Derivation and Analysis of Piecewise Constant Conservative Approximation for Anisotropic Diffusion Problems

Transcription

1 Derivation Analysis of Piecewise Constant Conservative Approximation for Anisotropic Diffusion Problems A. Agouzal, Naïma Debit o cite this version: A. Agouzal, Naïma Debit. Derivation Analysis of Piecewise Constant Conservative Approximation for Anisotropic Diffusion Problems <hal v> HAL Id: hal Submitted on 4 Apr 009 HAL is a multi-disciplinary open access archive for the deposit dissemination of scientific research documents, whether they are published or not. he documents may come from teaching research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Derivation Analysis of Piecewise Constant Conservative Approximation for Anisotropic Diffusion Problems A. Agouzal a, N. Debit b a Université de Lyon ; Université Lyon ; CNRS, UMR508, Institut Camille Jordan, 43 blvd du novembre 98, F-696 Villeurbanne-Cedex, France. b Université de Lyon ; Université Lyon ; CNRS, UMR508, Bât ISIL, 5 Bd Latarjet, F-696 Villeurbanne-Cedex, France. Abstract A variational approach to derive a piecewise constant conservative approximation of anisotropic diffusion equations is presented. A priori error estimates are derived assuming usual mesh regularity constraints a posteriori error indicator is proposed analyzed for the model problem. Key words: 99 MSC: 65N30 Finite elements, conservative approximation, a posteriori error estimate Introduction Various phenomena in scientific fields such as geoscience, oil reservoir simulation, hydrogeology, biology..., are generally modeled by anisotropic diffusion equations. he usual discretization schemes of this equations are finite difference, finite element or finite volume methods. he last are piecewise constant conservative approximation are actually very popular in oil engineering, the reason probably being that complex coupled physical phenomena may be discretized on the same grids (see for instance [9] references therein). But the well known five point on rectangles four point schemes on triangles are not easily adapted to heterogeneous anisotropic diffusion operators, so an enlarged stencil scheme which hles anisotropy on meshes satisfying an orthogonality property was proposed analyzed in [,6,7]. Let us recall that a huge literature exists in engineering study setting. However, even though these schemes perform well in the number of cases, their convergence addresses: agouzal@univ-lyon.fr (A. Agouzal), Naima.Debit@univ-lyon.fr (N. Debit). Preprint submitted to Elsevier Science 4 April 009

3 analysis often seems out of reach, unless some additional geometrical conditions are imposed. Moreover, actually in several applications the discretization meshes are imposed by engineering computing considerations, therefore we have to deal with unstructured meshes. A motivation for this work was to construct such a piecewise constant approximation for anisotropic diffusion problems which could satisfy the two assumptions : First, the resulting formulation is well-defined on general unstructured meshes, assuming usual finite element mesh regularity constraints. Secondly, the given scheme leads to stard algebraic system, for which we can use the existing efficient numerical solvers. his last point is of major importance in the coupling of physical models, from the implementation point of view good adaptivity properties. An outline of the paper is as follows. Among all the developments we briefly introduce the functional framework some usual notations. In second section, we introduce the numerical scheme for the anisotropic diffusion problem main approximation analysis results are given. A focus is made on the treatment of an additional reaction term. Finally, in section three we propose analyze an a posteriori error indicator for the diffusion model problem. Functional Framework some notations Let ω be a bounded polygonal domain of IR. We denote by H s (ω) the usual Sobolev space W s, (ω) (see e.g []), endowed with the norm. s,ω H s 0(ω) is the closure of D(ω) in H s (ω). For the semi-norm, we use the notation. s,ω. We introduce the set H(div, ω) of vector fields p (L (ω)) d div p L (ω). Equipped with the norm. H(div,ω) =. 0,ω + div. 0,ω, H(div, ω) is a Hilbert space. For any integer k, P k (ω) is the set of polynomials of degree less than or equal to k. Construction analysis of the numerical scheme Let denote a bounded polygonal domain of IR. We consider the anisotropic diffusion problem : div(k u) = f over, (.) u = 0 on Γ, with symmetric definite positive tensor K, assumed piecewise constant for simplicity, f L (). Let ( h ) be a family of triangulations of, by triangles, regular in the usual finite element sense [5]. For all in h, there exist reals d,e such that the bilinear form a : (H ( )) IR defined by p, q ( H ( ) ), a (p, q) = ( d,e e e ) ( ) p.n dγ q.n dγ. e

