A SYMMETRIC NODAL CONSERVATIVE FINITE ELEMENT METHOD FOR THE DARCY EQUATION

A SYMMETRIC NODAL CONSERVATIVE FINITE ELEMENT METHOD FOR THE DARCY EQUATION GABRIEL R. BARRENECHEA, LEOPOLDO P. FRANCA 1 2, AND FRÉDÉRIC VALENTIN Abstract. Tis work introduces and analyzes novel stable Petrov-Galerkin Enriced Metods (PGEM) for te Darcy problem based on te simplest but unstable continuous P 1 /P 0 pair. Stability is recovered inside a Petrov-Galerkin framework were element-wise dependent residual functions, named multi-scale functions, enric bot velocity and pressure trial spaces. Unlike te velocity test space tat is augmented wit bubble-like functions, multiscale functions correct edge residuals as well. Te multi-scale functions turn out to be te well-known lowest order Raviart-Tomas basis functions for te velocity and discontinuous quadratics polynomial functions for te pressure. Te enricment strategy suggests te way to recover te local mass conservation property for nodal-based interpolation spaces. We prove tat te metod and its symmetric version are well-posed and acieve optimal error estimates in natural norms. Numerical validations confirm claimed teoretical results. 1. Introduction Te Darcy equation arising in a porous media field belongs to te family of mixed problems [13] for wic numerical metods are limited by te coice of pair of approximation spaces. From classical stable elements as te Raviart-Tomas family (RT k ) [27], Brezzi-Douglas- Marini elements (BDM k ) [12], ig order stable elements given in [25, 5, 6] to more recent stabilized or least square finite element metods [26, 23, 14, 10, 11] te range of possibilities to tackle te Darcy equation as increased over te past years. Metods for tis problem sould combine stability and accuracy wile preserving pysical properties inerited from te continuous problem. Properties tat are only fulfilled by few of tem. To te best of our knowledge, symmetric stable nodal based finite element metods for te Darcy equation preserving mass locally remain an open problem (see [11] for a recent discussion). For example, least-squares finite element metods (cf. [10]) lead to a symmetric positive-definite system, but, in teir original nodal version tey are not locally mass conservative, and in [11] nodal unknowns for te velocity are forbidden. Furtermore, in Key words and prases. Darcy model; Enriced space; simplest element; Petrov-Galerkin approac. 1 Tis autor is supported by NSF/USA No. 0610039. 2 Tis autor is supported by CNPq /Brazil Grant No. 306255/2008-1and 304051/2006-3, FAPERJ/Brazil Grant No. E-26/100.519/2007, and by NSF/USA No. 0610039. 1

2 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN bot cases, te lowest order piecewise constant space for te pressure is not allowed. Tis possibility is considered in [25], but te degrees of freedom for te velocity are not nodal. Some of te previous requirements are satisfied by te so-called Petrov-Galerkin Enriced Metods (PGEM). PGEM ave been developed in [19, 20, 21] and furter analyzed in [18, 4]. Te metod is constructed by enricing polynomial functions wit two types of enancement: we add bubble functions to te test function and we add a special function to te trial function. Te latter depends on te residual of te polynomial part over edges, and tus, it is no longer a bubble-like function. Tis gives us a Petrov-Galerkin framework. To get stabilized metod forms of PGEM we use static condensation, tanks to te use of bubbles as test functions [3, 8]. Number and type of degrees of freedom stay uncanged wereas basis functions incorporate unsolved sub-scales modifying teir form yet preserving te polynomial basis function support. Interested readers can find a review on te subject in [2]. Wen applied to te Darcy equation, te Petrov-Galerkin approac leads to different finite element metods [7]. One of tem is obtained by searcing te velocity solution into a subspace of te Raviart-Tomas space built wit te Raviart-Tomas interpolation over te linear continuous trial functions. Te space for te pressure stays untouced. Te underlying PGEM appears to be stable for te simplest pair of interpolation spaces P 1 /P 0 wile preserving te mass-conservative feature, a desirable property for porous media practitioners. Performance of PGEM over several numerical tests given in [7] attests its stability and accuracy wile keeping loss of local mass negligible. Based on te previous considerations, te current work introduces a variant of te strategy proposed in [7] and leads to a final metod wic is symmetric, locally mass conservative and wose degrees of freedom are piecewise constants for te pressure and nodal values for te velocity. Indeed, we keep te trial space for te velocity and pressure as in [7], but te test space is built differently: it is first mapped using te Raviart-Tomas interpolation operator and ten enanced wit bubble functions, an approac wic allows te static condensation procedure. Tis new perspective opens te door to two new finite element metods, one of tem fulfilling all te requirements of symmetry, nodal degrees of freedom for te velocity and locally mass conservative. Bot metods prove to be well-posed and acieve optimal error estimates in natural norms. Since te starting point of our approac is a Petrov-Galerkin metod, te terminology PGEM is still used in tis work, even if te final metods differ from te ones presented in [7]. Finally, te approac suggests a general way of rendering some finite element metods locally mass conservative. We end tis introduction by summarizing te plan and main results of tis paper:

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 3 In Section 2 we introduce te new finite element metods, namely, te non-symmetric (13) and its symmetric counterpart (16). We ten derive te metods in a PGEM framework in Section 2.1, and in Section 2.2 we explicit te local problems solved by our enricment basis functions (cf. (35) and (37)). In particular, we recover te local basis of te RT 0 space as te solution of (35)-(36). Finally, in Section 2.3 we prove tat te discrete enanced solution is locally mass conservative; Section 3 is devoted to te error analysis. We analyze in detail metod (16) (te analysis of (13) is treated in Teorem 9, Section 3.3). Well posedeness and consistency error are proved first (cf. Lemmas 3 and 4) followed by convergence results (see Teorems 7 and 8 in Section 3.2). Furtermore, we use te caracterization of te RT 0 interpolation operator as te solution of (35)-(36) to obtain an alternative proof for te classical RT 0 error estimate (see Corollary 6); Te numerical tests are in Section 4 were two analytical solutions confirm teoretical results; Conclusions and future perspectives are drawn in Section 5; Finally, we relax te assumption on te source term g (initially assumed piecewise constant) to propose in Appendix A an error estimate for a smoot datum g (cf. Teorem 10). 1.1. Some notations. Tis section introduces definitions and notations used trougout. In wat follows, Ω denotes an open bounded domain in R 2 wit polygonal boundary Ω, and x = (x 1, x 2 ) is a typical point in Ω. As usual, L 2 (Ω) is te space of square integrable functions over Ω, L 2 0 (Ω) represents functions belonging to L2 (Ω) wit zero average in Ω, and H div (Ω) is composed by functions tat belong to L 2 (Ω) 2 wit divergence in L 2 (Ω). Te space H div 0 (Ω) stands for te space of functions belonging to H div (Ω) wic ave normal component vanising on Ω. From now on we denote by {T } a family of regular triangulations of Ω built up using triangles K wit boundary K composed by edges F. Te set of internal edges of te triangulation T is denoted by E. Te caracteristic lengt of K and F are denoted by K and F, respectively, and := max{ K : K T } > 0, and due to te mes regularity tere exists a positive constant C suc tat F K C F, for all F K. Also, for eac F = K K E we coose, once and for all, an unit normal vector n wic coincides wit te unit outward normal vector wen F Ω. Te standard outward normal vector at te edge F wit respect to te element K is denoted by n K F. Moreover, for a function q, one

