COMPACT 9-POINT FINITE DIFFERENCE METHODS WITH HIGH ACCURACY ORDER ANDOR M-MATRIX PROPERTY FOR ELLIPTIC CROSS-INTERFACE PROBLEMS

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COMPACT 9-POINT FINITE DIFFERENCE METHODS WITH HIGH

ACCURACY ORDER AND/OR M-MATRIX PROPERTY FOR ELLIPTIC

CROSS-INTERFACE PROBLEMS

QIWEI FENG, BIN HAN AND PETER MINEV

Abstract. In this paper we develop ﬁnite diﬀerence schemes for elliptic problems with piecewise

continuous coeﬃcients that have (possibly huge) jumps across ﬁxed internal interfaces. In contrast with

such problems involving one smooth non-intersecting interface, that have been extensively studied,

there are very few papers addressing elliptic interface problems with intersecting interfaces of coeﬃcient

jumps. It is well known that if the values of the permeability in the four subregions around a point of

intersection of two such internal interfaces are all diﬀerent, the solution has a point singularity that

signiﬁcantly aﬀects the accuracy of the approximation in the vicinity of the intersection point. In the

present paper we propose a fourth-order 9-point ﬁnite diﬀerence scheme on uniform Cartesian meshes

for an elliptic problem whose coeﬃcient is piecewise constant in four rectangular subdomains of the

overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting

point of the two lines of coeﬃcient jumps is a grid point, such a compact scheme, involving relatively

simple formulas for computation of the stencil coeﬃcients, can even reach sixth order of accuracy.

Furthermore, we show that the resulting linear system for the special case has an M-matrix, and prove

the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical

experiments demonstrate the sixth (for the special case) and at least fourth (for the general case)

accuracy orders of the proposed schemes. In the general case, we derive a compact third-order ﬁnite

diﬀerence scheme, also yielding a linear system with an M-matrix. In addition, using the discrete

maximum principle, we prove the third order convergence rate of the scheme for the general elliptic

cross-interface problem.

1. Introduction and problem formulation

Interface problems arise in many applications such as modeling of underground waste disposal,

oil reservoirs, composite materials, and many others. Some approaches to the solution of the el-

liptic interface problem with a smooth non-intersecting interface of coeﬃcient jumps were provided

by the immersed interface methods (IIM, see [7–9, 14, 16, 19, 20, 24, 29, 32] and references therein)

and matched interface and boundary methods (MIB, see [13, 30, 31, 33, 34]). Both methods belong

to the class of the ﬁnite diﬀerence methods (FDM). Elliptic interface problems with intersecting

interfaces appear in many applications, but perhaps the most notorious example is the modeling

of geological porous media ﬂows (see e.g. [1, 3–6, 15, 17, 18, 23, 25, 27]). A classical problem of

this type is formulated by the Society of Petroleum Engineers, the so-called SPE10 problem (see

https://www.spe.org/web/csp/datasets/set02.htm). Here we consider a 2D simpliﬁcation of this

problem that involves interface intersections of vertical straight lines and horizontal straight lines,

so that the permeability coeﬃcient in the four subregions in the vicinity of an intersection point has

diﬀerent values. Even for this relatively simple cross-interface problem, the only compact 9-point ﬁ-

nite diﬀerence method in the literature, that we are aware of, is the scheme in [2], that is third-order

consistent, and uses special non-uniform meshes. To our knowledge, the convergence rate of this

scheme has never been proven. We should also note here that the diﬃculty of the problem is usually

2010 Mathematics Subject Classiﬁcation. 65N06, 35J15, 76S05, 41A58.

Key words and phrases. Cross-interfaces, compact 9-point ﬁnite diﬀerence methods, explicit formulas, M-matrix

property, theoretical convergence, discrete maximum principle.

