The Lowest-order Weak Galerkin Finite Element

Method for the Darcy Equation on Quadrilateral and

Hybrid Meshes

Jiangguo Liua, Simon Tavenerb, Zhuoran Wangc

a Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874,USA, liu@math.colostate.edu

b Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874,USA, tavener@math.colostate.edu

c Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874,USA, wangz@math.colostate.edu

Abstract

This paper presents the lowest-order weak Galerkin finite element method forsolving the Darcy equation on quadrilateral and hybrid meshes consisting ofquadrilaterals and triangles. In this approach, the pressure is approximatedby constants in element interiors and on edges. The discrete weak gradientsof these constant basis functions are specified in local Raviart-Thomas spacesRT[0] for quadrilaterals or RT0 for triangles. These discrete weak gradientsare used to approximate the classical gradient when solving the Darcy equa-tion. The method produces continuous normal fluxes and is locally mass-conservative, regardless of mesh quality. It exhibits expected convergencein pressure, velocity, and flux when the quadrilaterals are asymptoticallyparallelograms. Implementation is straightforward and results in symmetricpositive-definite discrete linear systems. Numerical experiments and com-parison with other existing methods are also presented.

Key words: Darcy equation, hybrid meshes, lowest order finite elements,quadrilateral meshes, weak Galerkin

Preprint submitted to Elsevier March 7, 2017

1. Introduction

This paper concerns with finite element methods for boundary value prob-lems of the Darcy equation

∇ · (−K∇p) ≡ ∇ · u = f, x ∈ Ω,

p = pD, x ∈ ΓD,

u · n = uN , x ∈ ΓN ,

(1)

where Ω ⊂ R2 is a bounded (polygonal) domain, p the unknown pressure, K apermeability matrix that is uniformly symmetric positive-definite, f a sourceterm, pD, uN respectively Dirichlet and Neumann boundary data, n the out-ward unit normal vector on ∂Ω, which has a nonoverlapping decompositionΓD ∪ ΓN .

Efficient and robust solvers for the Darcy equation are fundamentallyimportant for numerical simulations of flow and transport in porous media.Some early work can be found in [14, 15, 30]. The two most importantproperties desired for Darcy solvers are local mass-conservation and normalflux continuity. For the continuous Galerkin (CG) methods, these can beachieved by postprocessing [13]. The discontinuous Galerkin methods arelocally mass-conservative by design, and postprocessing will provide normalflux continuity [7]. The mixed finite element methods (MFEMs) have bothproperties by design [8], but indefinite linear systems need to be solved.The enriched Galerkin methods attain these properties by enriching the CGapproximation spaces with discontinuous piecewise constants [31]. Otherresearch efforts can be found in [9].

Development of Darcy solvers on quadrilateral meshes is more challeng-ing than those for rectangular and triangular meshes. In [4, 5, 34], the Pi-ola transformation is utilized to maintain normal flux continuity on generalquadrilateral meshes.

The recently development weak Galerkin (WG) finite element methods[32] bring in new perspectives. When applied to the Darcy equation on tri-angular and rectangular meshes [22, 23], the WG methods satisfy the afore-mentioned two physical properties, have optimal order convergence rates, andresult in symmetric positive-definite discrete linear systems. Under certainconditions [20], the WG methods satisfy the discrete maximum principle.

However, Darcy solvers on rectangular meshes have limited use for realapplications. In certain cases, quadrilateral meshes or hybrid meshes consist-

2

ing of quadrilaterals and triangles are desired to accommodate complicateddomain geometry while keeping degrees of freedom low [1, 10, 37]. WGmethods have been developed for the elliptic equation on general polygonalmeshes [28, 27]. These methods can be applied to solve the Darcy equa-tion on quadrilateral and triangular meshes. But these WG methods involvea stabilization parameter, for which an optimal value is generally unknown.Although these WG methods offer a continuous normal flux, but it is unclearhow a numerical velocity can be calculated.

In this paper, we develop a new WG method for solving the Darcy equa-tion on quadrilateral and hybrid meshes. This new method uses the lowestorder approximants for pressure, namely, just constants inside elements andon edges. But the method is locally conservative and produces continu-ous normal fluxes, regardless of mesh quality. When the quadrilaterals areasymptotically parallelograms, this method has optimal order convergencein pressure, velocity, and flux.

The rest of this paper is organized as follows. In Section 2, we discussmainly construction of WG(Q0, Q0;RT[0]) finite elements on quadrilaterals.Section 3 presents the Darcy solver using the lowest order WG finite elements:WG(Q0, Q0;RT[0]) for quadrilaterals and WG(P0, P0;RT0) for triangles. Theproperties and errors of this Darcy solver are discussed in Section 4. Sec-tion 5 briefly discusses two other finite element methods developed for Darcyproblems, a mixed method based on the Piola transformation [5] and a WGmethod with stabilization [27]. (Recent work [3] constructs H(div) mixed fi-nite elements on quadrilaterals with minimal use of the Piola transformation,but this new method is not implemented here.) Section 6 presents numericalexperiments on this new WG method and comparison to the aforementionedtwo other methods. Section 7 concludes the paper with some remarks.

2. Lowest-order WG Elements on Quadrilaterals and Triangles

WG (P0, P0;RT0) finite elements on triangles and WG (Q0, Q0;RT[0]) fi-nite elements on rectangles have been used for solving the Darcy equationin [22]. In this section, we construct WG (Q0, Q0;RT[0]) finite elements forquadrilaterals. These will include the rectangular WG (Q0, Q0;RT[0]) ele-ments as a special case. This WG approach is different than the approachused in [5] which employs the Piola transformation to construct mixed finiteelements. The mixed finite element methods enforce normal flux continu-ity at the level of finite element spaces. The WG approach in this paper

3

uses a larger finite element space for gradients, but normal flux continuity isattained through the bilinear form and properties of discrete weak gradients.