4 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN denotes q its jump, defined by (see Figure 1): (1) and q = 0 if F Ω. q (x) := lim q(x + δn) lim q(x + δn), δ 0 + δ 0 Next, we denote by H0 div (K) te space wose functions belong to H0 div (Ω) wit support in K and vanising normal component on K, and L 2 0 (K) te space of functions wic belong to L 2 0 (Ω) wit support and zero mean in K. Ten, we can define te corresponding global spaces H div 0 (T ) := K T H div 0 (K) and L 2 0 (T ) := K T L 2 0 (K). Finally, (, ) D stands for te inner product in L 2 (D) (or in L 2 (D) 2, wen necessary), and s,d ( s,d ) te norm (seminorm) in H s (D) (or H s (D) 2, if necessary), and div,d te norm in H div (D). F n K K Figure 1. Te normal vector. 1.2. Preliminaries. In tis work we consider te following Darcy problem: Find (u, p) suc tat (2) σ u + p = f, u = g in Ω, u n = 0 on Ω, were σ = µ κ R+ is assumed constant in Ω, wit µ and κ denoting te viscosity and permeability, respectively. Here, u is te so-called Darcy velocity, p is te pressure, f and g are given source terms. We suppose f piecewise constant since it is usually related to te gravity force. Moreover, we assume tat te given data ave enoug regularity and te usual compatibility condition g = 0, Ω

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 5 olds. Remark. Wen we consider (2) wit a prescribed flux b on Ω suc tat g = b, Ω we can recover te omogeneous case since tere exists a function w b belonging to H div (Ω) suc tat w b n = b on Ω (cf. [22]), and tus we replace te rigt and side f by f σw b and g by g w b. Remark. In te more general case, σ can always be approximated by performing projections onto te piecewise constant space. Tis possibility as been considered in [7]. On te oter and, despite te fact tat te metods are presented in te two dimensional case, teir extension to te tree dimensional framework is straigtforward. Te standard symmetric mixed variational formulation associated wit (2) reads: Find (u, p) H0 div (Ω) L 2 0 (Ω) suc tat Ω (3) A s ((u, p), (v, q)) = F s (v, q) (v, q) H div 0 (Ω) L 2 0 (Ω), were A s ((u, p), (v, q)) := (σ u, v) Ω (p, v) Ω (q, u) Ω, F s (v, q) := (f, v) Ω (g, q) Ω. Te well-posedness of (3) follows from te classical Babuska-Brezzi teory for variational problems wit constraints (see [13] for more details). Remark. An equivalent and still well-posed non-symmetric version of (3) arises from adding te weak form of te second equation to te first one in (2). Te bilinear form and te linear form are now denoted by A(.,.) and F(.), respectively, and are given by (4) A((u, p), (v, q)) := (σ u, v) Ω (p, v) Ω + (q, u) Ω, F(v, q) := (f, v) Ω + (g, q) Ω. Next, te classical discrete mixed formulation of tis problem is: Find (u, p ) V Q suc tat (5) A s ((u, p ), (v, q )) = F s (v, q ) (v, q ) V Q, were V and Q are finite-dimensional approximations of H div 0 (Ω) and L 2 0(Ω), respectively. It is well known tat te pair of interpolation spaces for pressure and velocity must satisfy te discrete Babuska-Brezzi (or inf-sup) condition [13] in order to lead to a stable discrete version of problem (5). For te Darcy model containing a zero order term, tis restriction

6 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN as been proved to be unnecessary. In fact, in [24] standard continuous Lagrangian finite element spaces ave been proved to be stable and convergent. Te lowest order Raviart-Tomas space is one of te simplest examples of a stable mass conservative element, and is composed by te velocity space V RT0 := {v H div 0 (Ω) : v K RT 0 (K) K T }, were te local space RT 0 (K) is defined by (6) RT 0 (K) := P 0 (K) 2 + x P 0 (K), and (7) Q := {q 0 L 2 0 (Ω) : q 0 K P 0 (K) K T }. Hence, only te normal component of te velocity is continuous and te inf-sup condition is satisfied since V RT0 = Q. Associated wit te space RT 0 (K) tere exists a natural local interpolation operator π K : [H 1 (K)] 2 RT 0 (K), defined by (cf. [13, 17]) (8) π K (v) n = v n, for all F K, or, equivalently (9) F π K (v) := F K F F v n F ϕ F, were ϕ F is te Raviart-Tomas basis function given by (10) ϕ F (x) = ± F 2 K (x x F), and x F denotes te node opposite to te edge F. Hence, a global interpolation operator noted π : [H 1 (Ω)] 2 V RT0 follows by defining π(v) K = π K (v) in eac K T. Remark. Te sign before te Raviart-Tomas basis function ϕ F depends on weter te normal vector n on F K points inwards or outwards K. A lifting operator from L 1 (F) to V RT0 will be needed in te sequel, it is denoted by l and is suc tat l(q) := F E l F (q) were l F (q) = α F (11) σ were te coefficient α F is a given positive constant wic is independent of F and σ, but F q ϕ F, can vary wit F. We finally denote, for K T, l K (q) := l(q) K.