Research supported in part by Natural Sciences and Engineering Research Council (NSERC) of Canada under

grants RGPIN-2019-04276 (Bin Han), RGPIN-2017-04152 (Peter Minev), Westgrid (www.westgrid.ca), and Compute

Canada Calcul Canada (www.computecanada.ca).

arXiv:2210.01290v2 [math.NA] 17 Feb 2023

2 QIWEI FENG, BIN HAN AND PETER MINEV

exacerbated if the jumps of the permeability coeﬃcient across the diﬀerent interfaces are very large

(of several orders of magnitude). Details of the physical background of the elliptic interface problem

can be found in [28].

In practice, usually the permeability variation occurs on scales that are very small as compared to

the size of the medium, and therefore the solution of the interface problem is highly oscillatory. This

causes the appearance of the so-called pollution eﬀect in the error of its numerical approximation. In

order to obtain a reasonable low-order numerical solution to such problems one needs to employ a very

ﬁne, possibly nonuniform grid, that captures the small scale features. Therefore, the development of

higher-order compact approximations can help to reduce the computational costs. Compared to the

ﬁnite element or ﬁnite volume methods, the FDM does not require the integration of highly-oscillatory

or discontinuous functions. Furthermore, a compact 9-point scheme (in 2D) yields a sparse linear

system that can be solved very eﬃciently.

Finally we should mention, that even in the relatively simple case of elliptic interface problems

involving a smooth non-intersecting interface of coeﬃcient jumps, the theoretical proof of convergence

of the various proposed ﬁnite diﬀerence schemes is usually missing. The only exceptions are presented

in [21] using discrete maximum principle for a second order scheme, and [12] using numerically veriﬁed

discrete maximum principle for a fourth order scheme. The compact 9-point schemes considered in

the present paper possess the M-matrix property, that guarantees the discrete maximum principle

for the numerical solution. In turn, this property greatly facilitates the proof of their convergence

rate.

In this paper we develop numerical approximations to the following elliptic cross-interface problem:

Given the domain Ω := (l1, l2)×(l3, l4)with l1, l2, l3, l4∈R, then consider:











−∇ · a∇u=fin Ω\Γ,

[u] = φpon Γpfor p= 1,2,3,4,

[a∇u·~n] = ψpon Γpfor p= 1,2,3,4,

u=gon ∂Ω,

(1.1)

where the cross-interface Γis given by Γ := Γ1∪Γ2∪Γ3∪Γ4∪ {(ξ, ζ)}with

Γ1:= {ξ}×(ζ, l4),Γ2:= {ξ}×(l3, ζ),Γ3:= (ξ, l2)×{ζ},Γ4:= (l1, ξ)×{ζ},(ξ, ζ)∈Ω.

As usual, that the square brackets here denote the jump of the corresponding function, i.e. for

(ξ, y)∈Γpwith p= 1,2 (on the vertical line of the cross-interface Γ),

[u](ξ, y) := lim

x→ξ+u(x, y)−lim

x→ξ−u(x, y),[a∇u·~n](ξ, y) := lim

x→ξ+a(x, y)∂u

∂x(x, y)−lim

x→ξ−a(x, y)∂u

∂x(x, y);

while for (x, ζ)∈Γpwith p= 3,4 (i.e., on the horizontal line of the cross-interface Γ),

[u](x, ζ) := lim

y→ζ+u(x, y)−lim

y→ζ−u(x, y),[a∇u·~n](x, ζ) := lim

y→ζ+a(x, y)∂u

∂y (x, y)−lim

y→ζ−a(x, y)∂u

∂y (x, y).

Note that the interface curve Γ divides the domain Ω into 4 subdomains:

Ω1:= (l1, ξ)×(ζ, l4),Ω2:= (ξ, l2)×(ζ, l4),Ω3:= (ξ, l2)×(l3, ζ),Ω4:= (l1, ξ)×(l3, ζ).

See Fig. 1 for an illustration, where ap:= aχΩp,fp:= fχΩp, and up:= uχΩpfor p= 1,2,3,4.

To derive a compact 9-point scheme that approximates the cross-interface problem (1.1), we assume

that:

(A1) apis a positive constant in Ωp.