2.1. WG(Q0, Q0;RT[0]) Finite Elements on Quadrilaterals

Let E be a convex quadrilateral with center (xc, yc). Let X = x−xc, Y =y − yc. We denote

w1 =

[10

], w2 =

[01

], w3 =

[X0

], w4 =

[0Y

], (2)

and define a local Raviart-Thomas space as

RT[0](E) = Span(w1,w2,w3,w4). (3)

The Gram matrix of the above basis is obviously

GM =

|E| 0

∫EX 0

0 |E| 0∫EY∫

EX 0

∫EX2 0

0∫EY 0

∫EY 2

, (4)

where |E| is the quadrilateral area. This SPD matrix becomes a diagonalmatrix, when E is a rectangle. But in general, the integrals

∫EX,∫EY are

nonzero.

Now we consider five discrete weak functions φ0, φ1, φ2, φ3, φ4 on E andassume,

• φ0 takes value 1 in the element interior but is 0 on all four edges,

• φi, i = 1, 2, 3, 4 each take value 1 on a single edge but have value 0 onthe other three edges and in the element interior.

Note that φi = φi , φ∂i , i = 0, 1, . . . , 4, i.e., each of the above five discreteweak functions has two independent components, φi for element interior E,and φ∂i for element boundary E∂.

The discrete weak gradient ∇w,dφi, i = 0, 1, . . . , 4 is specified in RT[0](E)via integration by parts,∫

E

(∇w,dφi) ·w =

∫E∂

φ∂i (w · n)−∫Eφi (∇ ·w), ∀w ∈ RT[0](E). (5)

4

For i = 0, 1, . . . , 4, let

∇w,dφi = ci1w1 + ci2w2 + ci3w3 + ci4w4, (6)

where wj, j = 1, 2, 3, 4 are defined in (2). For each i = 0, 1, . . . , 4, thecoefficients cij, j = 1, . . . , 4 can be obtained by solving a 4 × 4 SPD linearsystem (5) with coefficient matrix (4). For the special case when E is arectangle [x1, x2]× [y1, y2], and the edges 1 through 4 are bottom, right, topand left respectively, we have [17]

∇w,dφ0 = 0w1 + 0w2 + −12(x2−x1)2

w3 + −12(y2−y1)2

w4,

∇w,dφ1 = 0w1 + −1y2−y1 w2 + 0w3 + 6

(y2−y1)2w4,

∇w,dφ2 = 1x2−x1 w1 + 0w2 + 6

(x2−x1)2w3 + 0w4,

∇w,dφ3 = 0w1 + 1y2−y1 w2 + 0w3 + 6

(y2−y1)2w4,

∇w,dφ4 = −1x2−x1 w1 + 0w2 + 6

(x2−x1)2w3 + 0w4.

(7)

AAAAAAAAAAAAAAA

tt

t

t

tQ0

Q0

CCCCCCCCCCCC

tt

t tP0

P0

Figure 1: Left : WG(Q0, Q0;RT[0]) element on a quadrilateral; Right : WG(P0, P0;RT0)element on a triangle.

2.2. WG (P0, P0;RT0) Finite Elements on Triangles

Let T be a triangle with vertices (xi, yi), i = 1, 2, 3 oriented counterclock-wise. Let (xc, yc) be the element center and let X = x− xc, Y = y − yc. We

5

define

w1 =

[10

], w2 =

[01

], w3 =

[XY

], (8)

and the local Raviart-Thomas space

RT0(T ) = Span(w1,w2,w3). (9)

The Gram matrix of the above basis is a diagonal matrix |T | 0 00 |T | 00 0 S

, (10)

where |T | is the triangle area, and

S =1

12((x1−x2)2 +(x2−x3)2 +(x3−x1)2 +(y1−y2)2 +(y2−y3)2 +(y3−y1)2).

We consider 4 discrete shape functions φ0, φ1, φ2, φ3 on the triangle andassume

• φ0 takes value 1 in the element interior but is 0 on all three edges,

• φi, i = 1, 2, 3 each take value 1 on a single edge but have value 0 onthe other two edges and in the element interior.

The discrete weak gradients, ∇w,dφi, i = 0, 1, 2, 3 are specified in RT0(T )using an equivalent definition to (5). Since the Gram matrix (10) is diagonal,each 3× 3 linear system can be solved immediately and

∇w,dφ0 = 0w1 + 0w2 + −2|T |S

w3,

∇w,dφ1 = y3−y2|T | w1 + x2−x3

|T | w2 + 2|T |3S

w3,

∇w,dφ2 = y1−y3|T | w1 + x3−x1

|T | w2 + 2|T |3S

w3,

∇w,dφ3 = y2−y1|T | w1 + x1−x2

|T | w2 + 2|T |3S

w3,

(11)

where wi, i = 1, 2, 3 are defined in (8).

6

3. The Lowest-order WG Scheme for the Darcy Equation

In this section, we present the weak Galerkin finite element scheme forsolving the Darcy equation on quadrilateral or hybrid meshes using thelowest-order WG finite elements discussed in the previous section. In otherwords, the pressure unknowns are just constants in element interiors andconstants on edges. The discrete weak gradients of these shape functionsare specified in RT[0] for quadrilaterals and RT0 for triangles. For ease ofpresentation, we focus on the numerical scheme for a quadrilateral mesh.

