summary:We derive a residual based a posteriori error estimate for the Stokes-Brinkman problem on a two-dimensional polygonal domain. We use Taylor-Hood triangular elements. The link to the possible information on the regularity of the problem is discussed

Burda, Pavel

Hasal, Martin

Institute of Mathematics AS CR, v. v. i.

PANM 17Pavel Burda; Martin HasalAn a posteriori error estimate for the Stokes-Brinkman problem in a polygonal domainIn: Jan Chleboun and Petr Přikryl and Karel Segeth and Jakub Šístek and Tomáš Vejchodský (eds.): Programs andAlgorithms of Numerical Mathematics, Proceedings of Seminar. Dolní Maxov, June 8-13, 2014. Institute ofMathematics AS CR, Prague, 2015. pp. 32--40.Persistent URL: http://dml.cz/dmlcz/702660Terms of use:© Institute of Mathematics AS CR, 2015Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This paper has been digitized, optimized for electronic delivery and stamped with digitalsignature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czPrograms and Algorithms of Numerical Matematics 17J. Chleboun, P. Přikryl, K. Segeth, J. Š́ıstek, T. Vejchodský (Eds.)Institute of Mathematics AS CR, Prague 2015AN A POSTERIORI ERROR ESTIMATE FOR THE STOKES-BRINKMAN PROBLEM IN A POLYGONAL DOMAINPavel Burda1, Martin Hasal21 Czech University of Technology, Faculty of Mechanical Engineering,Department of MathematicsKarlovo nám. 13, CZ-121 35 Praha 2, Czech Republicpavel.burda@fs.cvut.cz2 VŠB TU Ostrava17. listopadu 15, 708 03 Ostrava-Poruba, Czech Republicmartin.hasal@vsb.czAbstractWe derive a residual based a posteriori error estimate for the Stokes-Brinkmanproblem on a two-dimensional polygonal domain. We use Taylor-Hood triangularelements. The link to the possible information on the regularity of the problem isdiscussed.1. IntroductionIn the paper we try to contribute to the technique of a posteriori error esti-mates for the finite element solution of linearized flow problems. In this respect wenote that important results have already been obtained: concerning linear ellipticequations let us mention I. Babuška, W. C. Rheinboldt [2], I. Babuška, R. Durán,R. Rodŕıguez [3], concerning the Stokes problem e.g. M. Ainsworth, J. T. Oden [1],R. E. Bank, D. Welfert [5], C. Carstensen, S. Jansche [7], C. Johnson, R. Rannacher,M. Boman [12], R. Verfürth [15].The goal of this paper is to link the problem of a posteriori error estimates asmuch as possible to the information on the regularity of the solution.Let us illustrate it first on the Dirichlet problem for the Poisson equation−∆u = f in Ω,u = 0 on ∂Ω, (1)where Ω is a polygonal domain in R2. Let uh be the finite element solution of (1),with linear triangular elements. Let us denotee = u− uh,32the approximation error, andR(uh) = f +∆uh,the residual. Following the technique of K. Eriksson et al. [10], we first express theerror by means of product of residual and solution of the dual problem, then use theGalerkin orthogonality and get the estimate of the error, in the L2-norm:‖e‖20 ≤∑K∈Th{‖R(uh)‖0,K‖ϕ− πhϕ‖0,K +∑l∈∂K∥∥∥∥12[[∂uh∂n]]l∥∥∥∥0,l‖ϕ− πhϕ‖0,l}, (2)where ϕ is the solution of the dual problem−∆ϕ = e