ICES REPORT A Generalized Mimetic Finite Difference Method and Two-Point Flux Schemes over Voronoi Diagrams

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1 ICS RPORT July 2015 A Generalized Mimetic Finite Difference Method and Two-Point Flux Schemes over Voronoi Diagrams by Omar Al-Hinai, Mary F. Wheeler, Ivan Yotov The Institute for Computational ngineering and Sciences The University of Texas at Austin Austin, Texas Reference: Omar Al-Hinai, Mary F. Wheeler, Ivan Yotov, "A Generalized Mimetic Finite Difference Method and Two-Point Flux Schemes over Voronoi Diagrams," ICS RPORT 15-17, The Institute for Computational ngineering and Sciences, The University of Texas at Austin, July 2015.

2 A Generalized Mimetic Finite Difference Method and Two-Point Flux Schemes over Voronoi Diagrams Omar Al-Hinai and Mary F. Wheeler Center for Subsurface Modeling, Institute for Computational ngineering and Sciences, The University of Texas at Austin, 201 ast 24th Street, Austin, T 78712, USA. ohinai@ices.utexas.edu mfw@ices.utexas.edu Ivan Yotov Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA. yotov@math.pitt.edu June 24, 2015 Abstract We propose a generalization of the Mimetic Finite Difference (MFD) method that extends the family of convergent schemes to include Two-Point Flux Approximations (TPFA) over general Voronoi meshes. We prove first-order convergence in pressure and flux, and superconvergence of the pressure under further restrictions. We construct H(Ω; div) basis functions that correspond to the generalized form over the square element. We present numerical results that support the theory. 1 Introduction Finite Volume Methods (FVM) are among the most widely used techniques for the numerical solution of partial differential equations, especially those governing fluid flow problems [23]. The popularity of FVM is due to the fact that they are locally conservative by construction, fast and robust for a wide range of problems, and can preserve the maximum principle. FVM can be defined over a wide variety of mesh types, including triangular/tetrahedral, quadrilateral/hexahedral and Voronoi type meshes [5, 40, 23]. The reader is referred to [21, 20] for work 1

3 on unstructred grids and full tensor permeabilities, and [1] for distorted grids and multi-point flux approximations (MPFA). FVM can be related to other discretization methods such as the Finite Difference Method and the Finite lement Method. Of note is the connection established in [42] for Mixed Finite lements (MF), where the lowest order Raviart-Thomas element (RT 0 ) [41] over rectangles was shown to be equivalent to cell-centered finite differences. The technique demonstrated how to reduce the number of degrees of freedom for RT 0 and produce a linear symmetric positive definite system for the pressure variable. This connection enabled a proof of convergence for FVM based on the theory of MF [48]. Many authors have subsequently built on these ideas, producing further connections and more theoretical results. An extension for full-tensor coefficients was developed in [3, 4]. Superconvergence for both the scalar and vector unknowns has been shown in [39, 18, 22]. Further extensions include the use of the lowest order Brezzi-Douglas-Marini element (BDM 1 ) [11] for connecting MF with MPFA [50, 28, 49]. A similar connection based on broken RT 0 elements was developed in [31, 32]. While there has been success in demonstrating the connection between MF and FVM, some types of FVM have not been connected with MF. An example is the two-point flux approximation (TPFA) over Voronoi diagrams and K-orthogonal grids (grids aligned with the eigenvectors of the permeability tensor) [23]. Connecting such a method to MF would require the construction of discrete H(Ω; div) spaces over convex polyhedra with an arbitrary number of faces. While such spaces have been constructed via a local triangulation of the cells [33], it remains uncertain if such an approach can be reduced to TPFA. The work done in [13, 14] on the Mimetic Finite Difference (MFD) method opens up some possibilities. The authors of [13] use theoretical tools from MF to demonstrate convergence of the MFD method over a very general set of polyhedral cells. This is accomplished by avoiding the explicit construction of the velocity variable on the interior, and directly solving for flux degrees of freedom on the faces of the cells. The definition of the MFD method in [14] allowed for a reduction to TPFA for the case of regular polyhedra and rectangles. Later work in [19] extended the MFD method to include TPFA for acute triangular cells. The authors of [37] provided generalizations of the MFD method that included TPFA over centroidal Voronoi diagrams as well as MPFA methods over general polyhedra; see also related work in [30]. The objective of our work is to provide a further generalization of the MFD method that allows for TPFA type methods over general Voronoi diagrams. The reduction to TPFA results in a faster solution to the MFD method as well as solutions that satisfy the maximum principle. We propose a modified form of the consistency condition (S2) in [13], which we call here ( S2). The modification is inspired by the consistency conditions proposed by [19] and [37]. As a consequence, the method also relates the MFD method to the point-centered FVM. This is accomplished without the use of dual grids or local triangulations of the cells. Section 2 defines the new generalization of the MFD method. The stability of the new method is shown in Section 2.1, which is followed by first-order convergence proofs for the velocity in Section 2.2 and pressure variables in Section 2.3. Conditions for superconvergence of the method are presented in Section 2.4, which includes superconvergence with special quadrature in Section Novel H(Ω; div) shape functions that satisfy the superconvergence criteria are presented in Section We then discuss matrix construction in Section 3, and the relation to TPFA schemes on Voronoi grids in Section 4. Numerical results confirming the theory are demonstrated in Section 5. 2

4 The MFD method has found widespread application in many areas of numerical modeling. The central theme is to create discretizations that mimic the fundamental properties of the underlying partial differential equation. The MFD method has been used to solve elliptic and parabolic equations and has found applications in the areas of gas dynamics [45], Maxwell s equations [27, 9] and Stokes flow [16]. It is also possible to apply the MFD method to cell-faced [13] and nodal discretizations [10]. The MFD method can be used both to define higher-order methods [47], and on meshes with curved faces [36]. More recent work in higher order methods is in the related area of the Virtual lement Method [15]. arlier examples of the MFD method referred to as the Support Operator Method (SOM) can be found in [44, 45, 26]. 2 A Generalized MFD Method Consider the homogeneous Dirichlet problem written in mixed form, u = K p in Ω, u = f in Ω, p = 0 on Ω. (2.1) In porous media flow, u is Darcy velocity, p is pressure and K is the permeability tensor. The domain Ω R d (d = 2, 3) is assumed to be polyhedron with Lipschitz boundaries and is divided into a non-overlapping, conformal partition T h of elements. In three dimensions, is a polyhedron with planar polygonal faces. In two dimensions, is a polygon. For the remainder of this work, we refer to elements as polyhedra, with the understanding that the results also apply directly to the two-dimensional case. We make the following standard assumptions for the mesh used in previous work [13, 37]: 1. lements and faces are shape-regular and non-degenerate; 2. lements are star-shaped polyhedra; 3. Faces are star-shaped polygons. We define the following mesh related quantities: 1. N Q is the total number of elements in the mesh; 2. N is the total number of faces in the mesh; 3. is the volume of ; 4. h is the diameter of ; 5. h = max T h h ; 6. e is the area of face e; 7. n e is a fixed unit normal on face e; 3