Let Eh be a shape-regular quasi-uniform quadrilateral mesh that consistsof convex quadrilaterals. Let ΓDh be the set of all edges on the Dirichletboundary ΓD and ΓNh be the set of all edges on the Neumann boundaryΓN . Let Sh be the space of discrete shape functions on Eh that are degree 0polynomials in element interiors and also degree 0 polynomials on edges, S0

h

be the subspace of functions in Sh that vanish on ΓDh .

3.1. Quadrilateral mesh

We seek ph = ph, p∂h ∈ Sh such that p∂h|ΓDh

= Q∂h(pD) (the L2-projection

of Dirichlet boundary data into the space of piecewise constants on ΓDh ) and

Ah(ph, q) = F(q), ∀q = q, q∂ ∈ S0h, (12)

where

Ah(ph, q) :=∑E∈Eh

∫E

K∇w,dph · ∇w,dq, (13)

and

F(q) :=∑E∈Eh

∫E

fq −∑γ∈ΓN

h

∫γ

uNq∂. (14)

The major steps in the assembly of the global stiffness matrix for thisnew WG finite element scheme on a quadrilateral mesh are as follows.

(i) On each element,

(a) The Gram matrices (4) of the local RT[0] bases are computed.

(b) The coefficients (6) defining the discrete weak gradients of the fiveWG basis functions are computed.

7

(c) The 5 × 5 element stiffness matrices corresponding to the termK∇w,dph · ∇w,dq in (13) are constructed.

(ii) The element stiffness matrices are assembled into the global stiffnessmatrix.

The overall procedure is similar to those for rectangular and triangularmeshes as discussed in [22]. In Matlab, each of the above steps can beimplemented mesh-wise, using the strategies discussed in [25], which is muchmore efficient than looping over the elements. This new WG solver has beenincluded in the Matlab code package DarcyLite, which can be found on thefirst author’s webpage.

After the numerical pressure ph is solved from (12), we compute its dis-crete weak gradient ∇w,dph and then −K∇w,dph. Note that −K∇w,dph maynot be in RT[0]. Then we perform a local L2-projection Rh from L2(E)2 toRT[0](E) on any quadrilateral element E. This way, we obtained an element-wise numerical velocity uh:

uh = Rh(−K∇w,dph). (15)

Note that this local L2-projection Rh can be omitted when K is a constantdiagonal matrix.

3.2. Hybrid mesh

The above lowest-order WG finite element scheme can be easily extendedto a hybrid mesh that consists of quadrilaterals and triangles. Note that thepressure unknown is just a constant inside each element, no matter whetherit is a quadrilateral or a triangle. The pressure unknown is also a constanton each edge. Then the discrete weak gradient is specified in RT[0] whenthe element is a quadrilateral or RT0 when the element is a triangle. Thesediscrete weak gradients will be used in (12), which is solved for the numericalpressure ph. Then a numerical velocity can be obtained from (15) by the localL2-projections on quadrilaterals or triangles.

4. Properties and Convergence of the Lowest-order WG Scheme

Theorem 1 (Local mass conservation). Let E be any quadrilateral ortriangular element. There holds∫

E

f =

∫E∂

uh · n. (16)

8

Proof. For simplicity, let E be a quadrilateral. In (12), take a test functionq so that q|E = 1 but vanishes on all edges and inside all other elements.Then ∫

E

f =

∫E

(K∇w,dph) · ∇w,dq

=

∫E

Rh(K∇w,dph) · ∇w,dq

= −∫E

uh · ∇w,dq

= −∫E∂

q∂(uh · n) +

∫Eq(∇ · uh)

=

∫E∇ · uh

=

∫E∂

uh · n.

The equalities are provided by (12), the definition of projection Rh, the defini-tion of the discrete velocity field, the definition of the discrete weak gradient,the definition of the particular test function q, and finally the DivergenceTheorem on a vector function that is in the local RT[0] (RT0) space.

Theorem 2 (Continuity of bulk normal fluxes). Let γ be an edge sharedby two elements E1 and E2 (each could be a quadrilateral or a triangle) andn1,n2 be the constant outward unit normal vectors on γ for E1 and E2

respectively. Then ∫γ

u(1)h · n1 +

∫γ

u(2)h · n2 = 0. (17)

Proof. Consider equation (12) with a test function q = q, q∂ such thatq∂ = 1 on γ, but q∂ = 0 on all other edges and q = 0 in interior of any

9

quadrilateral or triangular element. Then from (13) and (14),

0 =

∫E1

(K∇w,dph) · ∇w,dq +

∫E2

(K∇w,dph) · ∇w,dq

=

∫E1

Rh(K∇w,dph) · ∇w,dq +

∫E2

Rh(K∇w,dph) · ∇w,dq

=

∫E1

(−u(1)h ) · ∇w,dq +

∫E1

(−u(2)h ) · ∇w,dq

= −∫γ

u(1)h · n1q

∂ +

∫E

1

u(1)h q −

∫γ

u(2)h · n2q

∂ +

∫E

2

u(2)h q

= −∫γ

u(1)h · n1 −

∫γ

u(2)h · n2,

using the definition of projection Rh, the definition of the discrete velocityfield, the definition of the discrete weak gradient, and the definition of theparticular test function q.