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 7 2. Te finite element metods We begin by introducing te standard finite element space V := [V ] 2 H div 0 (Ω) for te velocity variable, were (12) V := {v C 0 (Ω) : v K P 1 (K), K T }, wereas te pressure is discretized using te space Q defined in (7). We start by presenting te metods tat will be analyzed in tis work. First, te Petrov-Galerkin Enriced Metod reads: Find (u 1, p 0 ) V Q suc tat (13) B((u 1, p 0 ), (v 1, q 0 )) = F s (π(v 1 ), q 0 ), for all (v 1, q 0 ) V Q, were (14) B((u 1, p 0 ), (v 1, q 0 )) := A s ((π(u 1 ), p 0 ), (π(v 1 ), q 0 )) + (l( p 0 ), σ π(v 1 )) Ω F E τ F ( p 0, q 0 ) F, and π and l are te operators defined troug (8) and (11), respectively, and te coefficient τ F stands for (15) τ F := α F F σ. In Section 3 tis problem is proved to be well-posed for an appropriate coice of α F. Alternatively, a symmetric related formulation can also be derived and reads: Find (û 1, ˆp 0 ) V Q suc tat (16) B s ((û 1, ˆp 0 ), (v 1, q 0 )) = F s (π(v 1 ), q 0 ), for all (v 1, q 0 ) V Q, were (17) B s ((u 1, p 0 ), (v 1, q 0 )) := A s ((π(u 1 ), p 0 ), (π(v 1 ), q 0 )) F E τ F ( p 0, q 0 ) F. Remark. Te latter metod is based on te error analysis (see 3) wic points out tat we can remove from (13) te term (l( p 0 ), σ π(v 1 )) Ω witout introducing a loss of accuracy. Moreover, we recover te symmetric form of te so-called reduced PGEM metod presented in [7], by replacing te term (π(u 1 ), π(v 1 )) Ω by (u 1, v 1 ) Ω in (13). Tis reduced metod turns out to be optimally convergent [14], and in 2.3 we sow ow to render it locally mass conservative.

8 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN 2.1. Derivation of te metods. Te starting point towards our final metod is te following Petrov-Galerkin metod for (2): Find u := u 1 + u e V + H div 0 (Ω) and p := p 0 + p e Q L 2 0(T ) suc tat (18) A s ((u, p ), (v, q )) = F s (v, q ), for all v := π(v 1 ) + v b π(v ) H div 0 (T ) and for all q := q 0 + q e Q L 2 0(T ). Here π(v ) stands for te subspace of V RT0 built as te image of space V troug te operator π. Tis sceme is equivalent to te following system: for all (v 1, q 0 ) V Q and for all (v b, q e ) H div 0 (T ) L 2 0(T ) (19) (20) A s ((u, p ), (π(v 1 ), q 0 )) = F s (π(v 1 ), q 0 ), A s ((u, p ), (v b, q e )) = F s (v b, q e ). From now on, and just in order to derive te metod, we will assume tat g is a piecewise constant function (even if te metod is analyzed and implemented for more general functions g). Wit tis assumption in mind, starting from (20) and proceeding as in [7], te following strong problem is obtained for (u e, p e ): (21) (22) σ u e + p e = f σu 1, u e = p 0 + (Π K ( u 1 ) u 1 ) in K, σ u e n = α F p 0 + ( (f σu 1 ) n Π F ((f σu 1 ) n) ), F on eac F K E, and u e n = 0 on F Ω. Here, Π F and Π K stand for te L 2 - projection operators over te constant space, i.e, Π F (v) = 1 v and Π F F K(v) = 1 v. K K Finally, te constant p 0 is cosen in order to make (21)-(22) compatible, and is given by p 0 := 1 K 3 i=1 α Fi Fi σ F i p 0 n n K F i. Remark. For iger order velocity interpolation te vanising rigt and side term Π K ( u 1 ) u 1 left in te local problem (21) needs to be taken into account. Moreover, written in tis form, it will elp us to bound te consistency errors (see equation (62) below). On te oter and, since we ave assumed tat f is a constant (or piecewise constant) function, te divergence equation in (21) may be rewritten as follows (23) u e = p 0 + 1 σ ( (f σu 1) Π K ( (f σu 1 ))), wic is te form tat we will consider from now on.

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 9 Remark. Te boundary condition (22) assures te continuity of te normal component of te velocity on eac edge, tus keeping our approac conforming. Tis fact may be kept even for discontinuous coefficients, as it as been done in [7], were te mean value of σ as been included in te boundary condition. Now, let M K := (M u K, Mp K ) : H1 (K) 2 H0 div (K) L 2 0 (K) and D K := (DK u, Dp K ) : L 2 ( K) H div (K) L 2 0(K), defined as follows: (v e, η e ) := (M u K (v), Mp K (v)) is te solution of (24) σv e + η e = v, σ v e = v Π K ( v) in K, σ v e n = v n Π F (v n) on eac F K, and (w e, ξ e ) := (DK u (q), Dp K (q)) solves (25) σ w e + ξ e = 0, w e = 1 K σ w e n = α F F 3 i=1 α Fi Fi σ q on eac F K Ω. F i q n n K F i in K, Ten, using tese operators and (23), we can caracterize te solution (u e, p e ) = (u M e + u D e, pm e + p D e ) of (21)-(22) as follows (26) (27) (u M e, pm e ) = M K (f σu 1 ) K T, (u D e, p D e ) = D K ( p 0 ) K T. Next, we turn back to equation (19). First, since p e L 2 0 (K) and π(v 1) K R we obtain (28) (p e, π(v 1 )) K = 0 for all K T. Terefore, te problem (19) becomes: Find (u 1, p 0 ) V Q suc tat (29) A s ((u 1 + u e, p 0 ), (π(v 1 ), q 0 )) = F s (π(v 1 ), q 0 ) (v 1, q 0 ) V Q, were u e is caracterized wit respect to u 1 and p 0 by (26)-(27). It is also convenient to rewrite te problem above in an equivalent form integrating it by parts in eac K T (30) (q 0, u e ) K = τ F ( p 0, q 0 ) F. K T F E

10 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN Remark. Te term related to f in (26) vanises. Indeed, since f is constant in K ten f n Π F (f n) = 0 on eac edge F, and tere also exists a polynomial function q e belonging to L 2 0(K) suc tat q e = f in eac K T, wic leads to M u Kf = 0. Terefore, no enricing contribution comes from (26) but for te one related to u 1. Finally, based on te previous remark, and replacing (30) and (26)-(27) in (29), we arrive at te following final form of PGEM: Find (u 1, p 0 ) V Q suc tat A s ((u 1 σm u K(u 1 ), p 0 ), (π(v 1 ), q 0 )) + K T (D u K( p 0 ), σ π(v 1 )) K F E τ F ( p 0, q 0 ) F = F s (π(v 1 ), q 0 ), for all (v 1, q 0 ) V Q, wic is precisely te metod (13) since, as it will be sown in terms of te basis function in te next section, te following olds (31) π K (u 1 ) (I σm u K )(u 1) and l K ( p 0 ) D u K ( p 0 ). Next, te symmetric metod (16) follows by neglecting te non diagonal term (l( ˆp 0 ), σ π(v 1 )) Ω as tis term does not undermine convergence estimates (see Section 3). Remark. Following analogous steps and just replacing te forms A s (.,.) and F s (.) by A(.,.) and F(.), and switcing te velocity test space π(v ) to V in (18) we arrive at te following metod: Find (û 1, ˆp 0 ) V Q suc tat (32) A((π(û 1 ), ˆp 0 ), (v 1, q 0 )) + (l( ˆp 0 ), σ v 1 ) Ω + F E τ F ( ˆp 0, q 0 ) F = F(v 1, q 0 ), for all (v 1, q 0 ) V Q, wic is exactly te PGEM proposed in [7]. 2.2. Te local problems. Tis section is devoted to sow te relationsip between te local problems (26)-(27) and te Raviart-Tomas interpolation operator π and te lifting operator l. First, we decompose u 1 = 2 3 k=1 i=1 uk i ψk i, were uk i are te nodal values of u 1 and ψ k i, tat denotes te (vector-valued) at function. Ten, we look for solutions of (26) and (27) in te form (33) u D e K = 3 α Fj F j p 0 ϕ σ j and p D e K = j=1 3 α Fj F j p 0 η j, σ j=1