5 8. n e is the unit normal on face e pointing out of element ; 9. k is the number of faces of element. We note that the mesh assumptions imply that h d and e h d 1. (2.2) Let the space of discrete pressures Q h R N Q and fluxes h R N be defined, respectively, as Q h = {q h = {q } Th h = {v h = {v e } e T h such that q R}, such that v e R}. ach pressure degree of freedom (q ) represents the pressure on element. ach velocity degree of freedom (v e ) represents the flux normal to the face e facing out of element. The above definition for h results in two degrees of freedom per internal face in the mesh and a single degree of freedom for each boundary face. We require continuity of velocity in h, that is, for two adjacent elements a and b sharing a face e, let v e a + v e b = 0. The continuity of flux reduces the total number of degrees of freedom in h to N. In the following, we make use of the usual notation for Lebesgue spaces L p (Ω), Sobolev spaces W k,p (Ω) and Hilbert spaces H k (Ω), see [8]. For q L 1 (Ω), define a projection operator on Q h as (q I ) = 1 q dv. (2.3) For v (L s (Ω)) d, s > 2 and v L 2 (Ω), define the projection operator on h as (v I ) e = 1 v n e ds. (2.4) e Let the permeability tensor K be symmetric and positive definite, and let Let K denote a constant tensor over such that e K ij W 1, (Ω). (2.5) max K ij K,ij L ij () Ch. (2.6) Throughout the paper, C denotes a generic positive constant independent of h. positive definite, we also have that max K 1 ij ij Since K is K 1,ij L () Ch. (2.7) Following the standard steps of the MFD method [34], we define the discrete divergence operator DIV : h Q h as (DIV v h ) = DIV v := 1 v e e. (2.8) e The proof of the next lemma follow from (2.3), (2.4), (2.8), and the divergence theorem. 4

6 Lemma 1. For v (L s (Ω)) d, s > 2 and v L 2 (Ω), Next, define the scalar inner product as (DIV v I ) = ( v) I. (2.9) [p h, q h ] Qh := The flux bilinear form is defined as [u h, v h ] h := T h p q. T h [u, v ] = T h v T M u, (2.10) where u R k is the vector with components u e and M R k k is a given matrix. We discuss the construction of M in Section 3. This form approximates the continuous velocity bilinear form, that is, [u h, v h ] h K 1 u v dv. We assume that [, ] h satisfies: Ω (S1) (Stability) There exist two positive constants, s and S, such that, for all T h and ξ R k, and s ξ T ξ ξ T M ξ S ξ T ξ, (2.11) ξ T M T M ξ (S ) 2 2 ξ T ξ. (2.12) ( S2) (Consistency) For every element T h, for every linear function q 1 on, and for every v h (), [( K q 1 ) I, v ] = v e w e q 1 ds w q 1 DIV v dv, (2.13) e where the function w : R satisfies e w dv =, gw dv = x, where x is a point in and g : R d R d is the linear function x g(x, y, z) = y z, 5

7 and where the function w e : e R satisfies, w e ds = e, (2.14) e gw e ds = e x e, (2.15) e where x e is a point in the plane of e and g is the linear function defined on Cartesian coordinates (x, y) in the plane of e as ( ) x g(x, y) =. y The point x e may lie outside of e. Note that if x = C, the centroid of, then w = 1. Also, if x e = C e, the centroid of e, then w e = 1. We require that the functions w e and w be bounded by positive constants independent of the mesh, max w L T () w max, h max max w e L T h e (e) we max. Remark. The use of weighting functions in ( S2) is the only departure from the original (S2) condition in [13]. The addition of weighting functions to the integrals was inspired by the modification suggested in [19] and [37]. Define a discrete gradient operator G : Q h h as the adjoint of DIV, [G p h, u h ] h = [p h, DIV u h ] Qh. (2.16) Note that, due to the lack of a boundary term in (2.16), it imposes weakly the boundary condition p h = 0 on Ω. The numerical method can be represented in a weak saddle-point form. That is, find u h h and p h Q h such that [u h, v h ] h [p h, DIV v h ] Qh = 0, v h h, (2.17) [DIV u h, q h ] Qh = [f I, q h ] Qh, q h Q h. (2.18) Remark. The theoretical results presented here follow the exposition found in [37]. We have made modifications when necessary to adapt to the modified consistency condition ( S2). Since the stability analysis does not depend on the consistency condition ( S2), the stability results from [13] and [37] still hold. 2.1 Stability Analysis The analysis follows the classical approach to stability of mixed methods. We start by defining the following norms over the pressure and flux spaces, v h 2 h = [v h, v h ] h, q h 2 Q h = [q h, q h ] Qh. 6

8 The fact that h is a norm follows from s v e 2 [v, v ] S e e v e 2 T h, v h (), (2.19) which is a direct consequence of (2.11) in condition (S1). Note that (2.4) and (2.19) imply that for any v (H 1 ()) d, v I h, C v (H 1 ()) d. (2.20) It is easy to see that (2.11) and (2.12) in (S1) imply the continuity of the bilinear form, We also define discrete H(Ω; div) norm We clearly have that q h Q h and v h h, [u h, v h ] h S s u h h v h h. (2.21) v h 2 div = v h 2 h + DIV v h 2 Q h. [q h, DIV v h ] Qh q h Qh v h div. Let Z h represent the discrete divergence-free subspace of h, Z h = {v h h [DIV v h, q h ] Qh = 0, q h Q h }. Note that since DIV v h Q h, v h Z h implies that DIV v h = 0. This immediately implies the coercivity of the flux bilinear operator, [v h, v h ] h = v h 2 div v h Z h. (2.22) The inf-sup condition for the bilinear operator [, DIV ] Qh, has been shown in [37]. Theorem 1 (inf-sup). There exists a positive constant β independent of h such that, for any q h Q h, [DIV v h, q h ] Qh sup β q h Qh. (2.23) {v h h,v h 0} v h div The existence and uniqueness of the solution (v h, p h ) to (2.17) (2.18) follows from (2.22) and Theorem 1, using the general theory for saddle-point problems [12]. 2.2 Velocity Convergence In this section we establish a first-order error estimate for the velocity variable. We will utilize an approximation result from [8]: for any element and φ H 2 (), there exists a linear function φ 1 such that φ φ 1 H k () Ch m k φ H m (), k = 0, 1, m = 1, 2. (2.24) 7