in Ω,ϕ = 0 on ∂Ω, (3)πhϕ means the interpolant of ϕ. The sum in (2) is taken over all triangles in thetriangulation Th, the symbol[[∂uh∂n]]lmeans the jump of the normal derivative ∂uh∂nover the edge l of the triangle K.Let us now distinguish 3 cases:A) General polygonal domain Ω:Let hK be the largest side of the triangle K. The interpolation property togetherwith the (low) regularity of the dual problem (3) yield‖ϕ− πhϕ‖0,K ≤ CIhK‖ϕ‖1 ≤ CICRhK‖e‖0.Combining this with (2), we come to the a posteriori error estimate‖e‖0 ≤ CICR∑K∈ThhK{‖R(uh)‖0,K + h− 12K∑l∈∂K∥∥∥∥12[[∂uh∂n]]l∥∥∥∥0,l}. (4)B) Convex polygon Ω:Now the regularity of the dual problem (3) is higher, cf. R.B. Kellogg,J. E. Osborn [13], and together with the interpolation property it gives‖ϕ− πhϕ‖0,K ≤ CIh2K‖ϕ‖2 ≤ CICRh2K‖e‖0.Combining this with (2), we come to the more precise a posteriori estimate‖e‖0 ≤ CICR∑K∈Thh2K{‖R(uh)‖0,K + h− 12K∑l∈∂K∥∥∥∥12[[∂uh∂n]]l∥∥∥∥0,l}. (5)C) Nonconvex polygon Ω with known singularity:It is well-known that the solution near the nonconvex corner, in the local sphericalcoordinates, has the formu(r, ϑ) = rγw(ϑ),33where r is the distance from the corner, γ ∈ (0, 1). For instance, the case of theL-shaped domain with the interior angle ω = 32π gives γ = 23, cf. also [6]. Now theinterpolation together with the above regularity gives‖ϕ− πhϕ‖0,K ≤ CIh1+γ−εK ‖ϕ‖H1+γ−ε ≤ CICRh1+γ−εK ‖e‖0, ∀ε > 0,which, combined with (2), finally leads to the a posteriori estimate‖e‖0 ≤ CICR∑K∈Thh1+γ−εK{‖R(uh)‖0,K + h− 12K∑l∈∂K∥∥∥∥12[[∂uh∂n]]l∥∥∥∥0,l}, (6)valid ∀ε > 0. Of course, in (6) the parameter γ applies only in the nearest neighbor-hood of the corner.Comparing the estimates (4), (5), (6) we see that the a posteriori error estimatedepends significantly on the regularity of the problem. Having this in mind, we tryto derive the a posteriori error estimate for the Stokes-Brinkman problem.2. The Stokes-Brinkman modelLet Ω be a bounded Lipschitzian domain, Ω ⊂ R2, which consists of two parts:porous part Ωp and fluid part Ωf , Ω̄ = Ω̄p ∪ Ω̄f . The Stokes-Brinkman equationrepresenting a mathematical model of a single phase flow in a porous/free flow mediahas the following formνK−1v +∇p− ν∗∆v = f in Ω, (7)∇ · v = 0 in Ω, (8)v = w on ∂ΩD,∂v∂n− np = s on ∂ΩN , (9)where v is the vector of velocity, P is the pressure, f is the vector of external force,n is the outward-pointing normal to the boundary, ν∗ is the effective viscosity andν - the physical viscosity - is a uniform constant in the entire domain Ω. K isa symmetric permeability tensor, which in Ωp is equal to the Darcy permeability ofthe porous media. Note that with the choice ν∗ = 0 in the vugular region Ωp, theequation (7) reduces to the problem of Darcy’s law. On the other hand by choosingkij → ∞ (or very large) in fluid domain Ωf , the equation (7) reduces to the problemof Stokes flow (here ν∗ is taken equal to the physical fluid viscosity ν). Thus, theStokes or Darcy’s equations can be obtained by suitable choices of the parameters ν∗and K by defining them in vugular and rock matrix regions, respectively.In the porous region (K < ∞) it is known [14], that for