Discontinuous finite volume methods for

DOI: 10.1002/nme.5171 Discontinuous finite volume methods for the stationary Stokes–Darcy problem Gang Wang, Yinnian He*,†and Rui Li School of Mathematics and Statistics, Xi’an Jiaotong

Int J Numer Meth Engng (2015)

Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/nme.5171

Discontinuous finite volume methods for the stationary

Stokes–Darcy problem Gang Wang, Yinnian He*,†and Rui Li

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

SUMMARY

In this paper, we analyze discontinuous finite volume methods for the stationary Stokes–Darcy problemthat models coupled fluid flow and porous media flow The discontinuous finite volume methods arecombinations of finite volume method and discontinuous Galerkin method with three interior penaltytypes (incomplete symmetric, nonsymmetric, and symmetric), briefly, using discontinuous functions as trialfunctions in the finite volume method Optimal error estimates in broken H1norm are obtained for thethree discontinuous finite volume methods Optimal error estimates in the standard L2norm are derived forthe symmetric interior penalty discontinuous finite volume method Numerical experiments are presented toconfirm the theoretical results with non-matching meshes across the common interface of Stokes region andDarcy region Copyright © 2015 John Wiley & Sons, Ltd

Received 16 July 2015; Revised 22 October 2015; Accepted 5 November 2015

KEY WORDS: discontinuous Galerkin; finite volume method; Stokes equation; Darcy law; non-matching

meshes; optimal error estimates

1 INTRODUCTIONThe discontinuous Galerkin (DG) finite element method was originally introduced for a linear hyper-bolic problem by Reed and Hill in 1973 [1] Because of the use of discontinuous functions, DGmethod has the advantages of a high order of accuracy, high parallelizability, local mass conserva-tion, allowed hanging nodes, and easy handling of complicated geometries Since then, the study

of DG methods has been an active research field Recognizing that the interelement continuity ofapproximating functions could be weakly imposed, in the 1970s, Arnold [2] and Wheeler [3] intro-

duced interior penalty DG method for elliptic and parabolic problems Recently, Arnold et al [4]

provided a framework for the analysis of a large class of DG methods The framework has givenprofound understanding and comparison of most of the DG methods that have been proposed overthe past three decades for the numerical treatment of elliptic problems Well-known shortcoming of

DG methods is a large number of degrees of freedom For the coercivity of bilinear term, we need

to select large enough penalty parameters according to the specific problems More research worksconcerning DG methods can be found in [5–11] As another classical discretization technique forsolving the partial differential equations, finite volume method (FVM) deals with integral equationsobtained over a control volume on a dual mesh The attractive property of FVM is the local con-servation of a quantity of interest, such as mass, momentum, or energy Because of its physicalconservation property and simplicity, FVM is widely used in computational fluid mechanics andother applications But we cannot achieve a high order of accuracy when using FVM Recently,Cui [12] established a general framework for analyzing the class of FVMs for the Stokes equations.Under the framework, optimal convergence order for velocity and pressure in various Sobolev norms

*Correspondence to: Yinnian He, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China.

† E-mail: heyn@mail.xjtu.edu.cn

is obtained in a natural and systematic way Many researchers have focused on this area and gainedsome significant results; some works can be found in [13–16] for elliptic problem, in [17–20] forStokes and Navier–Stokes problems, and in [21, 22] for some overviews.

Combining DG and FVM, Ye [23] proposed a discontinuous finite volume method (DFVM) for

the second-order elliptic problem and obtained a priori error estimates in a mesh-dependent norm.

Like we expect, combining the benefits of DG and FVM, such method using discontinuous wise polynomials for the trial functions has the flexibility, a high order of accuracy of DG, andthe simplicity and conservative property of FVM The fascinating nature of DFVM reflects on thesmaller conservation control volume that is less than half the size of control volume applied inthe existing FVMs The localizability of the discontinuous element and its dual partition in DFVMshould provide an advantage for parallel computing Then Ye [24] spread DFVM to the Stokes prob-lem, using discontinuous P 1  P 0 approximations of velocity and pressure DFVM has been used

piece-to solve elliptic problem with adaptive technique or other applications [25–29] Recently, there are

a few papers applying DFVM to Stokes problem [30–32] Kumar [33] has applied DFVM to solvenonlinear problem