9 We will also utilize the trace inequalities [2]: ( ) χ H 1 (), χ 2 L 2 (e) C h 1 χ 2 L 2 () + h χ 2 H 1 (), (2.25) ( ) v (H 1 ()) d, v n e 2 L 2 (e) C h 1 v 2 (L 2 ()) + h v 2 d (H 1 ()). (2.26) d A combination of (2.24) and (2.25) implies that for any φ H 2 (), there exists a linear function φ 1 such that φ φ 1 2 L 2 (e) Ch3 φ 2 H 2 (). (2.27) The main result of this section is the following theorem bounding the velocity error. Theorem 2 (Velocity stimate). For the exact solution (u, p) of (2.1) and the MFD approximation (u h, p h ) solving (2.17) (2.18), assuming that p H 2 (Ω), and u (H 1 (Ω)) d, there exists a positive constant C, independent of h, such that u I u h h Ch( p H 2 (Ω) + p H 1 (Ω) + u (H1 (Ω)) d). (2.28) Proof. Set v h = u I u h. Note that, using Lemma 1, Thus, DIV v h = DIV (u I u h ) = f I f I = 0. u I u h 2 h = [u I u h, v h ] h = [u I, v h ] h [p h, DIV v h ] Qh = [u I, v h ] h. (2.29) Let p 1 be a piecewise linear function such that p 1 = p 1. By adding and subtracting ( K p 1 ) I, [u I, v h ] h = [u I + ( K p 1 ) I, v h ] h }{{} I 1 [( K p 1 ) I, v h ] h }{{} I 2. (2.30) 8

10 We now bound I 1 and I 2. Starting with I 1, I 1 C (u + K p 1 ) I h v h h (using (2.21)) ( ) 1/2 ( C ((u + K p 1 ) I ) e ) 2 v h h (using (2.19)) = C C C C Ω e ( Ω e ( Ω e ( Ω e ( Ω ( 2 1/2 1 (u + e K p 1 ) n e ds) ) v h h (using (2.4)) e ) ) 1/2 ( e (u + K p 1 ) n e 2 L 2 (e) v h h ) ( (u + K ) 1/2 p 1 ) n e 2 L 2 (e)) h v h h (using (2.2)) ( h 1 (u + K p 1 2 (L 2 ()) d + h u 2 (H 1 ()) d ) h ) 1/2 v h h (using(2.26)). (2.31) Taking the first term, we add and subtract K p 1, u + K p 1 (L 2 ()) d where the last inequality follow from, using (2.24), K (p p1 ) (L 2 ()) d + (K K ) p 1 (L 2 ()) d C ( h p H2 () + h p 1 (L2 ()) d ) (using (2.6) and (2.24)) C ( h p H 2 () + h p H 1 ()), (2.32) p 1 (L 2 ()) d p (L 2 ()) d + (p p1 ) (L2 ()) d C p H 1 (). Combining (2.31) and (2.32), we obtain I 1 Ch ( p H2 (Ω) + p H1 (Ω) + u (H 1 (Ω)) d ) vh h. Since DIV v h = 0, using condition ( S2), the expression for I 2 becomes I 2 = v e w e p 1 ds. (2.33) Ω e e Due to the continuity of p, we can subtract it from each element face in the summation above, 9

11 obtaining I 2 = Ω Ω e ( C Ω e v e e w e (p 1 p) ds e 1/2 we max v e p 1 p L2 (e) ) 1/2 v e 2 h p H 2 () (using (2.2) and (2.27)) e Ch p H2 (Ω) v h h (using (S1)). (2.34) Combining the bounds on I 1 and I 2 with (2.29) and (2.30) gives the desired estimate Weighted Projection Operator We also study the operator u Iw h such that, (u Iw ) e = 1 w e u n e ds. e Note that this weighted operator appears in the consistency condition ( S2) and therefore it can be expected that the numerical flux u h provides an improved approximation to u Iw, compared to u I. We study both errors in the numerical results. We have the following result. Corollary 1. Under the assumptions of Theorem 2, Proof. The triangle inequality implies u Iw u h h Ch( p H2 (Ω) + p H1 (Ω) + u (H1 (Ω)) d). u Iw u h h u Iw u I h + u I u h h. The second term is bounded in Theorem 2. Letting, ū n e = u n e ds, for the first term we have, u Iw u I 2 h S (u Iw u I ) e 2 e = S ( ) 2 e 2 w e (u ū) n e e e S (we max ) 2 ((u ū) n e e ) 2 e e e e Ch 2 u 2 (H 1 (Ω)) d(using arguments similar to Lemma 3.14 in [49]). Combining the above bounds implies the statement of the corollary. 10

12 2.3 Pressure Convergence We start by defining a weighted operator I w such that (p Iw ) = 1 w p dv. Note that the operator is exact for constant functions. We first establish a pressure estimate with respect to the weighted operator ( ) Iw. We then establish a bound for the regular projection operator ( ) I. Theorem 3. Let (u, p) be the exact solution to (2.1), and let (u h, p h ) be the MFD method s numerical approximation (2.17)-(2.18). Assuming that p H 2 (Ω), and u (H 1 (Ω)) d there exists a constant C independent of h, such that Proof. From Theorem 1 we know that Adding and subtracting (p 1 ) Iw, we get p Iw p h Qh Ch( p H2 (Ω) + p H1 (Ω) + u (H1 (Ω)) d). (2.35) p Iw p h Qh 1 [DIV v h, p Iw p h ] Qh sup. β v h h,v h 0 v h div [DIV v h, p Iw p h ] Qh = [DIV v h, (p p 1 ) Iw ] Qh [DIV v h, p h ] Qh + [DIV v h, (p 1 ) Iw ] Qh. Using (2.17) for the second term and condition ( S2) for the third term, we have We can bound I 3 by [DIV v h, p Iw p h ] Qh = [DIV v h, (p p 1 ) Iw ] Qh [u h, v h ] h }{{}}{{} I 3 I 4 u e w e p 1 ds [( K p 1 ) I, v h ] + e } {{ } I 5 e } {{ } I 6. [DIV v h, (p p 1 ) Iw ] Qh = (DIV v h, w (p p 1 )) L2 () DIV v h L 2 () w (p p 1 ) L 2 () DIV v h L2 () w L () (p p 1 ) L2 () w max ( ) 1/2 ( DIV v h 2 L 2 () p p 1 2 L 2 () Ch p H1 (Ω) DIV v h Qh (using (2.24)). ) 1/2 11

13 xpression I 5 is identical to I 2 in (2.33), implying I 5 Ch p H2 (Ω) v h h. Taking I 4 and I 6, and adding and subtracting u I, we have I 4 + I 6 = [( K p 1 ) I + u I, v h ] h [u I u h, v h ] h, = Ĩ4 + Ĩ6. xpression Ĩ4 is identical to I 1 in (2.30), giving the bound Ĩ4 Ch( p H2 (Ω) + p H1 (Ω) + u (H 1 (Ω)) d) v h h. xpression Ĩ6 is bounded by the velocity estimate in Theorem 2, Ĩ6 C u I u h h v h h Ch( p H 2 (Ω) + u (H 1 (Ω)) d) v h h. Combining the bounds on I 3 - I 6 gives the desired result. We can now derive a bound based on the L 2 projection operator ( ) I defined in (2.3). Corollary 2. Under the assumptions of Theorem 3, Proof. We start with p I p h Qh Ch( p H 2 (Ω) + p H 1 (Ω) + u (H1 (Ω)) d). (2.36) p I p h Qh p I p Iw Qh + p Iw p h Qh. The second term is bounded by Theorem 3. For the first term, we use thebramble-hilbert Lemma [8] and that the operator ( ) Iw is exact for constants. Letting p = 1 p dv, we have p I p Iw Qh = ( ( ( ( ) ) 2 1/2 1 w ( p p) dv ( ) ) 2 1/2 1 w ( p p) dv ( ) ) 2 1/2 1 w L () ( p p) dv w max p p L2 (Ω) Ch p H1 (Ω). Combining the above bounds implies the statement of the corollary. 12

