0 05 1 15 2 25 3minus1

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1minus2

02468

10

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

2005

x 10minus3

minus2minus1

01234

minus1 0 1 2 3

minus1minus05

005

1

x 10minus3

minus15minus1

minus050

051

15


minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

log(

Erro

r)

35 4 45 5 55 6 653log(N12)










5 Conclusion


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018










5 Conclusion


Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018




















Journal of












Journal of







Volume 2018






Volume 2018








a numerical method for solving elliptic interface problems using...

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