moderately small per-meabilities and pore fractions, the diffusive term ν∗∆v, where ν∗ takes values closeto the fluid viscosity ν, intoroduces only a small perturbation of the velocity andpressure fields in comparison with a pure Darcy law with ν∗ = 0. In [14] it is shownthat Stokes-Brinkman equation with the choice ν∗ = ν in the porous region is veryclose to the solution of coupled Stokes and Darcy’s equations.34The advantage of Stokes-Brinkman model is usage of uniform equations for porousand free flow domains. Boundary conditions between these two domains are repre-sented by K. This approach makes it possible to model heterogeneous material.Moreover, by a numerical point of view, it is easier to solve a monolithic system suchas Stokes-Brinkman, in contrast to a coupled Darcy-Stokes system which requiresan additional iterative scheme. Also, near the interface, Stokes-Brinkman equationsallow us to avoid the typical grid refinement issues necessary for solving the interfacebetween Darcy and Stokes region. On the other hand usage of Taylor-Hood elementsfor the whole domain requires big load of memory.3. Weak formulation of Stokes-Brinkman equationsIn what follows we denote G = K−1 and assume G is symmetric.For the weak formulation we denoteH1E := {u ∈ H1(Ω)2|u = w na ∂ΩD}, (10)H1E0 := {v ∈ H1(Ω)2|v = 0 na ∂ΩD}. (11)Now the weak form of the Stokes-Brinkman problem reads:Find v ∈ H1E0 and p ∈ L20(Ω) such thatν∗∫Ω∇v : ∇v∗ + ν∫ΩvTGv∗ −∫Ωp∇ · v∗ =∫Ωf · v∗ ∀v∗ ∈ H1E0 , (12)∫Ωq∇ · v = 0 ∀q ∈ L20(Ω). (13)Here L20(Ω) is the space of L2 functions having mean value zero.On the space V =(H10(Ω)2 × L20(Ω))we define the bilinear formA({v, p}, {v∗, p∗})= ν∗∫Ω∇v : ∇v∗ + ν∫ΩvTGv∗ −∫Ωp∇ · v∗ −∫Ωp∗∇ · v (14)where (., .)0 means the scalar product in L2.In what follows we assume w = 0, i. e. only zero Dirichlet condition on the wholeboundary ∂Ω. Problem (12), (13) can be written as follows: find {v, p} ∈ V , suchthatA({v, p}, {v∗, p∗})= (f , v∗)0, ∀{v∗, p∗} ∈ V. (15)4. Finite element approximationWe suppose Ω to be a polygon, for simplicity. Let Th be regular [11] triangulationsof Ω. Let Xh, Mh be the finite element spaces of Taylor-Hood elements (cf. e.g.F. Brezzi, M. Fortin [4]), i.e.Xh = {v ∈ H10 (Ω)2, v/T ∈ P2(T )2, T ∈ Th},Mh = {p ∈ L20(Ω), p/T ∈ P1(T ), T ∈ Th}.35These satisfy the Babuška-Brezzi condition [4]. The finite element approximation ofthe Stokes-Brinkman problem consists in finding {vh, ph} ∈ Xh ×Mh such thatA({vh, ph}, {v∗h, p∗h})= (f , v∗h)0 , ∀{v∗h, p∗h} ∈ Xh ×Mh. (16)5. A posteriori error estimateWe follow the idea of K. Eriksson et al. [10] who proved the a posteriori errorestimate for the Poisson equation. We define the residual components by the relationsR1{vh, ph} = f + ν∗∆vh − νGvh −∇ph, R2{vh, ph} = div vh. (17)Next we study the properties of the errorsev = v − vh , ep = p− ph ,where {v, p} is the exact solution of (15), {vh, ph} is the approximate solution definedin (16). The V norm of {ev, ep} is‖{ev, ep}‖2V = (ev, ev)1 + (ep, ep)0 =∫Ω(ev · ev +∇ev : ∇ev) +∫Ωepep.By the Poincaré-Friedrichs inequality, cf. [9], as ev ∈ H10 (Ω)2(ev, ev)1 ≤ CP∫Ω∇ev : ∇ev (18)5.1. Dual