14 2.4 Superconvergence of Pressure In this section we prove second-order convergence for p h p Iw Qh. Note that p h p I Qh is not in general superconvergent, i.e., there may not be second-order convergence at the centroids of the elements. This is confirmed by numerical experiments (Section 5). We require the existence of two lifting operators R, R : h () H(; div) defined locally over each element and satisfying the following properties: R (v I 0) = v 0, for all constant vectors v 0, (2.37) R (v ) = DIV v in, (2.38) R (v ) = w DIV v in, (2.39) R (v ) n e = w e v e on e, (2.40) R (v ) (L 2 ()) d C v h,, (2.41) R (v ) (L 2 ()) d C v h,. (2.42) Lemma 2. The lifting operators R and R define a bilinear form, [u, v ] = K 1 R (u ) R (v ) (2.43) that satisfies condition ( S2). Proof. We can demonstrate the result by observing that for all linear q 1 and all v h (), [( K q 1 ) I, v ] = K 1 R (( K q 1 ) I ) R (v ) dv = K 1 K q 1 R (v ) dv (using (2.37)) = q 1 R (v ) n e ds q 1 R (v ) dv (integration by parts) e = e e v e e w e q 1 ds w q 1 DIV v dv (using (2.40) and (2.39)). (2.44) We will need the following result proved in Lemma 3.3 in [37]: Lemma 3. For any v (H 1 ()) d, let v 0 be a constant vector representing the L 2 projection of v on. Then, there exists a constant C independent of h such that v I v0 I h, Ch v (H1 ()) d. (2.45) 13

15 The lifting operator R will not in general map constants to constants (as in (2.37)). However, we associate R with a tensor T with the property that for a constant vector v 0, We require that the operator T is bounded: Note that (2.40) implies that for a constant vector v 0, R (v I 0) = T v 0. (2.46) T v (L2 ()) C v d (L 2 ()) d. (2.47) T v 0 n e = w e v 0 n e on e, which, combined with (2.14) implies that (T v 0 v 0 ) n e L 2 (e) Ch v 0 n e L 2 (e), where C depends on w e H 1, (e), which is independent of h, due to (2.14) (2.15). Since T can be defined to be in a finite dimensional space, the above inequality and a scaling argument imply that T v 0 v 0 (L2 ()) Ch v 0 d (L2 ()) d. (2.48) We have the following results. Lemma 4. For all v (H 1 ()) d, R (v I ) v (L 2 ()) Ch v d (H 1 ()) d, (2.49) R (v I ) v (L 2 ()) Ch v d (H 1 ()) d. (2.50) Proof. Let v 0 be the L 2 () projection of v. Using (2.37), (2.41) and (2.45), we have where we also used that R (v I ) v (L 2 ()) R (v I v0) I d (L 2 ()) + v d 0 v (L 2 ()) d Ch v (H 1 ()) d, Similarly, using (2.42), (2.45), (2.46), (2.48) and (2.51), v v 0 (L2 ()) Ch v d (H1 ()) d. (2.51) R (v I ) v (L 2 ()) R d (v I v0) I (L 2 ()) + T v d 0 v 0 (L 2 ()) + v d 0 v (L 2 ()) d Ch v (H 1 ()) d. Theorem 4. Assuming the existence of lifting operators R and R with properties (2.37) (2.42) and the choice of velocity bilinear form (2.43), then the the solution p h to (2.17) (2.18) satisfies p h p Iw Qh Ch 2 ( p H2 (Ω) + p H1 (Ω) + u (H 1 (Ω)) d + f H 1 (Ω)). (2.52) 14

16 Proof. We start by defining ϕ and ψ such that, ϕ = K ψ in Ω, (2.53) ϕ = p h p Iw in Ω, (2.54) ψ = 0 on Ω. (2.55) In the above, by abuse of notation, p h p Iw is identified with a piecewise constant function. Let ϕ h = ϕ I, and note that by Lemma 1, ϕ h satisfies DIV ϕ h = p h p Iw. We require H 2 -regularity of problem (2.55). Conditions can be found in [24], which in this case can be satisfied by assuming convexity of Ω and using (2.5). As a result we have, We have ψ H 2 (Ω) C p h p Iw L 2 (Ω) = C p h p Iw Qh. (2.56) p h p Iw 2 Q h = [p h p Iw, DIV ϕ h ] Qh = [u h, ϕ h ] h p R(ϕ h ) dv (using (2.17) and (2.39)) Ω = [u h, ϕ h ] h + p R(ϕ h ) dv (integration by parts) Ω = [ ] K 1 (R (u ) u) R 1 (ϕ ) dv + ( K K 1 )u R (ϕ ) dv = [ 1 ( K K 1 )(R (u ) u) R (ϕ ) dv + K 1 (R (u ) u) R (ϕ ) dv ] + ( K 1 )u R (ϕ ) dv = J 1 + J 2 + J 3 For J 1, using (2.7), we have, where for the second step, we used K 1 J 1 Ch R(u h ) u (L2 (Ω)) d R(ϕ) (L2 (Ω)) d Ch 2 ( p H 2 (Ω) + p H 1 (Ω) + u (H 1 (Ω)) d) p h p Iw Qh, R(u h ) u (L 2 ()) C R(u d h u I ) (L 2 (Ω)) + d R(uI ) u (L 2 (Ω)) d C u h u I h + R(u I ) u (L2 (Ω)) d (using (2.41)) Ch( p H2 (Ω) + p H1 (Ω) + u (H 1 (Ω)) d) (by Theorem 2 and (2.49)), (2.57) 15

17 as well as R(ϕ h ) (L2 (Ω)) d C ϕ h h (using (2.42)) C ϕ (H1 (Ω)) d (using (2.20)) C ψ H2 (Ω) (using (2.5)) For J 2, we start by adding and subtracting ϕ, J 2 = K 1 (R(u h ) u) ( R(ϕ h ) ϕ) dv + Ω = J 21 + J 22. For J 21, using (2.50) and (2.57), we have C p h p Iw Qh (using (2.56)). (2.58) Ω K 1 (R(u h ) u) ϕ dv J 21 Ch 2 ( p H 2 (Ω) + p H 1 (Ω) + u (H 1 (Ω)) d) p h p Iw Qh. We can bound J 22 by noting that, J 22 = K 1 (R(u h ) u) K ψ dv (using (2.53)) Ω = (R(u h ) u) ψ dv Ω = (R(u h ) u)ψ dv Ω = (f I f)(ψ ψ I ) dv. From here we have by the Bramble-Hilbert Lemma, Ω J 22 Ch 2 f H1 (Ω) p h p Iw Qh For J 3, we add and subtract ϕ, obtaining J 3 = [ 1 ( K K 1 )u ( R (ϕ ) ϕ) dv + = J 31 + J 32. Using (2.7) and (2.50) and an argument similar to (2.58), J 31 is bounded by, For expression J 32, we have, J 32 = 1 ( K K 1 )u K ψ dv = (K K ) = (K K ) J 31 Ch 2 u (L2 (Ω)) d p h p Iw Qh. 1 K u K 1 K ψ dv 1 K (u u 0) ψ dv + (K K ) ( K 1 ] K 1 )u ϕ dv 1 K u 0 ( ψ ( ψ) 0 ) dv 16