Approximation accuracy of a finite element scheme is affected by meshquality, which can be defined through element diameters and certain angles.Let E be a quadrilateral, θ1 be the angle between the outward unit normalvectors on two opposite edges, θ2 be the angle for the other two edges. LetσE = max|π − θ1|, |π − θ2| and hE be the diameter of E. A quadrilat-eral mesh Eh is asymptotically parallelogram [5], provided that there exists apositive constant C such that σE/hE ≤ C for all E ∈ Eh.

Any polygonal domain can be partitioned into a family of asymptoticallyparallelogram, shape-regular quadrilateral meshes [4] (p. 918). This can beaccomplished by starting from any mesh of convex quadrilaterals and per-forming regular refinement. This implies that asymptotically parallelogramquadrilateral meshes are adequate for practical use.

To measure errors in pressure, velocity, and flux for this new WG finiteelement scheme, we use the following norms (see also [33, 32]),

‖p− ph‖2 =∑E∈Eh

‖p− ph‖2L2(E), (18)

‖u− uh‖2 =∑E∈Eh

‖u− uh‖2L2(E)2 , (19)

‖(u− uh) · n‖2 =∑E∈Eh

∑γ⊂E∂

|E||γ|‖u · n− uh · n‖2

L2(γ). (20)

10

Proposition (Convergence in pressure, velocity, and flux). Let p bethe exact solution of the Darcy problem (1) and u = −K∇p. Assume theexact solution has regularity p ∈ H1+s(Ω) and u ∈ Hs(Ω)2 for some s ∈ (0, 1].When the quadrilaterals are asymptotically parallelograms, there holds

‖p− ph‖ ≤ Ch, ‖u− uh‖ ≤ Chs, ‖(u− uh) · n‖ ≤ Chs, (21)

where C > 0 is a constant that is independent of the mesh size h.

A complete account of convergence analysis involves the following com-ponents.

1. Investigation of the approximation properties of the local Raviart-Thomasspaces RT[0] and the effect of quadrilateral mesh quality on approxima-tion capacity.

2. Establishing the connection among local Raviart-Thomas spaces, theglobal Raviart-Thomas space, and discrete weak gradients.

3. Establishing an error equation for the finite element solution and theprojection of the exact pressure solution.

4. Analyzing low the regularity case (s < 1).

This requires analysis techniques different from those used in [27]. Due topage limitation here, we will present the error analysis in a separate work.

5. Other FEMs for Darcy on Quadrilateral or Hybrid Meshes

5.1. Mixed Finite Element Methods on Quadrilateral Meshes

The primal pressure variable and its flux can be approximated simulta-neously by rewriting the Darcy equation (1) as a system of first-order partialdifferential equations as follows [8]

K−1u +∇p = 0, x ∈ Ω,

∇ · u = f, x ∈ Ω,

p = pD, x ∈ ΓD,

u · n = uN , x ∈ ΓN .

(22)

Mixed finite element methods satisfy the local mass conservation by de-sign. The Piola transformation is utilized to enforce normal flux continuityfor vector shape functions defined on quadrilaterals.

11

Define

HN,0(div,Ω) = v ∈ L2(Ω)2 : divv ∈ L2(Ω),v|ΓN = 0,

HN,uN (div,Ω) = v ∈ L2(Ω)2 : divv ∈ L2(Ω),v|ΓN · n = uN.

The mixed variational formulation based on (22) is: Find u ∈ HN,uN (div,Ω)and p ∈ L2(Ω) such that the following holds

(K−1u,v)Ω − (p,∇ · v)Ω = −〈pD,v · n〉ΓD , ∀v ∈ HN,0(div,Ω),−(∇ · u, q)Ω = −(f, q)Ω, ∀q ∈ L2(Ω).

(23)

In this formulation, the Neumann condition is essential, whereas the Dirichletcondition becomes natural.

MFEMs are based on H(div)-conforming finite elements and the inf-supcondition [8]. Denote V = H(div; Ω),W = L2(Ω). Let Vh ⊂ V and Wh ⊂W be a pair of finite element spaces (for the flux and the primal variablerespectively) satisfying the inf-sup condition. Furthermore, let Uh = Vh ∩HN,uN (div; Ω) and V 0

h = Vh ∩ HN,0(div; Ω). A mixed finite element schemecan be formulated as: Find uh ∈ Uh and ph ∈ Wh such that

∑E∈Eh

(K−1uh,v)E −∑E∈Eh

(ph,∇ · v)E = −∑γ∈ΓD

h

〈pD,v · n〉γ, ∀v ∈ V 0h ,

−∑E∈Eh

(∇ · uh, q)E = −∑E∈Eh

(f, q)E, ∀q ∈ Wh.

(24)

Let E = P1P2P3P4 be a convex quadrilateral with four vertices Pi(xi, yi), i =1, 2, 3, 4 that are oriented counterclockwise. The bilinear mapping from thereference element E = [0, 1]2 to the quadrilateral E can be expressed as

x = a1 + a2x+ a3y + a4xy,y = b1 + b2x+ b3y + b4xy,

(25)

where x = (x, y) ∈ E and x = (x, y) ∈ E. The eight coefficients ai, bi(i =1, 2, 3, 4) can be directly calculated using the vertex coordinates. The Jaco-bian matrix of the bilinear mapping is

J =

[a2 + a4y a3 + a4x,b2 + b4y b3 + b4x

]. (26)

12

The Jacobian determinant is a linear polynomial in x, y:

J = det(J) =

∣∣∣∣ a2 a3

b2 b3

∣∣∣∣+

∣∣∣∣ a2 a4

b2 b4

∣∣∣∣ x+

∣∣∣∣ a4 a3

b4 b3

∣∣∣∣ y. (27)