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 11 and (34) u M e K = σ 2 3 u k i ϕk i and p M e K = σ 2 3 u k i ηk i. k=1 i=1 k=1 i=1 Here, indexes j and i are representing edges and local nodal numeration, respectively. Next, by replacing (33) and (34) in te local problems (27) and (26) respectively, and factoring out te coefficients it follows tat te multi-scale basis functions (ϕ j, η j ) and (ϕ k i, ηk i ) must satisfy te following well-posed local Darcy problems (35) (36) σ ϕ j + η j = 0, ϕ j n = { 1 if j = i, 0 oterwise ϕ j = F j K n nk F j in K, on eac F i K, and (37) σ ϕ k i + η k i = ψ k i, ϕ k i = 0 in K, (38) σ ϕ k i n = ψ k i n Π F (ψ k i n) on eac F K. Te enricment functions emanating from te problem (35)-(36) are noting but te well known basis functions of te space RT 0 (K), i.e, te lowest order Raviart-Tomas approximation of H div (K) defined by (6), and tey are given by (cf. (10)) (39) ϕ j (x) = ± F j 2 K (x x F j ) for j = 1, 2, 3. Consequently, (40) η j (x) = σ ( F j x 2 ) 2 K 2 x x F j + C j for j = 1, 2, 3, were te constant C j is set up so K η j = 0. Te solution of (37)-(38) can also be analytically computed, providing (41) σϕ k i = ψk i 3 Π F (ψ k i n) ϕ j = ψ k i π K(ψ k i ), j=1 were π K as been defined in (9). A similar local problem as been used in [15] to obtain multiscale basis functions for te Darcy problem wit oscillating coefficients (see also [1] for te extension to porous media wit stocastic coefficients).

12 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN 2.3. Te local mass conservation feature: a general strategy. Computing numerical solutions troug PGEM (13) and te symmetric metod (16) does not assure local mass conservative velocity field if only linear part of solution u 1 (or û 1 ) is considered. Regarding te first metod, te required feature is acieved by locally updating u 1 wit u D e given by (27). In fact, since discontinuous pressure interpolations are used it emerges from (13) tat (see [7] for a related idea) K (u 1 + u D e ) = K g, for all K T. Remark. Summing up, we see tat in order to obtain a stable pair of interpolation spaces wit discrete velocity field locally mass-conservative, it is fundamental to enric te linear part of te discrete velocity u 1 wit an element of te Raviart-Tomas space V RT0, namely, te multi-scale function u D e computed from (33). We stress te fact tat te computation of (33) follows directly from te discrete solution, witout te need of any extra local computation. Moreover, te exact velocity u is approximated in eac K T by u = u 1 + u M e + u D e 2 3 = u k i ψ k i σ = k=1 3 l=1 i=1 2 k=1 3 u k i ϕ k i + i=1 ( ) Π Fl (u 1 n) + τ Fl Π Fl ( p 0 ) = π K (u 1 ) + l K ( p 0 ), ϕ l 3 l=1 α Fl σ F l p 0 ϕ l so, as expected, te continuity of te normal velocity component troug te internal edges is assured, but not te tangential one. Now, it turns out tat suc local mass recovering is not only restricted to metods arising exactly from te enancing approac. For example, stabilized metods based on pressure jumps as te one presented in [14], or te symmetric metod (16) are elegible to recover te local mass conservation feature adding l K ( p 0 ) to te computed velocity field, were p 0 is te constant pressure solution. We illustrate tis fact for te symmetric formulation (16). Coosing v 1 = 0 in (16) and q 0 = 1 in K and K / K in K (were K K = F E ), we obtain after integration by parts and te definition of l K tat (û 1 + l K ( ˆp 0 )) K (û K K 1 + l K ( ˆp 0 )) = K û 1 K û K K 1 + τ F ( ˆp 0, q 0 ) F = g K g, K F K K K K K

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 13 and ten, following closely te arguments given in [7], we obtain tat te value (û K 1 l K ( ˆp 0 )) g vanises in eac K, and ence it must vanis on eac element. It is important to empasize once more tat l( ˆp 0 ) does not perturb too muc te solution in te sense tat te order of error estimates is still preserved (see Teorems 8 and 9 for details). 3. Error analysis In te sequel C denotes a generic positive constant, independent of or σ, wit values tat may vary in eac occurrence. Before performing an error analysis of (13) and (16), we need to consider interpolation inequalities to approximate variables. 3.1. Interpolation, stability and consistency results. We start by presenting te Clément interpolation operator (cf. [16, 22, 17]) C : H 1 (Ω) V (wit te obvious extension to vector-valued functions), satisfying, for all K T and all F E, (42) (43) v C (v) m,k C t m K v t,ω K v H t (ω K ), v C (v) 0,F C t 1 2 F v t,ωf v H t (ω F ), for t = 1, 2, m = 0, 1, were ω K = {K T : K K } and ω F = {K T : K F }. Now, in order to take into account te approximation of te pressure and te consistency error, we consider te L 2 (Ω) projection onto Q wic is denoted by Π : L 2 (Ω) Q. Tis projection satisfies (cf. [17]) (44) q Π (q) m,ω C 1 m q 1,Ω q H 1 (Ω), for m = 0, 1. Moreover, using te result above and te following local trace inequality: given K T, F K, tere exists C suc tat for all v H 1 (K) ( 1 ) (45) v 2 0,F C v 2 0,K + K v 2 1,K, K we obtain (46) [ F E F q Π (q) 2 0,F ] 1/2 C q 1,Ω. Moreover, we will systematically use te Raviart-Tomas interpolation operator π defined troug (9) as π(v) = F E Π F (v n F ) ϕ F, satisfying (see [17] or Corollary 6 for an alternative proof) (47) v π(v) 0,Ω C v 1,Ω,