(A2) The restriction of the solution upand of the source term fphas uniformly continuous partial

derivatives of (total) orders up to seven and ﬁve, respectively, in each Ωpfor p= 1,2,3,4.

(A3) The essentially one-dimensional functions φpand ψpin (1.1) on the interface Γphave uniformly

continuous derivatives of orders up to seven and six respectively for p= 1,2,3,4.

COMPACT FINITE DIFFERENCE DISCRETIZATIONS FOR ELLIPTIC CROSS-INTERFACE PROBLEMS 3

x=ξ

y=ζ

x=l1x=l2

y=l4

y=l3Γ2

Γ1

Γ3

Γ4

Ω1Ω2

Ω3

Ω4φ2

φ1

φ3

φ4

a1a2

a4ψ2

ψ1

ψ3

ψ4

u1u2

~n~n

Figure 1. An illustration for the model elliptic cross-interface problem in (1.1).

Figure 2. An illustration for uniform Cartesian grids of the model problem in (1.1)

The remainder of the paper is organized as follows. In Section 2.1, we derive a compact 9-point

scheme with sixth order of consistency for interior grid points in Theorem 2.1. For grid points near

the interface, as illustrated by Fig. 2, we have two cases:

Case 1: If the point (ξ, ζ) of intersection of the interfaces is a grid point (see the left panel of

Fig. 2), we derive in Section 2.2 a compact 9-point scheme that has a seventh order of consistency

at every grid point lying on the cross-interface. The stencil coeﬃcients are given in Theorems 2.2

and 2.3. In Section 3.1 we prove that this scheme is sixth-order accurate, using the discrete maximum

principle satisﬁed by it. The results of some numerical experiments, demonstrating the sixth-order

convergence rate of the scheme, are presented in Section 4.1.

Case 2: If (ξ, ζ) is not a grid point (see the right panel in Fig. 2), we derive in Section 2.2 a compact

9-point scheme with fourth order of consistency for every grid point neighboring the interface. Next

we show in Section 3.2 that this scheme does not satisfy the M-matrix property. Subsequently, we

obtain a compact scheme satisfying the M-matrix property, with a consistency order three at grid

points neighboring the interface points except for the vicinity of the intersection point (ξ, ζ), and

order two at grid points neighboring (ξ, ζ). Since the M-matrix property immediately implies that

the scheme satisﬁes a discrete maximum principle, this allows us to prove that the overall convergence

rate of the scheme is of order three. In Section 4.2 we provide some numerical results that seem to

suggest that the scheme given in Theorems 2.1, 2.4 and 2.5, that does not satisfy a discrete maximum

principle, is ﬁfth-order accurate.

In Section 5, we summarize the main contributions of this paper. Finally, in Section 6 we present

the proofs for the results stated in Sections 2 and 3.

2. High order compact 9-point schemes using uniform Cartesian grids

In this section, we present some compact ﬁnite diﬀerence schemes on uniform Cartesian grids for

the elliptic cross-interface problem in (1.1). To improve readability, the technical proofs of the results

stated in this section are deferred to Section 6.

4 QIWEI FENG, BIN HAN AND PETER MINEV

We start by introducing a uniform Cartesian mesh on the domain:

Ω := (l1, l2)×(l3, l4),with l4−l3=N0(l2−l1) for some positive integer N0,

containing the grid points:

xi:= l1+ih, i = 0, . . . , N1,and yj:= l3+jh, j = 0, . . . , N2, h := l2−l1

N1=l4−l3

N2,

where N1is a positive integer and N2:= N0N1. We also deﬁne (uh)i,j to be the value of the numerical

approximation uhof the exact solution uof the elliptic cross-interface problem (1.1), at the grid point

(xi, yj). For stencil coeﬃcients {Ck,`}k,`=−1,0,1with Ck,` ∈Rin the compact 9-point stencil centered

at a grid point (xi, yj), the discrete operator Lhacting on uhis deﬁned to be:

Lhuh:=

k=−1

`=−1

Ck,`(uh)i+k,j+`.(2.1)

Similarly, the action of the discrete operator Lhon the exact solution uis given by:

Lhu:=

k=−1

`=−1

Ck,`u(xi+kh, yj+`h).(2.2)

2.1. Compact 9-point stencils at interior points. The following compact FDM with a consis-

tency order six for (1.1) at interior points is well known in the literature (e.g., see [11,26]).