4 verifies p, q (P 0 ( )), a (p, q) = K p.q dx, where n is the unit normal outward to. Remark.. We could give explicit expressions of the parameters d,e. Let be a triangle with vertices a, b, c. he edge ab is denoted e θ e the opposite angle to e then, d,e = det(k) < K ac, bc > 4meas( ) Clearly if K = αid then d,e = α cotan(θ e). In the sequel, we denote by R x the unique element of (P 0 ( )) checking a (R x x, q) = 0, q (P 0 ( )). Let V h be the nonconforming finite element space defined by V h = {λ h L (); λ h P ( ), h ; e interior edge, e [λ h ] e dγ = 0 e, λ h dγ = 0} e where [λ h ] e denotes the jump of the function λ h across the edge e. he non-stard finite element approximation we propose for the model problem is the following : Find λ h V h such that µ h V h, K λ h. µ h = h h where h, f = f ( µ h + (x G R x). µ h ) dx, fdx x G is the barycenter of. meas( ) (.) First of all, by adapting stard arguments used in the analysis of nonconforming finite element approximation of elliptic problems [5], we can easily prove that the discrete problem has a unique solution; moreover, if the weak solution u of the model problem belongs to the Sobolev space H +s () with 0 < s, then: u λ h, h C h s u +s, + h f 0, + h dist (x G, R x) ), (.3) where h is the diameter of the triangle. 3

5 Let us set (x R x) h, p h = K. λ h f on. he key point of the construction of the scheme is that p h is an admissible field in the following sense : Lemma. he vector field p h satisfies : p h H(div, ) h div p h = f on. Proof. It is obvious that div p h = f, h. Moreover if e is an interior edge of h, e =, with, h v e h V h the associated basis function, i.e., for any edge σ of h, σ v e hdγ = δ e σ, the Kronecker delta. Let us denote by [p h.n] e the jump of the flux across the edge e. We have thus [p h.n] e = [p h.n] e vhdσ e = which yields p h H(div, ). e = i= i= i (p h. v e h + v e h div p h ) dx i ((K. λ h f (x G R x) ). v e h f v e h)dx = 0, In order to define the numerical scheme, we need to introduce some notations. Let λ h be the solution of the discrete problem (.); For any h σ edge of, we set: where u = F σ, = σ p h.n dγ; u σ, = λ h dγ meas(σ) σ ( λ h + ) meas( ) λ h.(x G R x) dx + ρ,h 4 f, ρ,h = meas( ) a (x R x, x R x). Lemma. With the notations given above, one has the following scheme,, h, F σ, = meas( ) f σ F σ, + F σ, = 0, σ, (.4) d,σ F σ, + u = u σ,, σ, u σ, = 0, if σ. 4

6 Proof. First, we have Indeed, we have : σ F σ, = meas( ) f = fdx, h. div p h dx = p h.n dx = f dx = meas( ) f. e σ And for any interior edge σ, F σ, +F σ, = 0 is obvious since p h H(div, )). Let q h R 0 ( ) = (P 0 ( )) +xp 0 ( ) such that q h.n σ = q h.n e = 0, edge e σ. By one h, we have, B := a (p h, q h ) on the other h, if we set λ h q h.n = d,σ meas(σ) F σ, u σ,.meas(σ) A = a ( div p h (x R x), q h ), we get a (p h, q h ) λ h = meas( ) λ h q h.n = a (p h, q h ) = a (p h div p h (x R x), q h ) λ h dx, λ h.q h λ h.q h + A λ h div q h dx λ h div q h dx = a ( λ h, q h ) λ h.q h + A λ h div q h dx = a ( λ h, q h div q h (x R x)) λ h.q h + A λ h div q h dx ( = q h div q ) h x R x). λ h λ h.q h + A λ h div q h dx ( = λ h + ) λ h(x G R x) div q h + A. However, since div q h = q h.n dγ = meas(σ), we obtain But we have also, B = meas(σ) ( λ h + ) λ h(x G R x) + A 5