14 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN for all v H 1 (Ω) 2 and (48) v π(v) 0,Ω C v 1,Ω, for all v H 1 (Ω) 2 suc tat v H 1 (Ω). Now, we define te following mes dependent norm ( ) 1/2 (49) (v, q) := σ v 2 0,K + τ F q 2 0,F, K T F E and we present an interpolation result in tis norm. Lemma 1. Let us suppose tat (v, q) H 1 (Ω) 2 H 1 (Ω). Ten, tere exists C suc tat (50) ( σ (v π(c (v)), q Π (q)) C v 1,Ω + 1 q 1,Ω ). σ Proof. From te definition of te norm, (42) and (46) tere follows tat (v π(c (v)), q Π (q)) 2 = σ v π(c (v)) 2 0,Ω + F E τ F q Π (q) 2 0,F 2 σ ( v π(v) 2 0,Ω + π(v C (v)) 2 0,Ω) + F E τ F q Π (q) 2 0,F (51) C ) (σ 2 v 21,Ω + 2 σ q 2 1,Ω + 2σ π(v C (v)) 2 0,Ω. Next, from its definition it is easy to prove tat te Raviart-Tomas operator π satisfies (see [9] for a related result and Lemma 5 for an alternative proof) (52) π(v) 0,Ω C ( v 0,Ω + v 1,Ω ), for all v H 1 (Ω) 2, and ten te result follows applying (52) and (42) in (51). Before eading to stability, an auxiliary result is stated next. Lemma 2. Let π be te Raviart-Tomas interpolator, ten tere exists a positive constant C 1 suc tat, for all v 1 V and q 0 Q, it olds (53) (l( q 0 ), σ π(v 1 )) Ω C 1 { F E τ F q 0 2 0,F }1 2 σα π(v1 ) 0,Ω, were α := max{α F : F E }.

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 15 Proof. In order to prove (53) we first note tat l( q 0 ) V RT0. Hence, using successively te Caucy-Scwarz inequality, (11) and ϕ F 0,K C 1 F we get (l( q 0 ), σ π(v 1 )) Ω K T l K ( q 0 ) 0,K σ π(v 1 ) 0,K K T C 1 F K α F σ K T F K C 1 { F E τ F q 0 2 0,F F q 0 ϕ F 0,K σ π(v 1 ) 0,K τ F 1 2 F q 0 0,F σ π(v 1 ) 0,K } 1 2 σ α π(v 1 ) 0,Ω, and te result follows. We are ready to prove PGEM are well-posed. Lemma 3. Te bilinear forms B s (.,.) defined in (17) satisfies B s ((v 1, q 0 ), (v 1, q 0 )) = (π(v 1 ), q 0 ) 2 (v 1, q 0 ) V Q. Moreover, assuming α 1 C 2 1 2 were C 1 is te positive constant from Lemma 2, te bilinear form B(.,.) defined in (14) satisfies, B((v 1, q 0 ), (v 1, q 0 )) 1 2 (π(v 1), q 0 ) 2 (v 1, q 0 ) V Q. Hence, te problems (13) and (16) are well-posed. Proof. Te first equality follows directly from te definition of te bilinear form B s (.,.). For te second one, we recall tat (54) B((v 1, q 0 ), (v 1, q 0 )) = σ π(v 1 ) 2 Ω + (l( q 0 ), σ π(v 1 )) Ω + F E τ F q 0 2 0,F. As for te second term, we use Lemma 2 to obtain }1 2 (l( q 0 ), σ π(v 1 )) Ω C 1 { F E τ F q 0 2 0,F σα π(v 1 ) 0,Ω F E 1 2 C2 1 α 2 τ F q 0 2 0,F + 1 2 σ π(v 1) 2 0,Ω,

16 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN Hence by successively applying inequality above into (54) and assuming α 1 C 2 1 2 it olds B((v 1, q 0 ), (v 1, q 0 )) σ 2 π(v 1) 2 0,Ω + F E (1 C2 1 α 2 2 1 2 (π(v 1), q 0 ) 2. ) τ F q 0 2 0,F Te continuity of bilinear forms is straigtforward and tus te well-posedness of (13) and (16) stems from te Necas Teorem [17]. Remark. We remark tat te metric (π(v 1 ), q 0 ) defines a norm in te space V Q since te Raviart-Tomas interpolation operator is injective wen restricted to V. On te oter and, we remark tat te ypotesis on α is not really restrictive, since it only applies for coarse meses. For a sufficiently refined mes te coice for α is essentially unlimited. Neiter of te metods proposed in te previous section are formally consistent as points out te next result. Lemma 4. Let (u, p) H div 0 (Ω) [H 1 (Ω) L 2 0 (Ω)] be te weak solution of (3), (u 1, p 0 ) te solution of (13) and (û 1, ˆp 0 ) te solution of (16), respectively. Ten, B ( (u u 1, p p 0 ), (v 1, q 0 ) ) = K T (σ M u K (u), σ π K(v 1 )) K, B s ( (u û1, p ˆp 0 ), (v 1, q 0 ) ) = K T (σ M u K (u), σ π K(v 1 )) K, for all (v 1, q 0 ) V Q. Proof. Te result follows from te definition of B(.,.) and B s (.,.), and noting tat p = 0 a.e. across all te internal edges 3.2. Error estimates for te symmetric formulation. We begin tis section by proving te following tecnical result concerning te operator M u K. Lemma 5. Let v H 1 (K) 2. Ten, tere exists a constant C suc tat M u K(v) 0,K C σ 1 K v 1,K. Proof. Let v H 1 (K) 2, and w := M u K (v). Ten, from te definition of Mu K satisfies (cf. (24)), w (55) σ w + ξ = v, σ w = v Π K ( v) in K, σ w n = v n Π F (v n) on eac F K,

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 17 were ξ L 2 0 (K). Multiplying te first equation by w and integrating it by parts we arrive at (56) σ w 2 0,K = (v, w) K + 1 σ (ξ, v Π K( v)) K 1 σ (ξ, v n Π F (v n)) F F K = (v, w) K + 1 σ (ξ Π K(ξ), v Π K ( v)) K 1 σ (ξ Π F (ξ), v n Π F (v n)) F F K v 0,K w 0,K + 1 σ ξ Π K(ξ) 0,K v Π K ( v) 0,K + 1 σ F K ξ Π F (ξ) 0,F v n Π F (v n) 0,F. Next, from (55) it olds ξ 0,K v 0,K +σ w 0,K and ence, using te local trace result (45), te approximation property of te projection operators Π F and Π K, and te inequality above to obtain σ w 2 0,K v 0,K w 0,K + C σ K ξ 1,K v 1,K (57) v 0,K w 0,K + C K σ ( v 0,K + σ w 0,K ) v 1,K C σ 1 ( v 2 0,K + 2 K v 2 1,K ) + σ 2 w 2 0,K, and ten we ave proved tat (58) M u K (v) 0,K C σ 1 ( v 0,K + K v 1,K ). Finally, let us denote v 0 = Π K (v). Since v 0 is a constant in eac element, tere olds tat M u K (v 0) = 0 and ten, from (58) it follows tat M u K (v) 0,K = M u K (v v 0) 0,K C σ 1 ( v v 0 0,K + K v 1,K ), and te result follows using te approximation properties of te projection. Te previous lemma results in an alternative proof of te following classical interpolation error estimate: Corollary 6. Tere exists C suc tat v π K (v) 0,K C K v 1,K, for all v H 1 (K) 2.