18 An application of (2.6) and the Bramble-Hilbert Lemma gives J 32 Ch 2 u (H 1 (Ω)) d p h p Iw Qh Combining all the expressions gives the desired result Quadrature An alternative approach to pressure superconvergence is to use an approximate quadrature rule for the velocity bilinear form. In this case, we will restrict [, ] to symmetric form. Define σ (K 1 ; u, v ) as σ (K 1 ; u, v ) = [u, v ] K 1 R (u ) R (v ) dv. (2.59) Theorem 5. Assume the existence of lifting operators R and R with properties (2.37) (2.42) and that the choice of a symmetric velocity bilinear form [, ] satisfies for all u, v (H 1 ()) d σ (K 1, (ui ), (v I ) ) Ch 2 u (H1 ()) d v (H 1 ()) d. (2.60) Then the solution p h to problem (2.17) (2.18) satisfies p h p Iw Qh Ch 2 ( p H 2 (Ω) + p H 1 (Ω) + u (H1 (Ω)) + f d H 1 (Ω)). Proof. We utilize again the auxiliary problem (2.53) (2.55) and recall that ϕ h = ϕ I. We have p h p Iw 2 Q h = [p h p Iw, DIV ϕ h ] Qh = [u h, ϕ h ] h p R(ϕ h ) dv (using (2.17) and (2.39)) Ω = [u h, ϕ h ] h + p R(ϕ h ) dv (integration by parts) Ω = [u h, ϕ h ] h K 1 u R(ϕ h ) dv ± K 1 R(u I ) R(ϕ h ) dv Ω Ω = [u h u I, ϕ h ] h + σ (K 1 ; (u I ), (ϕ I ) ) + K 1 (R(u I ) u) R(ϕ h ) dv Ω = J 4 + J 5 + J 6. For J 4 we have J 4 = [u h u I, ϕ h + ( K ϕ 1 ) I ] h [u h u I, ( K ϕ 1 ) I ] h = J 41 + J 42. For J 41, using the same method as for I 1 in Theorem 2, we have J 41 Ch u h u I h ψ H 2 (Ω) Ch u h u I h p Iw p h Qh. 17

19 y V 2 e2 V 3 ~ x V 1 e3 ~ y e1 V 4 e4 Figure 1: The reference element for the lifting operator R. Since the bilinear form is symmetric, we can directly apply the argument for I 2 from Theorem 2 to term J 42 : J 42 Ch u h u I h ψ H2 (Ω) Ch u h u I h p Iw p h Qh. Combining J 41 and J 42, and using the velocity estimate from Theorem 2, we obtain J 4 Ch 2 ( p H 2 (Ω) + p H 1 (Ω) + u (H1 (Ω)) p d) piw h Qh. For J 5, assumption (2.59) gives J 5 Ch 2 u (H 1 (Ω)) d ϕ (H 1 (Ω)) d Ch 2 u (H1 (Ω)) p d piw h Qh. Term J 6 is bounded similarly to term J 2 in Theorem 4: J 6 Ch 2 ( u (H1 (Ω)) + f d H 1 (Ω)) p Iw p h Qh. Combining all the terms gives the desired result Lifting Operators for a Square lement We present an example of lifting operators R and R satisfying (2.37)-(2.42) over a square reference element = [0, 1] 2, see Figure 1, where x = ( x, ỹ) T, x e2 = x e4 = x, x e1 = x e3 = ỹ. For operator R, we use the standard lowest order Raviart-Thomas (RT 0 ) interpolant: R (v ) = ( v3 + (v 1 v 3 )x v 4 + (v 2 v 4 )y ), 18

20 where v i = v ei. It is well known [12] that the RT 0 interpolant satisfies (2.37), (2.38) and (2.41). The lifting operator R is defined as R (v ) = ( (v3 + (v 1 v 3 )W x (x))w y (y) (v 4 + (v 2 v 4 )W y (y))w x (x) ), with W x (0) = 0, W x (1) = 1, W y (0) = 0, W y (1) = 1, x W x = w x, (2.61) y W y = w y, (2.62) where the functions w x (x) and w y (y) are the weighting functions on the horizontal and vertical faces, respectively. The corresponding volume weighting function w is w (x, y) = w x (x)w y (y). In this form, the lifting operator R satisfies conditions (2.39) and (2.40): and R (v ) n 1 x=1 = v 1 w y, R (v ) n 3 x=0 = v 3 w y, R (v ) n 2 y=1 = v 2 w x, R (v ) n 4 y=0 = v 4 w x, R (v ) = w x w y (v 1 v 3 ) + w x w y (v 2 v 4 ) = w DIV v. We now are left with defining functions w y and w x, as well as W x and W y. We construct a function of the form and w x is of the form W x (x) = a 2 x2 + bx, w x (x) = ax + b. Functions w x and w y can be constructed directly by satisfying condition (2.14), (2.15), that is: w x ds = 1, (2.63) xw x ds = x. (2.64) 19

21 Note that this choice satisfies conditions (2.61) for W x, since 1 = 1 0 w x ds = a 2 + b = W x(1). quations (2.63) (2.64) result in the linear system ( ) ( 1/2 1 a 1/3 1/2 b which has a solution ) ( = 1 x ), a = x, b = 4 6 x. The form for w y is similar. The resulting function w satisfies, w dv = 1 and ( x y ) ( x w = ỹ ) = x. It is easy to check that (2.42) holds and that (2.46) holds with ( ) wy 0 T =. 0 w x Therefore the lifting operators satisfy all conditions needed for the pressure superconvergence in Theorem 4 with velocity inner product defined by (2.43). For the alternative approach to pressure superconvergence in Theorem 5, consider the case of diagonal permeability with ( ) kx 0 K =. 0 k y We can define the velocity inner product via (2.10) with a diagonal quadrature matrix M : M (1,1) = 1 x, M (2,2) = 1 ỹ, M (3,3) = x, M (4,4) = ỹ. (2.65) k x k y k x k y Note that the positivity of the entries ensures satisfaction of condition (S1) and that condition ( S2) is satisfied by the above choice of inner product. Notice that for constant u 0 we have σ ( K 1 ; u 0, v ) = 0, (2.66) 20