Theorem 2.1. Consider (xi, yj)∈Ωwith all 9 points (xi±h, yj±h)∈Ωpfor some p∈ {1,2,3,4}.

Assume that up:= uχΩpand fp:= fχΩphave uniformly continuous partial derivatives of (total)

orders up to seven and ﬁve, respectively, in Ωp. Let the discrete operator Lhbe deﬁned in (2.1) with

the stencil coeﬃcients

C0,0= 20, C0,−1=C0,1=C−1,0=C1,0=−4, C1,1=C1,−1=C−1,1=C−1,−1=−1.

Then the compact 9-point ﬁnite diﬀerence scheme h−2Lhuh=1

a(xi,yj)Fwith

F:= 6f(xi, yj) + h2

2h∂2f

∂2x(xi, yj) + ∂2f

∂2y(xi, yj)i+h4

60 h∂4f

∂4x(xi, yj) + ∂4f

∂4y(xi, yj)i+h4

∂4f

∂2x∂2y(xi, yj),

has a sixth order of consistency at the interior grid point (xi, yj)for −∇ · (a∇u) = f.

2.2. Compact 9-point stencils at grid points on the interface. In this subsection, we now

discuss how to ﬁnd a compact FDM of a consistency order seven at grid points (xi, yj) lying on

the cross-interface (i.e., Case 1 in Section 1, see the left panel of Fig. 2). Note that all interfaces

Γ1,...,Γ4are open intervals and hence they do not contain the intersection point (ξ, ζ).

In order to devise the explicit formulas for the coeﬃcients of the compact scheme, we need some

notations and deﬁnitions. Recall that ap:= aχΩpis a positive constant, fp:= fχΩp, and up:= uχΩp

for p= 1,2,3,4. For (x∗

i, y∗

j)∈Ωpwith p= 1,2,3,4, using only the information of upin Ωp, we can

deﬁne

u(m,n)

p:= ∂m+nup

∂mx∂ny(x∗

i, y∗

j), f(m,n)

p:= ∂m+nfp

∂mx∂ny(x∗

i, y∗

j).

For (x∗

i, y∗

j)=(ξ, y∗

j)∈Γpwith p= 1,2, we deﬁne φ(n)

p:= dnφp(ξ,y)

dny|y=y∗

jand ψ(n)

p:= dnψp(ξ,y)

dny|y=y∗

while for (x∗

i, y∗

j) = (x∗

i, ζ)∈Γpwith p= 3,4, we similarly deﬁne φ(n)

p:= dnφp(x,ζ)

dnx|x=x∗

iand ψ(n)

p:=

dnψp(x,ζ)

dnx|x=x∗

i. Note that φp, ψpin (1.1) are essentially 1D functions deﬁned on the line segment Γp.

Deﬁne N0:= N∪ {0}and for M∈N0, we deﬁne the following index subsets of N2

0as follows:

ΛM:= {(m, n)∈N2

0:m+n≤M},Λ1

M:= {(m, n)∈ΛM:m= 0,1},Λ2

M:= ΛM\Λ1

M.(2.3)

COMPACT FINITE DIFFERENCE DISCRETIZATIONS FOR ELLIPTIC CROSS-INTERFACE PROBLEMS 5

The illustrations for Λ1

7and Λ2

7are shown in Fig. 12 in Section 6. We shall also deﬁne the following

bivariate polynomials (which will be used in our compact FDMs later):

GM,m,n(x, y) :=

`=0

(−1)`xm+2`yn−2`

(m+ 2`)!(n−2`)!, HM,m,n(x, y) :=

1+bn

`=1

(−1)`xm+2`yn−2`+2

(m+ 2`)!(n−2`+ 2)!,(2.4)

where bxcis the ﬂoor function representing the largest integer less than or equal to x∈R.