18 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN Proof. Te result follows from te previous lemma and te fact tat v π K (v) = σm u K (v). Teorem 7. Let (u, p) H0 div (Ω) H 2 (Ω) 2 H 1 (Ω) L 2 0 (Ω) be te solution of (2), and (û 1, ˆp 0 ) V Q te solution of metod (16). Ten, defining û := π(û 1 ), te following error estimate olds (59) (u û, p ˆp 0 ) C ( σ u 2,Ω + 1 σ p 1,Ω ). Proof. Let (v 1, q 0 ) = (C (u), Π (p)). From te triangle inequality we ave (60) (u π(û 1 ), p ˆp 0 ) (u π(v 1 ), p q 0 ) + (π(v 1 û 1 ), q 0 ˆp 0 ). Te first term is easily estimated using Lemma 1. Next, let us estimate te second term on te rigt and side. For tat, we use te coercivity of B s (.,.) (cf. Lemma 3) and te consistency result (cf. Lemma 4) to obtain (π(v 1 û 1 ), q 0 ˆp 0 ) 2 = B s((v 1 û 1, q 0 ˆp 0 ), (v 1 û 1, ˆp 0 q 0 )) = B s ((u v 1, p q 0 ), (v 1 û 1, ˆp 0 q 0 )) σ 2 K T (M u K (u), π K(v 1 û 1 )) K = σ (π(u v 1 ), π(v 1 û 1 )) Ω + (p q 0, (π(v 1 û 1 ))) Ω + (ˆp 0 q 0, (π(u v 1 ))) Ω + F E τ F ( p q 0, ˆp 0 q 0 ) F K T σ 2 (M u K (u), π K(v 1 û 1 )) K. Next, from te properties of te projection operator Π it olds (61) (p q 0, (π(v 1 û 1 ))) Ω = 0,

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 19 and ten, integrating by parts and using te Caucy-Scwarz s inequality we arrive at (π(v 1 û 1 ), q 0 ˆp 0 ) 2 σ π(u v 1) 0,Ω π(v 1 û 1 ) 0,Ω + F E ( ˆp 0 q 0, (u v 1 ) n) F + F E τ F p q 0 0,F ˆp 0 q 0 0,F + σ 2 K T (M u K(u), π K (v 1 û 1 )) K σ π(u v 1 ) 0,Ω π(v 1 û 1 ) 0,Ω + C F E ˆp 0 q 0 0,F 3 2 F u 2,ωF + F E τ F p q 0 0,F ˆp 0 q 0 0,F + σ 2 K T (M u K(u), π K (v 1 û 1 )) K σ π(u v 1 ) 0,Ω π(v 1 û 1 ) 0,Ω + C F E σ 1 2 τ 1 2 F ˆp 0 q 0 0,F F u 2,ωF + F E τ F p q 0 0,F ˆp 0 q 0 0,F + σ 2 K T (M u K (u), π K(v 1 û 1 )) K C { σ π(u v 1 ) 2 0,Ω + σ 2 u 2 2,Ω + F E τ F p q 0 2 0,F }1 2 (π(v 1 û 1 ), q 0 ˆp 0 ) + σ 2 K T (M u K (u), π K(v 1 û 1 )) K. Te last term on te rigt and side is estimated next. Using Lemma 5 we may bound te consistency term as follows (62) (M u K (u), π K(v 1 û 1 )) K M u K (u) 0,K π K (v 1 û 1 ) 0,K K T K T C σ 1 u 1,Ω π(v 1 û 1 ) 0,Ω. Collecting all te above results it follows from Lemma 1 tat 1 2 (π(v 1 û 1 ), q 0 ˆp 0 ) 2 C ( σ u 2,Ω + σ p 1,Ω ) (π(v 1 û 1 ), ˆp 0 q 0 ), and ence (59) follows. Next, we analyze te error in te H div (Ω) norm for te velocity and te L 2 (Ω) norm for te pressure.

20 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN Teorem 8. Let (u, p) and (û 1, ˆp 0 ) be te solutions of (3) and (16), respectively, and û := π(û 1 ). Under te ypotesis of Teorem 7, tere exists C suc tat (63) (64) (65) u û div,ω C ( u 2,Ω + 1 σ p 1,Ω ), u û l( ˆp 0 ) div,ω C ( u 2,Ω + 1 σ p 1,Ω), p ˆp 0 0,Ω C ( σ u 2,Ω + p 1,Ω ). Proof. First, let v 1 := C (u); ten (u û 1 ) 2 0,Ω = ( (u û 1 ), (u û 1 )) Ω (66) = ( (u û 1 ), (u v 1 )) Ω + ( (u û 1 ), (û 1 v 1 )) Ω. Next, from Lemma 4 (considering q 0 := (û 1 v 1 ) Q and v 1 = 0) we get ( (u û 1 ), (û 1 v 1 )) Ω = F E τ F ( p ˆp 0, (û 1 v 1 ) ) F F E τ F p ˆp 0 0,F (û 1 v 1 ) 0,F (67) F E τ F γ p ˆp 0 2 0,F + γ F E τ F (û 1 v 1 ) 2 0,F. Next, using te local trace result (45), (42) and te mes regularity to obtain γ τ F (û 1 v 1 ) 2 0,F C γ α [ ] K 1 K σ (û 1 v 1 ) 2 0,K F E K T (68) C γ σ C γ σ K T K T [ ] (u û 1 ) 2 0,K + (u v 1) 2 0,K [ (u û 1 ) 20,K + C2K u 22,ωK ]. Hence, coosing γ = σ 4C in (68) and using ab (a2 /4) + b 2, te mes regularity and (42) again, (66) and (67) become (u û 1 ) 2 0,Ω 1 4 (u û 1) 2 0,Ω + (u v 1 ) 2 0,Ω + τ F γ p ˆp 0 2 0,F + C γ ] [ (u û 1 ) 20,K + C 2K u 22,ωK σ F E K T C σ 1 F E τ F p ˆp 0 2 0,F + C 2 u 2 2,Ω + 1 2 (u û 1) 2 0,Ω (69) C σ 1 (u û, p ˆp 0 ) 2 + C 2 u 2 2,Ω + 1 2 (u û 1) 2 0,Ω,