22 which can be observed by setting u 0 = (1, 0) I and v = (1, 0, 0, 0): ( ) ( ) K 1 R (u 0 ) R (v ) dv = K 1 1 Wx w y dv 0 0 = 1 x = [u 0, v ]. k x Due to non- The same argument can be extended to all other combinations of u 0 and v. symmetry of σ, in general σ ( K 1 ; v, u 0 ) 0. However, we can proceed by first defining an operator ˆT, ( ) ŵx 0 ˆT =, 0 ŵ y with xŵ x ds = 1 x, e ŵ x ds = 1, e and similarly for ŵ y. We now observe that for any v and u 0, [v, u 0 ] = K 1 R (v ) ˆT R (u 0 ) = K 1 R (v ) ˆT T u 0, (2.67) since for v = (1, 0, 0, 0) and u 0 = (1, 0) we have K 1 R (v ) ˆT T u 0 = and for v = (0, 0, 1, 0) and u 0 = (1, 0), K 1 R (v ) ˆT T u 0 = kx 1 xŵ x w y = 1 x k x = [v, u 0 ], kx 1 (1 x)ŵ x w y = x k x = [v, u 0 ], and similarly for all other combinations of v and u 0. We note that on any T h, since ŵ x = ŵ y =, and using (2.47), we have that for all constant vectors v 0, ˆT T v 0 T v 0 (L2 ()) 2 Ch v 0 (L2 ()) 2. (2.68) 21

23 Lemma 5. The choice of velocity inner product (2.65) satisfies condition (2.60). Proof. Given u (H 1 ()) d and v (H 1 ()) d, let u = u I and v = v I. We have σ (K 1 1 ; u, v ) = σ ( K ; u, v ) + ( K 1 )R (u ) R (v ) dv = J 1 + J 2. For J 1 we have, using (2.66) and (2.67), J 1 = [u, v ] K 1 R (u ) R (v ) dv, = [u u 0, v v 0 ] K 1 R (u u 0 ) ( R (v ) ˆT T v 0 ) dv = [u u 0, v v 0 ] K 1 R (u u 0 ) R (v v 0 ) dv K 1 R (u u 0 ) (T v 0 ˆT T v 0 ) dv. Using Lemma 3, Lemma 4, (2.68) and (2.48), we obtain K 1 J 1 Ch 2 u (H 1 ()) 2 v (H 1 ()) 2. xpression J 2 can be divided into 1 J 2 = ( K K 1 )R (u u 0 ) R (v ) dv 1 + ( K K 1 )R (u 0 ) R (v v 0 ) dv + ( K 1 )R (u 0 ) R (v 0 ) dv K 1 = J 21 + J 22 + J 23 (2.69) We can bound J 21 + J 22 using (2.7), (2.41)-(2.42) and (2.51), J 21 + J 22 Ch 2 v (H1 ()) 2 u (H 1 ()) 2. (2.70) For expression J 23, we have, setting K 1 T = 1 K 1 T and using (2.6), J 23 = = ( K K) ( K K) K 1 K 1 u 0 K 1 T v 0 dv u 0 (K 1 T K 1 T )v 0 dv Ch 2 u (L 2 ()) 2 v (L 2 ()) 2. (2.71) Combining the above expressions gives the desired result. 22

24 3 Matrix Construction We have explicitly defined the DIV operator, as well as the pressure inner product [, ] Qh. What remains is to define, for the general case, the velocity bilinear [, ] h. Recall that this bilinear is defined relative to the local bilinears, [u h, v h ] h = T h [u h, v h ]. The discretization yields a convergent solution if the inner product satisfies the stability (S1) and consistency ( S2) conditions. The second condition is the main focus of constructing the appropriate inner product. Define matrix R R k d as R = e 1 (x e1 x ) T. e k (x ek x ) T, (3.1) and matrix N R k d as N = ( K n e1 ) T. ( K n ek ) T, (3.2) see Figure 2. Condition ( S2) can be satisfied with M defined to satisfy M N = R. (3.3) Note that by setting w = 1 and w e = 1 we retrieve the original definition of the MFD method [14], in which case, x corresponds to the centroid of element, and x e would correspond to the centroid of face e. Also note that we do not require explicit construction of the weighting functions in order to build the linear system. The authors of [14] demonstrate how to construct M to satisfy (3.3). This is done by first defining C R k k d such that In [14], they note that without weights, so they define M 0 as N T C = 0. N T R = K (3.4) M 0 = 1 R 1 K RT. 23

25 The result is that, for any positive-definite U R (K d) (K d), M = M 0 + C U C T, (3.5) satisfies (3.3). Due to the modification ( S2) of condition (S2), equality (3.4) no longer holds. However, we can still proceed by following the same form proposed by [34], Doing so, we see that (3.5) implies (3.3), since M 0 = R (R T N ) 1 R T. M 0 N = R (R T N ) 1 R T N = R. Note that R T N automatically reduces to in the case when no boundary points are shifted. In general, R T N may not always be invertible, in which case we can use an alternative form for M 0 found in [37], 1 K M 0 = R (N T N ) 1 N T. 4 Finite Volume Methods and the MFD Method Finite Volume Methods (FVM) are based on an application of the divergence theorem to each element in the domain, u dv = u n e ds 1 e u e. e e Finite volume discretizations are distinguished by how the flux (u e ) is approximated. In the case of porous media applications, the flux of the fluid is a function of the fluid pressure. ach element in the domain has a single, piecewise constant pressure degree of freedom (p ), and each face in the mesh has a single flux representing the normal component of the velocity across that face (u e ). The flux is then related to the pressure via a function u e = F (p h ). The choice of function F is what distinguishes a particular FV method. When the velocity at face e is approximated using two adjacent pressures, we have a two-point flux approximation (TPFA). When more than two pressures are used, the method is referred to as a multi-point flux approximation (MPFA). The MFD method, much like the related MF method, produces a saddle-point problem with both velocity and pressure unknowns: ( M DIV DIV 0 ) ( ) uh = p h e ( 0 f I One can observe the relation between FVM and the MFD method by explicitly expressing the velocity unknown as a function of pressure: u h = M 1 DIV p h = F MFD (p h ). ). 24

26 n 2 x n 3 x r 3 r 2 r 1 x n 1 r 4 x n 4 Figure 2: The vectors r are constructed by connecting x with the appropriate point x e. These vectors make up the rows of matrix R, ( e 1 r1 T,... e 4 r4 T ). The rows of matrix N are (( K n 1 ) T,...( K n 4 ) T ). The nature of the relation between p h and u h is a direct consequence of the structure of M 1. In general, the matrix M 1 is dense, causing the velocity at every face to be a function of all the pressure unknowns. However, constructing a diagonal matrix M leads trivially to a diagonal M 1. In this case, due to the structure of matrix DIV, velocities are a linear function of two pressures, resulting in a TPFA scheme. There is a simple geometric criterion that indicates when diagonality can be achieved in the MFD method. Recall that matrix M is a global matrix formed from the summation of local matrices, M, M = M. Due to the consistency condition ( S2), M must satisfy the relation M N = R. (4.1) Recall from (3.1) that R is a k d matrix with rows corresponding to the vectors e (x e x ) T. From (3.2), the rows of matrix N correspond to the K-normal components to face e, see Figure 2. When the rows of N are collinear to the rows of R, a simple scaling of the rows of N satisfies (4.1), i.e., a diagonal matrix M suffices. Therefore, our objective is to construct matrices R with rows collinear to the rows of N. The original MFD method sets the point x to the centroid of the cells and the points x e to the the centroids of the faces. Our generalization allows the point x to be shifted on the interior of the cell, and the points x e to be shifted on the plane of the faces. By shifting these points, we can establish collinearity of the rows of R and N, resulting in a diagonal matrix M. Next, we show that this can be achieved for general Voronoi meshes when K is a scalar function. A Voronoi diagram is a tessellation of R d relative to a set of points known as generators. Definition 1. Voronoi Diagram. Given a set of generating points, V = {V i R d }, we define 25