By the Piola transformation, a vector function v on E is transformed intoa vector function on E as

v(x) =J(x)

J(x)v(x). (28)

The Piola transformation maintains the normal fluxes for the vector basisfunctions on E that are mapped from the vector basis functions on E, andthus plays a fundamentally important role in constructing H(div) finite el-ements on quadrilaterals [5, 8, 33]. Moreover, it can be verified that ([8]p.140)

∇ · v =1

J(x)∇ · v. (29)

H(div)-conforming finite elements on quadrilaterals based on the Piolatransformation were investigated in [5] and necessary and sufficient conditionsfor optimal order approximations for vector fields and their divergences wereestablished. The Raviart-Thomas element pair (RT[0], Q0) on quadrilateralswas used to solve the Darcy or elliptic equations. First order convergencein velocity and pressure is obtained on general quadrilateral meshes pro-vided the problem has full elliptic regularity on a convex domain. However,there may be no convergence for velocity divergence on general quadrilateralmeshes. Our numerical results (Section 6 Example 2) indicate that optimalorder convergence in velocity (and flux) can still be obtained in low regularitycases.

An advantage of the mixed finite elements constructed using the Piolatransformation is the applicability to general quadrilateral meshes. However,an obvious restriction comes from solving indefinite linear systems or saddle-point problems. It is unclear whether the mixed element pair (RT[0], Q0) forquadrilaterals can be combined with the mixed element pair (RT0, P0) fortriangles to solve the Darcy (elliptic) equation on a hybrid mesh.

5.2. WGFEMs on Polygonal Meshes

In [27], a family of stabilized WG(Pk+1, Pk;Pdk ) (dimension d = 2, 3, inte-

ger k ≥ 0) finite element schemes have been developed for general polygonal

13

or polyhedral meshes. The schemes can be applied to distorted meshes thatcontain different types of polygons and polyhedra, e.g., a mix of trianglesand quadrilaterals in two dimensions. The lowest-order WG finite elementin this family in two dimensions is (P1, P0;P 2

0 ). The interior basis functionsare therefore one degree higher than the elements developed in this paper.Importantly, for the WG(P1, P0;P 2

0 ) elements in [27], the discrete weak gra-dients are specified in P 2

0 , whereas for the lowest-order WG elements in thispaper, the discrete weak gradients are specified in RT[0] or RT0 .

The WG (P1, P0;P 20 ) finite element scheme for the Darcy equation on a

quadrilateral mesh is: Let Eh be a quadrilateral mesh and define

Bh(ph, q) = Ah(ph, q) + Sh(ph, q), (30)

where Ah(ph, q) has the same form as the one shown in (13), but is defined forshape functions that are linear polynomials in element interiors and constantson edges. Here Sh is a stabilization term,

Sh(ph, q) = ρ∑E∈Eh

∑γ⊂E∂

h−1〈Q∂h(ph)− p∂h, Q∂

h(q)− q∂〉γ. (31)

and Q∂h is the local L2-projection into the space of constant functions on an

edge.

As shown in [27], the above WG scheme with stabilization is locally mass-conservative and produces continuous normal fluxes. Furthermore, it hasoptimal order convergence in pressure. However, an optimal value for thestabilization parameter ρ is unclear, as demonstrated by the numerical resultsin Section 6 Example 2. Computation of a velocity field was not discussed in[27] and hence requires further investigation. One possible approach wouldbe via a postprocessing procedure, applying an interpolation operator to thenormal flux on the edges to obtain a numerical velocity that is in the globalRaviart-Thomas space (a finite dimensional subspace of H(div,Ω)).

6. Numerical Experiments

In this section, we test the lowest-order WG finite element method onquadrilateral meshes and hybrid meshes. We compare this new WG methodwith the WG(P1, P0;P 2

0 ) method in [27] and with the mixed finite elementmethod in [5] based on the Piola transformation.

14

For quadrilateral mesh generation, see [29]. For requirements on quadri-lateral mesh quality, see [16, 36]. For Matlab implementation techniques forrelated finite element methods, see [2, 6, 11, 25].

Type I Type II Type III Type IV

Figure 2: Four types of meshes for numerical experiments

We shall consider these four types of meshes.

• Type I: Logically rectangular meshes. This family of meshes adoptedfrom [33] is asymptotically parallelogram. Specifically, the quadrilateralmesh points are

x = x+ 0.06 sin(2πx) sin(2πy),y = y − 0.05 sin(2πx) sin(2πy),

where (x, y) are the corresponding rectangular mesh points.

• Type II: Asymptotically parallelogram trapezoidal meshes. See [4, 5].

• Type III: Quadrilaterals meshes obtained from a triangular mesh. Aninitial quadrilateral mesh is obtained by refining a triangular mesh.Then finer meshes are obtained from regular refinement of quadrilat-eral meshes. This results in a family of asymptotically parallelogramquadrilateral meshes, as discussed in [4] (p.918).

• Type IV: Hybrid meshes consisting of quadrilaterals and triangles.

The quality of an initial Type III quadrilateral mesh may be poor, as is shownin Figure 2. But successive refinement improves mesh quality. More detailsabout conditions and quality of quadrilateral meshes can be found in [12].