In this paper, we study DFVM for solving Stokes–Darcy model describing the coupling fluidflow with porous media flow This model is composed of Stokes equations for the fluid flow andDarcy’s law for the porous media flow, coupling through three interface conditions A lot of methodshave been developed to numerically solve the Stokes–Darcy model, such as finite element methods[34], two-grid methods [35], DG methods [36], decoupled methods [37, 38], domain decompositionmethods [39, 40], and so on Compared with these existing methods, DFVM has the property oflocal mass conservation at a so-called diamond control volume level Because of using low-orderpolynomial as the trial function, DFVM is easy to code, but also, a high order of accuracy can beachieved Because discontinuous functions are used in the approximation, the number of degrees offreedom is larger, but still in acceptable range So, parallel computing is a good choice in the future

In this article, we obtained the optimal order error estimates corresponding the variables in L2normand broken H1 norm Some experiments are presented to illustrate the correctness of the theory.Our numerical examples show that non-matching meshes are allowed across the interface of Stokesdomain and Darcy domain

The rest of the paper is organized as follows In Section 2, we describe the Stokes–Darcymodel and give the abstract function spaces, the preparations of DG In Section 3, we discussthe DFVM for the Stokes–Darcy model and some lemmas that are used in the error analysis InSection 4, we present optimal order error estimation In Section 5, some experiments are made

to validate the correctness and applicability of the algorithms Throughout the paper, the letter Cdenotes a positive constant independent of the mesh size and may indicate different values at itsdifferent appearances

2 PRELIMINARY AND NOTATION

We consider a coupled Stokes–Darcy model in a bounded domain   Rd.d D 2 or 3/, consisting

of a fluid region f and a porous medium region p, with interface  D @f T

@p, as depicted

in Figure 1 Both f and p have Lipschitz continuous boundaries nf and np denote the unitoutward normal vectors on @f and @p, respectively, and i i D 1; 2; : : : ; d  1/ the unit

tangential vectors on the interface  Note that npD nf on 

In f, the fluid flow is assumed to be governed by the Stokes equations:

²

uf C rp D gf in f;

where  > 0 is the kinetic viscosity, uf denotes the fluid velocity, p denotes the kinematic pressure,

and gf denotes a general body force term that includes gravitational acceleration

In p, the flow is governed by the Darcy law:

Figure 1 A sketch of the porous medium domain p, fluid domain f, and interface .

where q is the specific discharge, gp is a sink/source term, K is the hydraulic conductivity tensor,

n is the volumetric porosity, upis the fluid velocity in p, and the hydraulic head ' D ´ C pp

g,the sum of elevation head plus pressure head, where ppis the pressure of the fluid in p,  is thedensity of the fluid, g is the gravitational acceleration, and ´ is the elevation from a reference level

Without loss of generality, we assume ´ D 0 Furthermore, we assume that K D diag.K; : : : ; K/

with K 2 L1.p/, K > 0, which implies that the porous media is homogeneous

where ˛ is a positive parameter depending on the properties of the porous medium Equation (2.4)

is the Beavers–Joseph–Saffman–Jones condition

For simplicity, we assume that the hydraulic head ' and the fluid velocity uf satisfy thehomogeneous Dirichlet boundary condition except on , that is,

uf D 0 on @f n ;

We will use the standard notations for the Sobolev spaces Hs.D/ equipped with their usual innerproducts ; /s;D, norms k  ks;D, and seminorms j  js;D, s > 0 The space H0.D/ coincides with

L2.D/, in which case the norm and inner product are denoted by k  kDand ; /D, respectively Let

L20.D/ denote the subspace of L2.D/ consisting of functions with mean integral value zero In thispaper, D denotes f or p Without loss of generality, if no specified, we will drop subscript Dthroughout the paper And jj  jjeis L2integration on edge e

Figure 2 An example of primal mesh and its dual mesh.