7 A = a ( div p h (x R x), q h ) = a ( div p h (x R x), q h div q h (x R x)) div q h a (x R x, x R x).f 4( ) = div q h dx. ρ,h 4.f = meas(σ) ρ,h 4.f. where which implies thus scheme (.4). ρ,h = B = meas( ) a (x R x, x R x) ( λ h + λ h(x G R x) + ρ ),h 4.f Using once more Lemma., we can derive the following a priori error estimate, Lemma.3 If the weak solution u of model problem (.) belongs to H +s (), 0 < s, one has: u u 0, h C h s u +s, + h f 0, + h dist (x G, R x) ). (.5). A focus on the treatment of an additional reaction term Let us consider the problem of diffusion-reaction equations: div( u) + cu = f over, u = 0 on Γ. (.6) where f L (), c L () with c 0, the following associated discrete problem : Find λ h V h such that µ h V h, λ h. µ h dx h + α c h = α h where for all h, (λ h + (x G R x). λ h ) (µ h + (x G R x). µ h ) dx f ( µ h + (x G R x). µ h ) dx (.7) 6

8 4 c = cdx, f = fdx, α =, meas( ) meas( ) 4 + c ρ λ h = meas( ) λ hdx ( analogously for µ h ), x G is the barycenter of. Using the same arguments as before, we have in this case : Lemma.4 Let λ h be the solution of the discrete problem. We introduce where hen we have p h = λ h (f 4 + c ρ + c (λ h + λ h (x G R x)))(x R x). u = c ρ F σ, = σ p h.n dγ, u σ = ( meas( ) ρ,h = λ h dγ meas(σ) σ ( λ h + ) ) λ h.(x G R x) + ρ,h 4 f, meas( ) a (x R x, x R x). F σ, + c u = meas( )f σ F σ, + F σ, = 0, σ, d,σ F σ, + u = u σ,, σ, u σ, = 0, if σ. 3 A posteriori error estimator for the diffusion model problem Usually, error estimators for adaptive refinement require exact discrete solutions (see [0] references therein), but in practical cases the exact solution is not available so we are in the presence of solvers error. In this subsection, we introduce a posteriori error estimator for solutions obtained by black-box solver, in this case we are in the presence of many source of errors : approximation, error solvers, post processing error...etc. We indicate the a posteriori error estimator for the diffusion model equation: u = f over, u = 0 on Γ. (3.) he solution is assumed to be obtained by any existing solver. he given estimator is valid also for equilibrium mixed finite element approximations with or without numerical 7

9 integration. Let h be a regular triangulation of by triangles, E is the set of all edges E I the set of all interior edges. Given h, ( ) is the union of all elements of h sharing a vertex with, ω is the union of all elements of h sharing an edge with E the set of all edges of. We consider the finite dimensional space V h = {v h H (), h v h P ( )} (3.) E h = {p h (L ()), h p h R 0 ( ) = (P 0 ( )) + xp 0 ( )} (3.3) M h = {v h L (), h v h P 0 ( )} (3.4) Let p h E h u h M h, for all h we set ε, (p h ) = sup v h V h ( ) ε, (p h ) = ε 3, (p h, u h ) = ( ) sup φ h V h ( ) sup q h E h ( ) p h v h dx ( ) ( ) v h fdx v h, ( ), (3.5a) p h curl φ h dx φ h, ( ), (3.5b) (p h q h + u h div q h )dx ω, (3.5c) q h H(div,ω ) where η, (p h ) = h f f 0, + (h l [p h.t l ] l 0,l), l E (3.5d) η, (p h ) = h p h 0,, (3.5e) V h ( ) = {v h H0( ( )), ( ) v h P ( )}, (3.6) E h ( ) = {q h E h H(div, ), ω q h = 0}. (3.7) h h l are the diameters of l respectively. he outward normal to an edge l of some h is written as n l = (n,l, n,l ) we set t l = (n,l, n,l ) for associated tangential direction. we denote by [p h.t l ] l the jump of p h.t l across the edge l. In the sequel, C, C, C are positive generic constants independent of h (which may change from one line to other). Remark 3.. Let us notice that () Since div p h = fdx = f on, meas( ) 8