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 21 and te result follows by applying Teorem 7 and extracting te square root. Now, we use te local mass conservation feature to prove (64). In fact, we get (u û 1 l K ( ˆp 0 )) = 0, K and ten, since (û 1 + l K ( ˆp 0 )) K R, we end up wit (u û 1 l K ( ˆp 0 )) 2 0,K = ( (u û 1 l K ( ˆp 0 )), (u û 1 l K ( ˆp 0 ))) K = ( (u û 1 l K ( ˆp 0 )), (u π(u))) K (70) C u 1,K (u û 1 l K ( ˆp 0 ) 0,K. As seen in Lemma 2 we can prove tat (71) l( ˆp 0 ) 0,Ω F E τ F ϕ F 0,Ω ˆp 0 C σ 1 2 and ten using (59) we obtain { F E τ F p ˆp 0 2 0,F }1 2, (72) u û l( ˆp 0 ) 0,Ω C ( u 2,Ω + 1 σ p 1,Ω), and (64) follows from (70) and (72). Finally, we consider te estimate for te pressure. From te continuous inf-sup condition (see [13]), tere exists a w H 1 0(Ω) suc tat w = p ˆp 0 in Ω and (73) w 1,Ω C p ˆp 0 0,Ω. Let w 1 = C (w). Since w 1 = (π(w 1 )), using Lemma 4 and recalling tat π K = I σm u K, we obtain p ˆp 0 2 0,Ω =( w, p ˆp 0) Ω =( (w w 1 ), p ˆp 0 ) Ω + ( w 1, p ˆp 0 ) Ω = K T [(w w 1, p) K + (w w 1, (p ˆp 0 ) I n) K ] + K T σ ((I σm u K )(u) π K(u 1 ), π K (w 1 )) K + σ 2 (M u K (u), π K(w 1 )) K [ ]1 C 2 K p 2 1,K + σ (u π(u 1), p ˆp 0 ) 2 2 K T [ 2 K w w 1 2 0,K + 1 F w w 1 2 0,F + π(w 1 ) 2 0,Ω K T F E ]1 2.

22 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN Now, using (42)-(43), te regularity of te mes, (52) and (73) we obtain [ 2 K w w 1 2 0,K + 1/2 τ 1 F w w 1 2 0,F + π(w 1) 0,Ω] 2 C w 1,Ω C p ˆp 0 0,Ω, K T F E ence, dividing by p ˆp 0 0,Ω and using Teorem 7, we arrive at [ ]1 p ˆp 0 0,Ω C σ (u π(û 1 ), p ˆp 0 ) 2 + σ2 2 p 2 2 1,Ω C [σ u 2,Ω + p 1,Ω ], and te result follows. Remark. Having assumed te coefficient σ to be constant in Ω (and ten independent of any small scale), te H 2 (Ω)-norm of te exact solution u does not blow up as is usual in Darcy problems wit igly oscillating coefficients (see [15] for furter details). 3.3. An error estimate for te metod (13). We end te error analysis by proving te following error result concerning te metod (13). Teorem 9. Let (u, p) H div 0 (Ω) H 2 (Ω) 2 H 1 (Ω) L 2 0(Ω) be te solution of (2), and (u 1, p 0 ), (û 1, ˆp 0 ) te solution of metods (13) and (16), respectively. Ten, defining u := π(u 1 ) and û := π(û 1 ), te following error estimate olds (u u, p p 0 ) 3 (u û, p ˆp 0 ). Proof. First, from Lemmas 3 and 4 and Lemma 2 it follows 1 2 (π(u 1 û 1 ),p 0 ˆp 0 ) 2 B((u 1 û 1, p 0 ˆp 0 ), (u 1 û 1, p 0 ˆp 0 )) =B((u û 1, p ˆp 0 ), (u 1 û 1, p 0 ˆp 0 )) + K T (σ M u K (u), σ π K(u 1 û 1 )) K =B s ((u û 1, p ˆp 0 ), (u 1 û 1, ˆp 0 p 0 )) + K T (l K ( p ˆp 0 ), σ π K (u 1 û 1 )) K + K T σ (M u K(u), σ π K (u 1 û 1 )) K = K T (l K ( p ˆp 0 ), σ π K ((u 1 û 1 )) K C 1 { F E τ F p ˆp 0 2 0,F }1 2 σα π(u1 û 1 ) 0,Ω C 2 1 α 2 F E τ F p ˆp 0 2 0,F + 1 4 σ π(u 1 û 1 ) 2 0,Ω (74) (u û, p ˆp 0 ) 2 + 1 4 (π(u 1 û 1 ), p 0 ˆp 0 ) 2 0,Ω

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 23 were we used α 1 and C C1 2 1 is te positive constant from Lemma 2. Te result follows 2 applying te triangular inequality. Remark. We end tis section by remarking tat te same analysis from Teorem 8 may be carried out to prove error estimates on u û div,ω and p ˆp 0 0,Ω as well. 4. Numerical experiments Now, we are interested in te numerical validation of te PGEM in its symmetric version (16). Te metod (13) beaves similarly as sown in Section 3.3. Two numerical tests wit available analytical solutions are performed and te teoretical results validated. Te assumed vanising boundary condition to generate te metods is adopted by te first numerical test, a property wic is no longer sared by te second case. In all te computations te value for α F as been set to one. 4.1. An analytical problem. Te domain is Ω = (0, 1) (0, 1) and we set σ = 1 and te exact pressure is given by p(x, y) = cos(2πx) cos(2πy). Next, te exact velocity is determined from te Darcy law and te boundary condition is taken to be its normal component on te boundary, tus b = 0. Consequently, te divergence velocity field is set as g = 8 π 2 cos(2πx) cos(2πy). MESH Figure 2. Mes for te analytical problem. In Figures 4-6 we report te errors on velocity and pressure in a sequence of structured meses. One observes optimal convergence of all quantities as 0 in teir respective

24 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN natural norms, wic are in accordance wit te teoretical results. Moreover, in Figure 4 we also plot te error π(u) π(u 1 ) 0,Ω wic is smaller tan u u 1 0,Ω. Here we denote [ 1/2. p p 0 := F τ E F p p 0 0,F] 2 First, we adopt te structured mes described in Figure 2 wic contains 4096 triangular elements ( = 3.125 10 2 ). We depict in Figure 3 te isolines (free of oscillations) of te pressure and u 1 obtained from (16). Furtermore, in Table 4.1 we study te local mass conservation feature of (16) weter we look at eiter u 1 or u 1 + u e. We define te quantities (75) K M e := max K T ( (u1 + u e ) g ) dx K K and M 1 := max K T ( u1 g ) dx, K and we find a loss of mass, as expected, wen just te linear part of te solution is used. Neverteless, we recover te local mass conservation property updating te linear velocity field by te multi-scale velocity u e. Similar results were obtained in [7] using te nonsymmetric PGEM (32). 0.25 0.125 6.25 10 2 3.125 10 2 1.5625 10 2 7.8125 10 3 M e 2.3 10 14 2.3 10 13 10 12 10 11 1.2 10 10 10 9 M 1 0.81 0.35 0.09 0.03 6.4 10 3 1.6 10 3 Table 4.1: Relative local mass conservation errors wit te symmetric metod (16).