Let Rf;h be a regular triangulation of f with diam.f/ 6 hf;K and hf;K the diameter ofelement Every triangle K 2 Rf;h is divided into three subtriangles by connecting the barycenter

of the triangle K to its corner nodes, as shown in Figure 2 Then we define the dual partitionTf;h

of the primal partitionRf;h to be the union of the triangles T as shown in Figure 2 for triangularmesh Respectively, we define primal meshRp;hand dual meshTp;hfor Darcy domain p withmesh size hp;K We define the mesh parameter hf D max

K2Tf;h

hf;K and hp D max

K2Tp;h

hp;K In thetheoretical analysis, we will drop subscript f for hf; hf;K, respectively, p for hpand hp;K And

hedenotes the length of edge e

The same as the usual practice, Pk.T / denotes the space of polynomials with degree less than orequal to k defined on T We first define two function spaces:

QhD®

q 2 L2.f/ W qjK2 P0.K/; 8K 2Rf;h

¯:Respectively, we define trial function space Xhand test function space Yhfor piezometric head '.x/:

f denote the set of all interior edges e inRf;h,ED

f denote the set of all boundary edges thatbelong to Dirichlet boundary inRf;h, andE

f denotes the set of all boundary edges that belong tointerface  inRf;h, so we haveEf DEo

For vectors v and n, let v ˝ n denote the matrix whose ij th component is vinj For two valued variables  and , we define  W D P2

matrix-i;j D1ij ij Let e be an interior edge shared bytwo elements K1and K2 inTh.here;Tf;horTp;h/, and let n1 and n2be unit normal vectors on epointing exterior to K1 and K2, respectively We define the average

q, vector v; and matrix , respectively.

¹qº D 12



j@K1C j@K2



@K1 n1C j@K2 n2:

We also define a matrix-valued jumpfor a vector asvD vj@K1˝ n1C vj@K2˝ n2on e If e is

an edge on the boundary of  here; f or p/, define

e

.vf  vf/ds D 0 8e 2 @K; 8vf 2 Vh; (2.9)

ifvfD 0; then vfD 0; (2.10)

k vf  vfkK 6C hKjvfj1;K; (2.11)

k vfke6kvfke 8e 2Ef: (2.12)And we have similar results for variable  These results can be found in [12]

3 DISCONTINUOUS FINITE VOLUME FORMULATIONS

Multiplying (2.1) by vf 2 Vhand q 2 Qh, respectively, we have

@T

pn  vfds D X

T 2Tf;hnZ

K

where n is the unit outward normal vector on @T

Let Tj 2Tf;h.j D 1; 2; 3/ be the triangles in K 2Rf;h, as shown in Figure 3 Then we have

T 2Tf;h

nZ

Aj C1CAj

ruf  nvfds

K2Rf;hnZ

where A4 D A1 Making use of the second and third terms of interface conditions (2.4), we have

K2Rf;hnZ

e

ruf  nvfds  X

e2EnZ

e

ruf  nvfds

e2EnZ

e

.p  g'/nf  vf 



˛p

i Ki

ufi

.i vf/ds:

e

.p g'/nfvf



˛p

i Ki

ufi

.i vf/ds:(3.5)

By same computation, we obtain that

Aj C1CAj

pn  vfds C X

e2Eo

f [ED fnZ

e

¹pº  Œvf

X

e2EnZ

e

pnf  vfds:

(3.6)Multiplying Darcy’s Equation (2.3) by  2 Yh, we have

T 2Tp;h

gZ

@T

Kr'  nds D X

T 2Tp;h

gZ

Aj C1CAj

Kr'  nds  X

K2Rp;h

gZ

Aj C1CAj

Kr' nds  X

e2Eo

p [ED p

gZ

e

X

e2E

g nZ

Aj C1CAj

ruf  n vfds  X

e2Eo

f [ED fnZ

Aj C1CAj

Kr'  nds  X

e2Eo

p [ED p

gZ

i Ki

ufi

.i vf/ds  X

e2E

g nZ

Aj C1CAj

pn  vfds C X

e2Eo

f [ED fnZ

e

¹pº  Œ vf (3.11)

C uf; '/; q/ D X

K2Rf;hnZ

K

r  ufqds  X

e2Eo

f [ED fnZ

Aj C1CAj

pn  vfds; (3.14)