27 the Voronoi tessellation T h = { i } as i = {x R d d(x, V i ) d(x, V j ) j i}. The set of generating points uniquely defines a Voronoi diagram. Note that, in this definition, the domain is infinite. For our purposes we will focus on what is known as a bounded Voronoi diagram, which is defined over a bounded domain Ω as follows. Definition 2. Bounded Voronoi Diagram. Given a domain Ω R d, and a set of generating points V = {V i R d V i Ω}, we define the bounded Voronoi tessellation T h = { i } of Ω by i = {x Ω d(x, V i ) d(x, V j ) j i}. Since we are only concerned with bounded domains, we refer to a bounded Voronoi diagram simply as a Voronoi diagram. ach cell i is called a Voronoi polygon/polyhedron. We say that two Voronoi cells (and their generating points) are adjacent if they share a non-trivial Voronoi face. It is a direct consequence of the definition of Voronoi diagrams that the line joining two adjacent generating points is always perpendicular to the common face between them and bisected by the plane of the face. For two adjacent points, V i and V j, with associated face e, we refer to the midpoint between them as b e. Bounded diagrams introduce certain challenges because fundamental properties of the Voronoi diagram are broken at the boundary. For example, a non-convex boundary could lead to nonconvex cells. An extreme case can be considered by selecting a single generating point in an arbitrarily shaped domain. In order to simplify the problem, we only consider convex boundaries. In addition, we require that the orthogonality of the diagram is maintained at the domain boundary. Lemma 6. For a Voronoi polyhedron with isotropic, piecewise constant permeability ( K = κ I), by setting x = V and x e = b e, a diagonal matrix M can be constructed that satisfies both conditions (S1) and ( S2). Proof. Let us denote the rows of R and N by Re T and Ne T, respectively. Since the line joining two adjacent generating points is orthogonal to the Voronoi face between them, the vectors R e are collinear to N e. That is, R e = e R e κ N e, where is the uclidean norm on R d. Therefore, a diagonal matrix M with choice of entries (M ) ii = ei Re i κ satisfies condition ( S2). Note that the point b e may fall outside of the boundary faces, see Figure 3. This, however, presents no problems for our definition, as we allow the function w e to be negative, which can shift the boundary point x e outside of the face. In porous media applications, it is common to use the so-called 2.5-dimensional Voronoi mesh. These diagrams are constructed by forming a two-dimensional Voronoi diagram. Cells are then vertically extruded into three-dimensional prisms. A 2.5-dimensional Voronoi mesh allows for slightly greater flexibility in permeability tensor when establishing TPFA. 26

28 Figure 3: A two-dimensional Voronoi diagram (solid line) and the lines joining adjacent Voronoi generating points (dashed line), which corresponds to the Delaunay triangulation. The dashed lines are always perpendicular to and bisected by the corresponding edges of the Voronoi diagram. The red circle illustrates an example of how the intersection of the two lines may occur outside of the face boundaries. Lemma 7. Given a mesh that is a Voronoi diagram in the x-y direction and orthogonally extruded in the z direction, let κ x 0 0 K = 0 κ y 0, 0 0 κ z with κ x = κ y. Then there exists a diagonal matrix M that satisfies (S1) and ( S2). Proof. In this case, a diagonal M can be constructed by setting for faces e facing in the x-y direction, and for faces on the z-plane. (M ) ii = e i R ei κ x (M ) ii = e i R ei κ z 5 Numerical Results In this section we present numerical results for our proposed generalization of the MFD method. The permeability tensor K is computed at the centroid of each cell. Given a set of points 27

29 z, T h, define ( ) 1/2 Pressrr(z ) = (p(z ) p ) 2. Note that ( ) 1/2 p Iw p h Qh = ((p Iw ) p ) 2 ( ) 1/2 (p(x ) p ) 2 = Pressrr(x ). Note that for linear p, p Iw Similarly, = p(x ), therefore the above is a second-order approximation. ( ) 1/2 p I p h Qh = ((p I ) p ) 2 ( ) 1/2 (p(c ) p ) 2 = Pressrr(C ), which is also a second order approximation. We can establish similar norms for the velocity errors. Given a set of points z e e, define ( Velrr(z e ) = ) 1/2 (u(z e ) n e u e ) 2. e Note that ( u Iw u h h ) 1/2 ((u Iw ) e u e ) 2 e ( ) 1/2 ((u(b e ) n e u e ) 2 e = Velrr(b e ). Note that this is a second order approximation, since for linear u n e on e, (u Iw ) e = u(b e ) n e. 28

30 Similarly, u I u h h ( ) 1/2 ((u I ) e u e ) 2 e ( ) 1/2 ((u(c e ) n e u e ) 2 = Velrr(c e ). e For each mesh, we test four different methods based on choices of points x and x e in ( S2): Case 1: x = C, x e = c e, Case 2: x = V, x e = c e, Case 3: x = C, x e = b e, Case 4: x = V, x e = b e. These four cases are illustrated in Figure 4. In all cases we compute the pressure errors for z = C, the centroid of, and z = V, the generating point for the Voronoi diagram and the velocity errors for z e = c e, the centroid of e, and z e = b e, the Voronoi bisection point. For Case 1 and Case 3, we have For Case 2 and Case 4, we have For Case 1 and Case 2, we have For Case 3 and Case 4, we have Pressrr(C ) = Pressrr(x ) p Iw p h Qh = p I p h Qh, Pressrr(V ) is not directly related to either error. Pressrr(C ) p I p h Qh, Pressrr(V ) = Pressrr(V ) p Iw p h Qh. Velrr(c e ) = Velrr(x e ) u Iw u h h = u I u h h, Velrr(b e ) is not directly related to either error. Velrr(c e ) u I u h h, Velrr(b e ) u Iw u h h. The theory predicts for Case 1 and Case 3 the following pressure convergence rates: Pressrr(C ) = O(h 2 ), Pressrr(V ) = O(h). 29