Stokes-Brinkman problemTo study the above norms we introduce the dual Brinkman-Stokes problem by−ν∗∆ϕv + νGϕv +∇ϕp = −∆ev in Ω, here ∆ev ∈ H−1(Ω)−div ϕv = ep in Ω, (19)ϕv = 0 on ∂Ω,which in a weak form is: find ϕv ∈ H1(Ω)2 and ϕp ∈ L20(Ω) such that(ν∗∇ϕv,∇v∗)0 + ν((Gϕv), v∗)− (ϕp,∇v∗)0 = (∇ev,∇v∗)0, ∀v∗ ∈ H10(Ω)2,(−div ϕv, p∗)0 = (ep, p∗)0, ∀p∗ ∈ L20(Ω), (20)or, using the notation (14)A({ϕv, ϕp}, {v∗, p∗}) = (∇ev,∇v∗)0 + (ep, p∗)0 , ∀{v∗, p∗} ∈ V . (21)By (18) and (20) where we put v∗ = ev, p∗ = ep, we get1CP(ev, ev)1 ≤ (∇ev,∇ev)0 = ν∗(∇ϕv,∇ev)0 + ν((Gϕv),ev)− (ϕp∇, ev)0= ν∗(∇ϕv,∇v)0 + ν((Gϕv)v)− (ϕp∇, v)0 − ν∗(∇ϕv,∇vh)0− ν((Gϕv)vh) + (ϕp∇, vh)0, (22)(ep, ep)0 = (ep,−div ϕv)0 = −(p∇,ϕv)0 + (ph∇,ϕv)0. (23)365.2. Estimation of the error by means of the residual and solution of thedual problemCombining (22), (23), and (19) we get (as CP ≥ 1)1CP{(ev, ev)1 + (ep, ep)0}≤ ν∗(∇v,∇ϕv)0 + ν((Gvϕv))− (p,∇ϕv)0 − (∇v, ϕp)0+∑K∈Th{−ν∗(∇ϕv,∇vh)0,K − ν((Gvhϕv)) + (ph,∇ϕv)0,K + (ϕp,∇vh)0,K}= (f ,ϕv)0 +∑K∈Th{(ν∗∆vh,ϕv)0,K −∫∂Kν∗∂vh∂nϕvds}− ν((Gvhϕv)) (24)−∑K∈Th{(∇ph,ϕv)0,K +∫∂Kphϕv · nds + (div vh, ϕp)0,K}=∑K∈Th(f + ν∗∆vh − ν((Gvhϕv))−∇ph,ϕv)0,K +∑K∈Th(div vh, ϕp)0,K−∑K∈Th∫∂Kν∗∂vh∂nϕvds+∑K∈Th∫∂Kphϕv · ndsIn view of (16) we also have∑K∈Th(f + ν∗∆vh − νGvh −∇ph, v∗h)0,K + (div vh, p∗h)0= (f , v∗h)0 +∑K∈Th{(−ν∗∇vh,∇v∗h)0,K − ν(Gvh, v∗h) +∫∂Kν∗∂vh∂nv∗hds}+ (∇ph, v∗h)0 −∑K∈Th∫∂Kphv∗h · nds+ (div vh, p∗h)0 (25)= 0 +∑K∈Th∫∂Kν∂vh∂nv∗hds−∑K∈Th∫∂Kphv∗h · nds, ∀{v∗h, p∗h} ∈ Xh ×Mh.This implies, taking v∗h = πhϕv, p∗h = πhϕp , the Clement interpolants, (cf. e.g. [8],p. 146) that∑K∈Th(f + ν∗∆vh − νGvh −∇ph, πhϕv) + (div vh, πhϕp)0−∑K∈Th∫∂Kν∗∂vh∂nπhϕvds−∑K∈Th∫∂Kphπhϕv · nds = 0 (26)Now subtracting zero in (26) from (24) we get371CP{(ev, ev)1 + (ep, ep)0}≤∑K∈Th(f + ν∗∆vh − νGvh −∇ph,ϕv − πhϕv)0,K + (div vh, ϕp − πhϕp)0−∑K∈Th∫∂Kν∂vh∂n(ϕv − πhϕv)ds+∑K∈Th∫∂Kph(ϕv − πhϕv) · nds (27)=∑K∈Th(f + ν∗∆vh − νGvh −∇ph,ϕv − πhϕv)0,K + (div vh, ϕp − πhϕp)0−∑K∈Th∑l∈∂K∫l(12[[ν∂vh∂n− phn]]l)(ϕv − πhϕv)ds,where we denoted[[ν∂vh∂n− phn]]l=(ν∂vh∂n− phn)/l+−(ν∂vh∂n− phn)/l−the jump along the common side l of two adjacent triangles. Then, using in turnthe Schwarz inequality, the interpolation properties of Xh, Mh (cf. e.g. [4]), and theestimate of the solution of the dual problem (19) (cf. [4]), we get the inequalities‖ev‖21 + ‖ep‖20≤ CP∑K∈Th{‖R1{vh, ph}‖0,K‖ϕv − πhϕv‖0,K + ‖R2{vh, ph}‖0,K‖ϕp − πhϕp‖0,K}+ CP∑K∈Th∑l∈∂K∥∥∥∥12[[ν∂vh∂n− phn]]l∥∥∥∥0,l‖ϕv − πhϕv‖0,l (28)≤ CPCI∑K∈Th{hK ‖R1{vh, ph}‖0,K ‖ϕv‖1 + ‖R2{vh, ph}‖0,K ‖ϕp‖0}+ CPCI∑K∈Th(hK)12∑l∈∂K∥∥∥∥12[[ν∂vh∂n− phn]]l∥∥∥∥0,l‖ϕv‖1≤ CPCICR∑K∈Th{hK ‖R1{vh, ph}‖0,K + ‖R2{vh, ph}‖0,K+∑l∈∂K(hK)12∥∥∥∥12[[ν∂vh∂n− phn]]l∥∥∥∥0,l}·{‖∆ev‖−1 + ‖ep‖0}.Using then the relations‖∆ev‖−1 ≡ supv∗∈H10,v∗ 6=0|(∆ev, v∗)0|‖v∗‖1= supv∗∈H10,v∗ 6=0|(∇ev,∇v∗)0|‖v∗‖1≤ supv∗∈H10,v∗ 6=0‖∇ev‖0 ‖∇v∗‖0‖v∗‖1≤ ‖∇ev‖0 ≤ ‖ev‖138we get, by (28){‖ev‖1 + ‖ep‖0}2≤ 2{‖ev‖21 + ‖ep‖20}≤ 2CPCICR∑K∈Th{hK‖R1{vh, ph}‖0,K+ ‖R2{vh, ph}‖0,K + h12K∑l∈∂K∥∥∥∥12[[ν∂vh∂n− phn]]l∥∥∥∥0,l}·{‖ev‖1 + ‖ep‖0}. (29)Upon cancelling{‖ev‖1 + ‖ep‖0}in (29) we finally get the following theorem:Theorem 1. Let Ω be a polygon in R2. Let Th be a family