Example 1 (Smooth solution). First, we consider a simple examplewith Ω = (0, 1)2, K = I2 (order 2 identity matrix), a known analytical

15

1/h ‖p− ph‖ ‖u− uh‖ ‖(u− uh) · n‖ min(ph) max(ph)

Type I meshes8 8.3889e-2 2.8087e-1 3.8999e-1 2.4644e-2 9.4861e-116 4.2299e-2 1.3818e-1 1.9030e-1 7.3817e-3 9.8536e-132 2.1196e-2 6.8734e-2 9.4556e-2 2.0891e-3 9.9605e-164 1.0604e-2 3.4320e-2 4.7204e-2 5.5932e-4 9.9897e-1

Conv.rate 0.994 1.010 1.015 N/A N/A

Type III meshes8 8.3751e-2 7.5352e-1 9.8838e-1 1.4350e-2 8.7929e-116 3.8217e-2 3.7097e-1 4.9134e-1 7.1831e-3 9.6817e-132 1.8595e-2 1.6574e-1 2.2054e-1 2.0653e-3 9.9358e-164 9.2692e-3 7.5206e-2 1.0084e-1 5.5556e-4 9.9883e-1

Conv.rate 1.058 1.108 1.097 N/A N/A

Type IV meshes8 7.2665e-2 2.7675e-1 3.3213e-1 1.3150e-2 9.6983e-116 3.6563e-2 1.3839e-1 1.6405e-1 3.3015e-3 9.9233e-132 1.8311e-2 6.9195e-2 8.1719e-2 8.1269e-4 9.9807e-164 9.1592e-3 3.4598e-2 4.0817e-2 2.0159e-4 9.9952e-1

Conv.rate 0.995 0.999 1.008 N/A N/A

Table 1: Example 1 (smooth solution): WG(Q0, Q0;RT[0]) results for three types of meshes

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

WG Numerical pressure and velocity

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

WG Numerical pressure and velocity

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 3: Example 1 (smooth solution): Profiles of numerical pressure and velocity fromthe lowest-order weak Galerkin method on a quadrilateral mesh (left) and a hybrid mesh(right). Both have mesh size h = 1/16.

16

solution for pressure p(x, y) = sin(πx) sin(πy) and a homogeneous Dirichletboundary condition on the entire boundary. It is easy to see that f(x, y) =2π2 sin(πx) sin(πy). We test this example on three different types meshes(Type I, III, IV). The pressure solution is infinitely smooth, so the approx-imation accuracy is dictated by mesh quality. The convergence in pressure,velocity, flux shown in Table 1 is first order as expected.

Example 2 (Low regularity solution). For this example, Ω = (0, 1)2,K = I2, a homogeneous Dirichlet boundary condition is posed on the entireboundary, and an analytical solution for pressure is specified as

p(x, y) = x(1− x)y(1− y)(√

x2 + y2)−(2−a)

,

where a ∈ (0, 1] is a regularity parameter. The right hand side function inthe PDE is derived accordingly. The pressure admits a corner singularityat the origin. The pressure solution is in Sobolev space p ∈ W 2,r(Ω) withr = 2/(2− a) + δ for any δ > 0 [35]. By the Sobolev embedding theorem, itis known [24] that

p ∈ H10 (Ω) ∩H1+a−δ(Ω), p /∈ H1+a(Ω),

for any small δ > 0. A more accurate characterization based on Besovspaces can be found in [24]. The regularity order is close to 1 + a. It isalso interesting to note that the pressure solution has a minimum 0 and a

maximum√

22+a

aa/(2 + a)2+a (attained at point ( a2+a

, a2+a

)).

For a = 0.4, the exact pressure maximum is 0.194783879417743. We testthe WG(Q0, Q0;RT[0]) method in this paper, the WG(P1, P0;P 2

0 ) methodwith stabilization in [27], and the mixed method (RT[0], Q0) (based on the Pi-ola transformation) in [5] on a family of asymptotically parallelogram trape-zoidal meshes used in [5].

Table 2 provides results obtained from using WG (Q0, Q0;RT[0]) on TypeII (asymptotically parallelogram trapezoidal) meshes. It is observed that thepressure error exhibits about 1st order convergence. The errors in velocityand flux exhibit slightly better than order 0.4 convergence. Overshoots forpressure maximum are observed.

Table 3 provides the numerical results obtained from using the WG(P1, P0;P 2

0 ) method studied in [27] on the same trapezoidal meshes. It canbe observed that the convergence rate in pressure error is about order 1.4

17

1/h ‖p− ph‖ ‖u− uh‖ ‖(u− uh) · n‖ min(ph) max(ph)8 2.2025e-02 3.7831e-01 1.1725e+00 2.6101e-03 2.3095e-0116 1.0418e-02 2.7950e-01 8.5877e-01 6.8901e-04 2.0163e-0132 5.1299e-03 2.0940e-01 6.3999e-01 1.7357e-04 1.9552e-0164 2.5591e-03 1.5788e-01 4.8109e-01 4.2083e-05 1.9489e-01128 1.2805e-03 1.1938e-01 3.6315e-01 1.0350e-05 1.9479e-01

Conv.rate 1.026 0.416 0.422 N/A N/A

Table 2: Example 2 (low regularity): Results of the lowest order WG(Q0, Q0;RT[0])method on asymptotically parallelogram trapezoidal meshes (Type II)

1/h ‖p− ph‖ min(ph) max(ph) ‖p− ph‖ min(ph) max(ph)ρ = 1 ρ = 10

8 9.950E-2 2.775E-3 7.848E-1 9.979E-3 2.627E-3 1.733E-116 3.586E-2 7.575E-4 5.651E-1 3.745E-3 7.003E-4 1.892E-132 1.320E-2 1.715E-4 4.165E-1 1.414E-3 1.703E-4 1.938E-164 4.925E-3 4.140E-5 3.110E-1 5.357E-4 4.172E-5 1.945E-1128 1.850E-3 1.023E-5 2.340E-1 2.031E-4 1.030E-5 1.947E-1