10 we have ε, (p h ) C ( ) h f f 0,, which is higher order perturbation of the error. () Let u h, ψ h V h ( ) q h E h ( ) be the unique solutions of the following respective problems (P ) Find u h V h ( ) such that v h V h ( ), u h v h dx = ( ) ( ) p h v h dx ( ) fv h dx, (P ) Find ψh V h ( ) such that φ h V h ( ), curl ψh curl φ h dx = ( ) ( ) p h curl φ h dx (P 3) Find qh E h ( ) such that s h E h ( ), (qh s h + div qh div s h )dx = (p h s h + u h div s h )dx. ω ω It is easy to see that u h, = ε, (p h ) ψ h, = ε, (p h ) q h H(div,ω ) = ε 3, (p h, u h ). We have the following error estimates, heorem 3. Let u H 0() be the weak solution of the model problem (3.), p = u, p h E h u h M h the solution of the given numerical scheme. hen there exists a positive constant C only depending on the minimum angle of h such that { ) u u h 0, + p p h 0, C (η, (p h ) + ε, (p h ) + ε, (p h )) h Moreover, we have ) } + (η, (p h ) + ε 3, (p h, u h )). h η i, (p h ) + ε i, (p h ) + ε 3, (p h, u h ) C ( p p h 0, ( ) + u u h 0, ( ) ) i= i= +C ω h f f 0, ) 9

11 Proof : First, Using Helmholtz-decomposition, we have e h = p p h = w + curl ζ, with w H0(), ζ H () w curl ζdx = 0. Let us remark that the orthogonality implies the following error decomposition : w, = e h 0, = w, + curl ζ 0,, (3.8) e h wdx curl ζ 0, = e h curl ζdx. (3.9) Now, let w I V h ζ I V h be continuous approximations of w ζ respectively such that : h, w w I 0, Cmeas( ) w, ( ), (3.0a) w I, C w,, (3.0b) l E, w w I 0,l Cmeas(l) w, (l), (3.0c) ( analogously for ζ ) where (l) is the union of the elements sharing l. Moreover we assume that the interpolation preserves boundary conditions, that is, w I V h H 0(). It is well known that such approximations exist (see [5], [8]). First, according to (3.9) we have by element-wise integration by parts, noting that w w I H 0(), w 0, = e h (w w I )dx + fw I dx p h w I dx = (f + div p h )(w I w)dx + fw I dx p h w I dx. From Cauchy s inequality from (3.0a) (3.0c), h ) (f + div p h )(w I w)dx h f + div p h 0, w,. h Using (3.0b), we obtain fw I dx hen we have ) ( p h w I dx C ε ), (p h ) w I, C ε, (p h ) w,. h h w 0, C h f + div p h 0, + ) ε, (p h ). (3.) h h Arguing as above, since p = u p h E h, by element-wise integration by parts we have e h curl(ζ ζ I )dx = (ζ ζ I )[p h.t l ] l dσ. l E l 0

12 By Cauchy s inequality (3.0c) we obtain Finally, since curl ζ 0, = ζ 0,, curl ζ 0, = ) e h curl(ζ ζ I )dx C h l [p h.t l ] l 0,l ζ 0,. (3.) l E e h curl ζ I dx = p h curl ζ I dx (3.3) e h curl ζdx = using (3.) (3.0b), we obtain e h curl(ζ ζ I )dx + e h curl ζ I dx, curl ζ 0, C h l [p h.t l ] l 0,l + ) ε, (p h ). (3.4) l E h Using the Helmholtz decomposition (3.8) together with the estimates (3.) (3.4), we get p p h 0, C h (η, (p h ) + ε, (p h ) + ε, (p h )) (3.5). Now, let P h u M h defined by h, P h u = udx on. meas( ) We have u P h u 0, Ch p 0, C(h p p h 0, + h p h 0, ). (3.6) On the other h, using the inf-sup condition we have u h P h u 0, C sup q h E h (u h P h u) div q h dx q h H(div,), since (u h P h u) div q h dx = (u h div q h dx + p h q h )dx + (p p h )q h dx we obtain easily that ) u h P h u 0, C ε 3, (p h, u h ) + p p h 0,. (3.7). h