31 C e3 b e3 C C e2 C e4 b e2 V b e4 C e1 b e1 Case 1 Case 2 Case 3 Case 4 Figure 4: The four cases are based on different locations for placing points x e and x. The point C represents the centroid of the cell, and the point c e represents the centroid of the face e. The point V represents the Voronoi generating point for cell, and the point b e represents the bisection point of the line joining two adjacent Voronoi generating points. For Case 2 and Case 4, we expect the following convergence rates: Pressrr(C ) = O(h), Pressrr(V ) = O(h 2 ). We expect at least O(h) for all velocity errors. Recall from Section 2.2 that we expect better accuracy for u Iw u h h compared to u I u h h, which translates to better accuracy for Velrr(c e ) in Cases 1 and 2 and for Velrr(b e ) in Cases 3 and Two-Dimensional Problems We present results confirming convergence of the modified MFD formulation for (2.1) over twodimensional meshes. We consider the following manufactured solution from [14] over the unit square domain Ω = [0, 1] 2, and a full permeability tensor p(x, y, z) = x 3 y 2 + x sin(2πxy) sin(2πy) + 1, K = ( (x + 1) 2 + y 2 xy xy (x + 1) 2 For all the example in this work, the matrix U is chosen to be Rectangular Grids U = trace(k ) I. We carry out the results on the meshes in Figure 6, see also Figure 7 for an example of the computed pressure solution and the analytic solution. These meshes are generated by perturbing ). 30

32 Cell Centers (C ) Voronoi Generating Points (V ) Figure 5: An illustration of the cell centered points and the Voronoi generating points used for the two-dimensional rectangular numerical experiments. In this case, the rectangular mesh is also a Voronoi diagram. evenly spaced points with spacing h by ξ l = lh + 3 sin(4πlh), ξ = x, y. 50 These points serve as the shifted location V for point x. The mesh is then constructed as a Voronoi diagram using these points as generating points (see Figure 5). This is equivalent to constructing a rectangular mesh in a point-centered fashion, where faces are placed midway between two adjacent points. The centroids of the cells formed by this procedure are the points C, see Figure 5. As predicted, we observe O(h 2 ) convergence for the pressure at the points x, i.e., at points C for Cases 1 and 3, and at points V for Cases 2 and 4. We observe a deterioration in the convergence rate of approximately O(h 3/2 ) at the points that are not used in the construction of matrix M. With respect to the velocity error, we observe superconvergence of at least O(h 3/2 ) in all cases. As expected, the convergence is somewhat better at the points x e, i.e. at points c e for Cases 1 and 2, and points b e for Cases 3 and Voronoi Grids We test the method using randomly generated Voronoi diagrams. The diagrams are constructed by selecting uniformly distributed random points in the domain as generating points. The mesh is refined by selecting a larger number of randomly generated points unrelated to the previous 31

33 Figure 6: The rectangular meshes used in the convergence study. Figure 7: The MFD method pressure solution (left) and the analytic pressure solution (right) for using rectangular meshes. 32

34 (Case 1: x = C, x e = c e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e (Case 2: x = V, x e = c e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e (Case 3: x = C, x e = b e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e (Case 4: x = V, x e = b e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e Table 1: Convergence rates for the four test cases with rectangular grids in two dimensions. The results demonstrate at least first-order convergence across the board, and second-order convergence of the pressure variable when the error is calculated at the point x, as predicted by theory. We consistently find a slightly better velocity approximation when calculating the velocity error at point x e. 33

35 Figure 8: The random Voronoi meshes used in the convergence study. mesh. xamples of the meshes used can be seen in Figure 8. An example of the computed pressure solutions along with the analytic solution can be seen in Figure 9. We tested the four different cases, and the results can been seen in Table 2. We observe second-order convergence occurs at the location of point x. Shifting of the point on the interior changes where the solution is most accurate. As predicted by the theory, we consistently maintain at least first-order convergence of both pressure and flux independent of shifting x and x e. 5.2 Three-Dimensional Problems We now consider problems defined in three-dimensions. We solve the following manufactured problem over the unit cube domain Ω = [0, 1] 3, and a full permeability tensor p(x, y, z) = x 3 y 2 z + x sin(2πxy) sin(2πyz) sin(2πz) + 1, K = 1 + y 2 + z 2 xy xz xy 1 + x 2 + z 2 yz xz yz 1 + x 2 + y 2 We follow the same four test cases outlined in the previous section. The matrix U is chosen to be U = trace(k ) I.. 34

36 (Case 1: x = C, x e = c e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e (Case 2: x = V, x e = c e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e (Case 3: x = C, x e = b e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e (Case 4: x = V, x e = b e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e Table 2: Convergence rates for the four test cases with random Voronoi grids. We consistently observe first-order convergence for all cases, as predicted by the theory. We also observe secondorder convergence of pressure when the error is calculated at x. 35

37 Figure 9: The MFD method pressure solution (left) and the analytic pressure solution computed at the cell centroids (right) using two-dimensional Voronoi diagrams Rectangular Grids We carry out the tests on the meshes in Figure 10. These meshes are generated by perturbing points evenly spaced by h by, ξ l = lh + 3 sin(4πlh), ξ = x, y, z. 50 The mesh is then constructed as a Voronoi diagram using these points as generating points. This is equivalent to constructing a rectangular mesh in a point-centered fashion, where faces are placed midway between two adjacent points. The results of these experiments are shown in Table 3. xamples of the pressure solution can been seen in Figure 11. Much like the twodimensional problem, in all cases we have established at least first-order convergence, and often the method exhibits superconvergence. We observe again a second-order convergence of the pressure at the point x, as well as more accurate velocities at the points x e Voronoi Grids We test the method over three-dimensional Voronoi meshes. The meshes are generated by randomly selecting points uniformly over the domain. An example of the produced meshes can be seen in Figure 12. The pressure solution can been seen in Figure 13. The convergence results are shown in Table 4. Much like the two-dimensional case, we find that the pressure solution is most accurate at the points x. 6 Conclusions The MFD method defines a family of discretizations on a very general set of polyhedral meshes. Solving over general polyhedra allows for better representation of complex geometric features. 36

38 Figure 10: The rectangular meshes used in the convergence study. The two meshes correspond to h = 1 16 and Figure 11: Comparison between the MFD method solution (left) and the exact solution evaluated at the cell centroids (right) over a three dimensional domain with full permeability tensor. 37

39 (Case 1: x = C, x e = c e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e (Case 2: x = V, x e = c e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e (Case 3: x = C, x e = b e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e (Case 4: x = V, x e = b e ) n Pressrr(C ) Conv. Pressrr(V ) Conv. Velrr(c e ) Conv. Velrr(b e ) Conv e e e e e e e e e e e e e e e e e e e e Table 3: Convergence rates for the four test cases with rectangular grids in three-dimensions. We consistently find at least first-order convergence for both pressure and velocity. Also note that the pressure solution is most accurate when the error is calculated using the x point. The velocity exhibits a slightly more accurate solution when the error is calculated at x e. 38

40 Figure 12: Unstructured three-dimensional Voronoi diagrams. The plot on the left is the full domain of the convergence test problem in which we can only see the outer faces of the boundary. The plot on the right is a slice of the mesh showing the inner structure of the diagram. Figure 13: Pressure solution over a three-dimensional Voronoi diagram. The plot on the left represents the MFD method s approximation to pressure, and the plot on the right is the analytic solution computed at the cell centroids. 39