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 25 PRESSURE VELOCITY Figure 3. Isolines of te pressure (left) and u 1 (rigt) using te symmetric metod (16). error 10 1 0.1 0.01 u u 1 0,Ω π(u) π(u 1 ) 0,Ω 2 0.001 1e-04 0.001 0.01 0.1 1 Figure 4. Convergence istory of u u 1 0,Ω and π(u) π(u 1 ) 0,Ω for te symmetric metod (16).

26 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN 100 (u u 1 ) 0,Ω 100 (u u 1, p p 0 ) 10 10 (u π(u 1 ), p p 0 ) error 1 error 0.1 1 0.01 0.001 0.01 0.1 1 0.1 0.001 0.01 0.1 1 Figure 5. Convergence istory of (u u 1 ) 0,Ω and (u u 1, p p 0 ), and (u π(u 1 ), p p 0 ) for te symmetric metod (16). 10 p p 0 0,Ω 10 p p 0 1 1 error 0.1 error 0.01 0.1 0.001 0.001 0.01 0.1 1 0.01 0.001 0.01 0.1 1 Figure 6. Convergence istory of p p 0 0,Ω and p p 0 for te symmetric metod (16).

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 27 4.2. A second analytical problem. Te problem is set up as in te first test, differing by replacing te previous exact pressure by p(x, y) = x3 y 3 y3 x. Once again, te velocity 3 is computed from te Darcy law and te boundary condition b is taken to be its no vanising normal component on te boundary. On te oter and, clearly te velocity field is divergence free (g = 0). We validate te symmetric metod (16) using a sequence of structured meses. Optimality is reaced watever te norm is considered as sown in Figures 8-10. Similar quadratic convergence is observed for u u 1 0,Ω and π(u) π(u 1 ) 0,Ω. Table 4.2 igligts tat te local mass conservation property is recovered as soon as u 1 is updated by u 1 + u e. 0.25 0.125 6.25 10 2 3.125 10 2 1.5625 10 2 7.8125 10 3 M e 3.1 10 15 1.5 10 14 1.9 10 13 7.1 10 13 5.6 10 12 7.6 10 11 M 1 0.14 0.08 0.04 0.02 0.01 5.3 10 3 Table 4.2: Relative local mass conservation errors wit te symmetric metod (16). Next, te structured mes of Figure 2 is once more adopted, and te solution is oscillationfree as it can be seen troug te isolines of p 0 and u 1 in Figure 7. PRESSURE VELOCITY Figure 7. Isolines of te pressure (left) and u 1 (rigt) using te symmetric metod (16).

28 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN 0.1 0.01 u u 1 0,Ω π(u) π(u 1 ) 0,Ω 2 error 0.001 1e-04 1e-05 0.001 0.01 0.1 1 Figure 8. Convergence istory of u u 1 0,Ω and π(u) π(u 1 ) 0,Ω for te symmetric metod (16). 1 (u u 1 ) 0,Ω 1 (u u 1, p p 0 ) (u π(u 1 ), p p 0 ) 0.1 0.1 error error 0.01 0.01 0.001 0.001 0.01 0.1 1 0.001 0.001 0.01 0.1 1 Figure 9. Convergence istory of (u u 1 ) 0,Ω and (u u 1, p p 0 ), and (u π(u 1 ), p p 0 ) for te symmetric metod (16).

A SYMMETRIC NODAL CONSERVATIVE FEM FOR THE DARCY EQUATION 29 0.1 p p 0 0,Ω 0.1 p p 0 0.01 error error 0.01 0.001 1e-04 0.001 0.01 0.1 1 0.001 0.001 0.01 0.1 1 Figure 10. Convergence istory of p p 0 0,Ω and p p 0 for te symmetric metod (16). 5. Conclusion New enriced finite element metods make te simplest pair of nodal based interpolation spaces stable for te Darcy model. It as been proved tat suc metods lead to optimal error estimates in natural norms in addition to be locally mass conservative. Suc fundamental property is recovered inside a general framework wic relies on updating te linear part of velocity wit a particular Raviart-Tomas function. Suc strategy can prevent oter jump-based stabilized finite element metods from local loss of mass wile keeping tem stable and accurate. Alternatives to deal wit iger order interpolations sould include additional control on te jumps of gradient of te pressure, in te form of a new enricment function leading to a term like (l( p 1 n ), q 1 ) Ω, a feature tat may be incorporated into te current Petrov-Galerkin framework. Appendix A. Te error for general g If we do not suppose tat g is a piecewise constant function, instead we admit tat g H 1 (Ω), ten a tird enricment function (u g e, pg e ) appears as te solution of (76) σ u g e + p g e = 0 in K, u g e = g Π K (g) in K, u g e n = 0 in K.

30 G.R. BARRENECHEA, L.P. FRANCA, AND F. VALENTIN Now, denoting (u g e, pg e ) = G K(g Π K (g)), ten, te original metod (13) becomes: Find (ũ 1, p 0 ) V Q suc tat for all (v 1, q 0 ) V Q it olds (77) B((ũ 1, p 0 ), (v 1, q 0 )) = F s (π(v 1 ), q 0 ) K T (G K (g Π K (g)), σ π K (v 1 )) K, were we recall tat Π K (g) = 1 K g. K Remark. Regarding te symmetric metod its rigt and side must be enanced wit te same contribution as well, and ten te following results also apply to suc version. Remark. Next, we see tat tere exists C > 0 suc tat (78) u g e 0,K C K g Π K (g) 0,K K T. Indeed, we remark tat multiplying (77) by u g e and integrating it by parts we obtain tat u g e 2 0,K = 1 σ ( pg e, u g e) K = 1 σ (pg e, u g e) K = 1 σ (pg e, g Π K (g)) K and te result follows. 1 σ pg e 0,K g Π K (g) 0,K C K σ pg e 1,K g Π K (g) 0,K = C K u g e 0,K g Π K (g) 0,K, Now, applying te Strang Lemma (cf. [17]) and (78) we arrive at K T (π(ũ 1 u 1 ), p 0 p 0 ) sup (G K (g Π K (g)), σ π K (v 1 )) K (π(v 1 ), q 0 ) (v 1,q 0 ) V Q 0 {0} C [ K T G K (g Π K (g)) 2 0,K ]1 2 C g Π K (g) 0,Ω C 2 g 1,Ω, and ten, as claimed, we see tat te error is not affected by te fact tat we projected g onto te piecewise constant space. Terefore, following te same strategy in 3 we can

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