13 By triangular inequality, using (3.5) (3.7), we have : { ) u u h 0, + p p h 0, C (η, (p h ) + ε, (p h ) + ε, (p h )) h ) } + (η, (p h ) + ε 3, (p h, u h )). h o indicate the efficiency of the a posteriori error estimator we follow Verfürth [0] show a local reverse up to higher order perturbations. For each h, we reset : ω = { h such that have a common edge}, h, f = fdx, meas( ) for all l E I, we denote by + the two elements of h sharing this edge. Let b be the stard bubble function on with max b =, as defined in [0]. hen norms. 0, b. 0, are equivalent on P 0 ( ), r 0, C b r (f f +div(p h p))dx C where r := f + div p h on. hen we have (b r ).(p h p)dx+c r 0, f f 0, r 0, C b r, p p h 0, + C r 0, f f 0,, Using the inverse estimate r b, Ch r 0,, we obtain h f + div p h 0, C p p h 0, + C h f f 0,. (3.8) Concerning the jump terms for l E E I, let b l be the stard bubble function on vanishing on \l such that max b l = (see [0]). hen again the norms. 0,l b l. 0,l are equivalent on P (l). Let l E I, then using the extension operator P : C 0 (l) C 0 ( + ) of [0], it follows that [p h.t l ] l 0,l C b l P ([p h.t l ] l )[p h.t l ] l dσ = C p h curl(b l P ([p h.t l ] l ))dx. l + Because of the inverse inequality the equality b l P ([p h.t l ] l ), + Ch l b l P ([p h.t l ] l ) 0, +, + p curl(b l P ([p h.t l ] l ))dx = 0, using Cauchy s inequality b l P ([p h.t l ] l ) 0, + Ch l [p h.t l ] l 0,l, we obtain h l [p h.t l ] l 0,l C p p h 0,+. (3.9)

14 Finally, it is easy to see that i =, ε i, (p h ) p p h 0, ( ). (3.0) Combining (3.8)-(3.0) we obtain η, (p h ) + ε, (p h ) + ε, (p h ) C p p h 0, + C h f f 0,. (3.) ω ( ) Introducing as above the bubble function b, we get p h 0, C p h b p h dx = C b p h (p h p)dx + C u.(b p h ). (3.) Since div(b p h )dx = 0 u h M h, by element-wise integration by parts we have u.(b p h )dx = (u h u) div(b p h )dx. (3.3) Using the inverse inequality div(b p h ) 0, Ch p h 0,, (3.)-(3.3), we obtain η, (p h ) C( p p h 0, + u u h 0, ). (3.4) Finally, it is easy to check that ε 3, (p h, u h ) C( p p h 0, ( ) + u u h 0, ( ) ). (3.5) Using the estimates (3.4)-(3.5), we get h, η, (p h ) + ε 3, (p h, u h ) C( p p h 0, ( ) + u u h 0, ( ) ). (3.6) Finally, using (3.) (3.6) concludes the proof. References [] R.A. Adams. Sobolev Spaces. Academic Press, New York (975). [] F. Brezzi, M. Fortin L.D. Marini. Error analysis of piecewise constant pressure approximations of Darcy s law. Comput. Methods Appl. Mech. Engrg. 95, no. 3-6, pp (006). [3] C. Carstensen R.H.W. Hoppe. Error reduction convergence for an adaptive mixed finite element method. Math. Comp. Vol 75, No 55, pp (006) [4] C. Carstensen R.H.W. Hoppe. Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. Vol 03, No, pp (006) [5] P. G. Ciarlet. Hbook of numerical analysis. Vol. II. Eds. Ciarlet, P. G. Lions, J.-L.. Finite element methods. Part. North-Holl, Amsterdam (99). 3

15 [6] R. Eymard,. Gallouët R. Herbin. Finite volume methods. Hbook of numerical analysis, Vol. VII, pp 73 00, North-Holl,Amsterdam 000. [7] R. Eymard,. Gallouët R. Herbin. A finite volume scheme for anisotropic diffusion problems. C.R. Acad. Sci. Paris, Ser I 339 (004). [8] Scott L. Ridgway Zhang Shangyou. Finite element interpolation of non smooth functions satisfying boundary conditions. Math. Comp., Vol 54, No 90, pp (990) [9] William J. Layton, Friedhelm Schieweck Ivan Yotov. Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. Vol 40, No 6, pp (003). [0] R. Verfürth. A review of a posteriori error estimation adaptive mesh-refinement techniques.wiley eubner, 3 (996) 4