To state our compact FDMs later, we shall use some auxiliary polynomials of h. For w∈R,

M∈N0and m, n ∈N0, we deﬁne the univariate polynomials of hwith the parameter wfor the

vertical interface Γ1or Γ2to be

M,m,n :=

`=−1

C−1,`GM,m,n(wh −h, `h),

H−,w

M,m,n :=

`=−1

C−1,`HM,m,n(wh −h, `h), H+,w

M,m,n :=

k=0

`=−1

Ck,`HM,m,n(wh +kh, `h).

(2.5)

u1u2

Γ2

Γ1

Γ3

Γ4

u1u2

u4Γ2

Γ1

Γ3

Γ4

u1u2

u4Γ2

Γ1

Γ3

Γ4

Figure 3. First panel: compact 9-point stencils in Theorem 2.2 with (xi, yj) =

(x∗

i, y∗

j)∈Γ1∪Γ2. Second panel: compact 9-point stencils with (xi, yj)=(x∗

i, y∗

j)∈

Γ3∪Γ4. Third panel: compact 9-point stencil in Theorem 2.3 with (xi, yj)=(x∗

i, y∗

j) =

(ξ, ζ). The grid point (xi, yj) is indicated by the red color.

We ﬁrst consider the compact discretization at grid points lying on the vertical interface line

Γ1or Γ2. The modiﬁcation of the scheme corresponding to the horizontal interfaces Γ3or Γ4is

straightforward and we will only brieﬂy mention it afterwards.

Theorem 2.2. Consider a grid point (xi, yj)such that (xi, yj)∈Γ1(see the ﬁrst panel of Fig. 3).

Assume that up:= uχΩpand fp:= fχΩphave uniformly continuous partial derivatives of (total)

orders up to seven and ﬁve, respectively, in each Ωpfor p= 1,2. Also assume that the essentially

one-dimensional functions φ1and ψ1on the interface Γ1have uniformly continuous derivatives of

orders up to seven and six, respectively. Let (x∗

i, y∗

j) := (xi, yj)and Lhbe the discrete operator in

(2.1) with the stencil coeﬃcients

C1,0=−4, C1,−1=C1,1=−1, C−1,−1=C−1,1=−α, C0,−1=C0,1=−2(1 + α),

C−1,0=−4α, C0,0= 10(1 + α), α := a1/a2>0.(2.6)

Then the compact 9-point ﬁnite diﬀerence scheme h−1Lhuh=h−1Fhas a seventh order of consistency

at the grid point (xi, yj)∈Γ1, where

F:= 1

a1X

(m,n)∈Λ5

f(m,n)

1H−,0

7,m,n +1

a2X

(m,n)∈Λ5

f(m,n)

2H+,0

7,m,n −

n=0

φ(n)

1G0

7,0,n −1

n=0

ψ(n)

1G0

7,1,n (2.7)

and H−,0

7,m,n, H+,0

7,m,n, G0

7,0,n, G0

7,1,n are deﬁned in (2.5).

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COMPACT9-POINTFINITEDIFFERENCEMETHODSWITHHIGHACCURACYORDERAND/ORM-MATRIXPROPERTYFORELLIPTICCROSS-INTERFACEPROBLEMSQIWEIFENG,BINHANANDPETERMINEVAbstract.Inthispaperwedevelopnitedierenceschemesforellipticproblemswithpiecewisecontinuouscoecientsthathave(possiblyhuge)jumpsacrossxedinternalinterfaces...

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COMPACT 9-POINT FINITE DIFFERENCE METHODS WITH HIGH ACCURACY ORDER ANDOR M-MATRIX PROPERTY FOR ELLIPTIC CROSS-INTERFACE PROBLEMS

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