Conv.rate 1.437 N/A N/A 1.404 N/A N/A

Table 3: Example 2 (low regularity): Results of WG(P1, P0;P 20 ) with stabilization [27] on

asymptotically parallelogram trapezoidal meshes (Type II)

1/h ‖p− ph‖L2 ‖u− uh‖ min(ph) max(ph)8 2.1071e-02 3.6181e-01 2.7100e-03 2.2682e-0116 1.0243e-02 2.7261e-01 7.0278e-04 2.0042e-0132 5.1021e-03 2.0666e-01 1.7033e-04 1.9528e-0164 2.5549e-03 1.5682e-01 4.1671e-05 1.9483e-01128 1.2799e-03 1.1897e-01 1.0298e-05 1.9478e-01

Conv.rate 1.010 0.401 N/A N/A

Table 4: Example 2 (low regularity): Results of MFEM based on the Piola transformation[5] on asymptotically parallelogram trapezoidal meshes (Type II)

for different stabilization parameter values ρ = 1 and ρ = 10. However,the maximum of the numerical pressure clearly depends on the choice of thestabilization parameter ρ. For ρ = 1, the method shows the expected con-vergence in pressure errors, but a fairly large overshoot is still observed fora fine mesh with h = 1/128. Increasing stabilization parameter value results

18

in increase of the condition number of the discrete linear system. Furtherinvestigation is needed to determine the optimal stabilization parameter andto compute element-wise velocity for this WG method.

Table 4 provides numerical results obtained from using the mixed finiteelement method in [5] on the same trapezoidal meshes. It can be observedthat the pressure error exhibits 1st order convergence and the velocity errorexhibits order 0.4 convergence. Overshoots in pressure maximum are alsoobserved.

Example 3 (Quadrilateral meshes in polar coordinates). In this exam-ple, we solve the Darcy equation on a circular sector domain that is describedin polar coordinates

Ω = (r, θ) : ra ≤ r ≤ rb, 0 ≤ θ ≤ π

2,

where 0 < ra < rb are respectively the inner and outer radii of an annulus.For simplicity, we assume the permeability matrix K has principal axes inthe radial and theta directions, i.e.,

K =

[cos(θ) − sin(θ)sin(θ) cos(θ)

] [λ 00 µ

] [cos(θ) sin(θ)− sin(θ) cos(θ)

],

or in Cartesian coordinates,

K =

λx2 + µy2

x2 + y2(λ− µ)

xy

x2 + y2

(λ− µ)xy

x2 + y2

λy2 + µx2

x2 + y2

.While the permeability matrix have the same form in both polar and Carte-sian coordinates when λ = µ, many applications exhibit strong anisotropy.For example in the biological problems described in [26, 19], the radial per-meability λ greatly exceeds the angular permeability µ.

The gradient operators in the Cartesian and polar coordinates are relatedas [

∂x∂y

]p =

[cos θ − sin θsin θ cos θ

] [1 00 1

r

] [∂r∂θ

]p.

Accordingly, the Darcy velocity is

v = −K

[∂x∂y

]p = −

[cos θ − sin θsin θ cos θ

][λ∂rµ

r∂θ

]p,

19

and the source term is

f =

[∂x∂y

]· v =

[cos θ − sin θsin θ cos θ

][ ∂r1

r∂θ

]· v.

We consider an exact pressure solution

p(x, y) = xy,

and calculate the right-hand side in the Darcy equation as

f(x, y) = −4(λ− µ)xy

x2 + y2.

All four sides of the sector domain are posed as Dirichlet boundary conditionswith data prescribed by the exact pressure solution.

For testing anisotropy in the radial and angular directions, we fix λ = 1and vary µ from 1 down to 10−3 or 10−6. Our numerical results demonstrategood approximation for pressure, velocity, and flux in the cases of highlyanisotropic permeability fields.

0 0.5 1 1.5 2

0

0.5

1

1.5

2WG Numerical pressure and velocity

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.5 1 1.5 20

0.5

1

1.5

2WG Numerical pressure and velocity

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 4: Example 3 (annual domain): Profiles of numerical pressure and velocity on aquadrilateral mesh with λ = 1 and nr = 8, nθ = 16 (Left: µ = 1; Right: µ = 0.1)

For numerical tests, we set nr as the number of uniform partitions in theradial direction and nθ = 2nr as the number of uniform partitions in theangular direction.

Numerical results applying WG (Q0, Q0;RT[0]) to this example with λ = 1and µ = 10−3 (strong anisotropy) are provided in Table 5. Nearly first

20

nr ‖p− ph‖ ‖u− uh‖ ‖(u− uh) · n‖ min(ph) max(ph)8 1.0207e-1 9.9727e-2 1.4104e-1 5.4044e-2 1.865816 5.1112e-2 4.9031e-2 6.9337e-2 2.5808e-2 1.935332 2.5564e-2 2.4373e-2 3.4468e-2 1.2621e-2 1.968164 1.2783e-2 1.2167e-2 1.7206e-2 6.2293e-3 1.9841

Conv.rate 0.999 1.011 1.011 N/A N/A

Table 5: Example 3 (annular domain): Numerical results for strong anisotropy (λ = 1, µ =10−3) and nθ = 2nr

order convergence can be observed in pressure and velocity errors and in theedgewise bulk flux. No overshoots or undershoots in pressure are observed.