F vf; / D X

T 2Tf;hnZ

T

gf vfdx C X

T 2Tp;h

gZ

T

gpdx; (3.15)

where ˛f > 0, ˛p > 0 are penalty parameters to be determined later In this article, we take

We will take advantage of the following trace inequalities in the later theoretical analysis Let T

be an element and e be an edge of T ; there exists a constant C that depends only on the minimumangle of T such that for any function u 2 H2.T /,

A uf; '/; vf; // D n.ruf; rvf/ C X

K2Rf;h

nZ

nZe

e2E

g nZe

e

g'nf vf  vf/C



˛p

i Ki

ufi

.i vfvf//ds

e2E

g nZ

Firstly, using Hölder inequality, Poincaré inequality, (3.20), and (2.12), we estimate

A uf; '/; vf; // and the interface integration in the bilinear form A1 uf; '/; vf; //,

1 A

g Z

1 0

e2Eo

f [ED f

Z

e

h e jr'j2ds

1 C

1 0

e2Eo

p [ED p

1 0

e2Eo

f [ED f

1 0

e2Eo

p [ED p

The sum of (3.28) and (3.29) yields the continuity 

Lemma 4 ([24])

For vf 2 V.h/;  2 X.h/; q 2 L20.f/,

C vf; /; q/ 6 jk.vf; /kj

0B

11C

gZ

1A

1A

>njjrvfjj2C gKjjrjj2C n X

e2Eo

f [ED f

e2Eo

f [ED fn

"22X

e2Eo

p [ED p

gKZ

A2 uf; '/; vf; //

D A1 uf; '/; vf; // C X

e2Eo

f [ED fnZ

gZ

e

(3.35)

Algorithm 2: The second discontinuous finite volume scheme for Stokes–Darcy problem seeks

uf;h; 'h/; ph/ 2 Vh Xh/  Qhsuch that

So, Algorithm 2 is unconditionally stable with respect to the choice of the penalty parameters.

Algorithm 3: The third discontinuous finite volume scheme for Stokes–Darcy problem seeks

uf;h; 'h/; ph/ 2 Vh Xh/  Qhsuch that

A3 uf;h; 'h/; vf; // C B vf; /; ph/ D F vf; / 8.vf; / 2 Vh Xh; (3.38)

C uf;h; 'h/; q/ D 0 8q 2 Qh: (3.39)Here, we define

A3 uf; '/; vf; //

D A1 uf; '/; vf; //  X

e2Eo

f [ED fnZ

gZ

e

(3.40)The continuity and coercivity of bilinear A2.; /, A3.; / can be obtained by means of the proof

of bilinear A1.; /

Remark 1

In the variational formulation (3.38)–(3.39), we take ˇf D ˇp D 1, corresponding to the SIPGmethod in the context of DG methods applied to the second-order linear elliptic problems Optimal

L2 error estimates can be derived for the SIPG method by the Aubin–Nitsche lift technique used

in the analysis of the classical DG finite element method So, in Section 4, we prove the optimal

L2error estimates for variational formulation (3.38)–(3.39) On the other two cases, they have beenobserved numerically on uniform meshes that L2 convergence rate is optimal while using linearapproximation functions Note that if using super-penalty techniques and taking ˇf D ˇpD 3, theoptimal L2error estimates can be achieved in the existing theoretical analysis of DG method

where ˇ is a positive constant depending only on f Define operators …1W V ! Vh, …2 W X !

Xh, …3W L20.f/ ! Qh; the following results can be seen in [24],

C vf  …1vf; /; q/ D 0 8q 2 Qh; 8vf 2 V (4.2)

jvf  …1vfjs;k 6C h2sjvfj2;k 8K 2Rf;h; s D 0; 1; 8vf 2 V (4.3)j  …2js;k6C h2sjj2;k 8K 2Rp;h; s D 0; 1; 8 2 X: (4.4)

Theorem 4.1

Let uf;h; 'h; ph/ 2 Vh; Xh/  Qh be the unique solution of (3.18)–(3.19) and uf; '; p/ 2.H2.f//2\ V  H2.p/ \ X  L20.f/ be the solution of (3.16)–(3.17); then there exists C > 0such that