Figure 4 shows profiles of numerical pressure and velocity for nr = 8, nθ =16, with fixed λ = 1 but two different µ values (1 and 0.1). As the ratio λ/µincreases, the radial dominance in numerical velocity becomes obvious.

This type of problem could be solved by isogeometric finite element meth-ods [19], an approach that requires different data structures. Here we demon-strate how the novel weak Galerkin FEMs could be used in a simple way. Onecould also use mixed finite element methods [5, 21] or multi-point flux mixedfinite element methods [33] on quadrilaterals, but the Piola transformationneeds to be used. The lowest order WG finite element method is easier byadopting the non-mapped approach and allows hybrid meshes comprised ofquadrilaterals and triangles.

Example 4 (Highly heterogeneous permeability fields). This exampleoriginates from [14] and has been tested by many researchers and also in [23]on rectangular and triangular meshes. Here we extend this example to atrapezoidal domain

Ω2 = (x, y) : 0 ≤ y ≤ x+ 1

2, 0 ≤ x ≤ 1,

and a circular sector domain

Ω3 = (r, θ) : 2 ≤ r ≤ 3,π

6≤ θ ≤ π

3.

For the unit square domain Ω1 = [0, 1]2, as discussed in [23], Dirichletboundary conditions are specified on the left and right sides, whereas Neu-mann boundary conditions are specified on the top and bottom sides:

p = 1, left; p = 0, right; −(K∇p) · n = 0, top or bottom.

21

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Numerical pressure and velocity

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) (b)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1WG Numerical pressure and velocity

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1.5 2 2.51

1.5

2

2.5

WG Numerical pressure and velocity

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(c) (d)

Figure 5: Example 4 (heterogeneous permeability fields): Pressure and velocity profiles.(a) Permeability profile on Ω1 = [0, 1]2; (b) Rectangular domain: WG(Q0, Q0;RT[0]) ona uniform rectangular mesh with h = 1/40; (c) Trapezoidal domain: WG (Q0, Q0;RT[0])on a quadrilateral with h = 1/40; (d) Circular sector domain: WG (Q0, Q0;RT[0]) on aquadrilateral with h = 1/40. Flow patterns differ slightly.

The trapezoidal domain Ω2 is mapped from the unit square with correspond-ing domain sides and boundary conditions. The circular sector domain Ω3 isalso a mapping image of the unit square (left side → inner circle, right side→ outer circle, etc.)

The new WG method for quadrilateral meshes is tested on all three do-mains. Figure 5 shows that flow patterns differ slightly. For the circularsector domain, flow paths in the radial direction are more pronounced. Butno overshoots or undershoots in numerical pressure are observed.

22

7. Concluding Remarks

We have developed an easily implemented and practically useful Darcysolver that can be used on quadrilateral meshes, triangular meshes, and hy-brid meshes consisting of quadrilaterals and triangles. This Darcy solver isdeveloped utilizing the novel ideas of weak Galerkin finite elements. Thepressure unknowns are constants inside (triangular and quadrilateral) ele-ment interiors and on edges. The discrete weak gradients of these constantbasis functions are computed inexpensively. On triangles and rectangles theyare obtained by direct calculations, and on general quadrilaterals they areobtained by solving 4× 4 symmetric positive-definite linear systems. Thesediscrete weak gradients are used to approximate the classical gradient in thevariational formulation. The finite element scheme results in a sparse sym-metric positive-definite discrete linear system that can be solved by stan-dard linear solvers. After the numerical pressure ph is solved, a local L2-projection of −K∇w,dph is conducted to obtain the numerical velocity. (TheL2-projection is not required if K is an element-wise constant scalar matrix.)The normal flux can then be computed in a straightforward manner.

This new approach has some obvious nice features, including

1. Local mass conservation and continuity of normal flux

2. Lowest-order approximants (constants inside elements and on edges)

3. Unified treatment of triangles, rectangles, and quadrilaterals

4. No necessity for stabilization

5. Sparse SPD global linear systems.

The limitation of the method is the “asymptotically parallelogram” as-sumption on quadrilaterals, although this assumption is widely adopted. Thisis not a serious limitation, since any polygonal domain can be partitioned intoa family of asymptotically parallelogram, shape-regular quadrilateral meshes[4] (p. 918).

Different than the existing work in [3, 5], convergence in divergence isnot a main concern of this paper. The main theme of this paper is aboutdevelopment of an easy-to-use Darcy solver on quadrilateral meshes. Thefocus is on local conservation, normal flux continuity, and convergence inpressure, velocity, and flux. Our results indicate that the unmapped approach(not using the Piola transformation) combined with the novel weak Galerkinmethodology can do a good job in this regard.

23

An extension of this lowest order weak Galerkin finite element methodto three-dimensional domains is not difficult but requires some attention todetails [18]. First, the faces of a hexahedron may not be planar. Integrationon a face or interior of the hexahedron requires mapping back to the unitsquare or the unit cube and involves higher computational costs. By con-trast, an integral on a quadrilateral can be simply split into integrals on twotriangles. More delicate handling is needed for fluxes on non-planar faces.

Acknowledgements: The first was partially supported by US NationalScience Foundation under grant DMS-1419077. The third author was par-tially supported by US National Science Foundation under grant DMS-1419077and a graduate fellowship from the Center of Interdisciplinary Mathematicsand Statistics at Colorado State University.

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