of regular triangulationsof Ω. Let {vh, ph} be the Taylor-Hood approximation of the solution {v, p} of theStokes-Brinkman problem. Then the error {ev, ep} satisfies the following a posterioriestimate‖ev‖1 + ‖ep‖0 ≤ 2CPCICR∑K∈Th{hK‖R1{vh, ph}‖0,K + ‖R2{vh, ph}‖0,K+ h12K∑l∈∂K∥∥∥∥12[[ν∂vh∂n− phn]]l∥∥∥∥0,l}. (30)where CP , CI , CR are positive constants, residuals R1 and R2 are defined in (17) .ConclusionsThe estimate in Theorem 1 applies to more general class of elements. Of course,for Taylor-Hood elements with continuous pressure the jumps of ph along the commonsides disappear.Let us note that for convex domains stronger regularity applies to the Stokesproblem, cf. [13], and better a posteriori error estimate may be expected.For nonconvex domains with corners we do not obtain so strong regularity asin [13], cf. e.g. [6], but still the a posteriori error estimate should be better thanin (30), as it was for the Poisson equation in (2).AcknowledgmentsThis work was supported by the IT4Innovations Centre of Excellence project,reg. no. CZ.1.05/1.1.00/02.0070, and by the grant Kontakt II number LH11004.References[1] Ainsworth, M. and Oden, J. T.: Aposteriori error estimators for the Stokes andOseen problems. SIAM J. Numer. Anal. 34 (1997), 228–245.[2] Babuška, I. and Rheinboldt, W.C.: A posteriori error estimates for the finiteelement method. Int. J. Numer. Meth. Eng. 12 (1978), 1597–1615.39[3] Babuška, I., Durán, R., and Rodŕıguez, R.: Analysis of the efficiency of ana posteriori error estimator for linear triangular finite elements. SIAM J. Numer.Anal. 29 (1992), 947–964.[4] Brezzi, F. and Fortin, M., Mixed and hybrid finite element methods. Springer,Berlin, 1991.[5] Bank, R.E. and Welfert, D.: A posteriori error estimates for the Stokes equa-tions: A comparison. Comp. Meth. Appl. Mech. Eng. 82 (1990), 323–340.[6] Burda, P.: On the F.E.M. for the Navier-Stokes equations in domains withcorner singularities. In: M. Kř́ıžek, P. Neittaanmäki, and R. Stenberg (Eds.),Finite Element Methods, Supeconvergence, Post-Processing and A PosterioriEstimates, pp. 41–52. Marcel Dekker, New York, 1998.[7] Carstensen, C. and Jansche, S.: A posteriori error estimates for the finite el-ement discretizations of the Stokes problem, Berichtsreihe Math. Sem. Kiel.Techn. Report 97–9, 1997.[8] Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland,Amsterdam, 1980.[9] Elman, H.C., Silvester, D., and Wathen A. J.: Finite elements and fast iterativesolvers: with applications in incompressible fluid dynamics. Oxford UniversityPress, New York, 1994.[10] Eriksson, K., Estep, D., Hansbo, P., and Johnson, C.: Introduction to adaptivemethods for differential equations. Acta Numerica, CUP (1995), 105–158.[11] Girault, V. and Raviart, P.G.: Finite element method for Navier-Stokes equa-tions. Springer, Berlin, 1986.[12] Johnson, C., Rannacher, R., and Boman, M.: Numerics and hydrodynamic sta-bility: towards error control in computational fluid dynamics. SIAM J. Numer.Anal. 32 (1995), 1058–1079.[13] Kellogg, R.B. and Osborn, J. E.: A regularity result for the Stokes problem ina convex polygon. J. Funct. Anal. 21 (1976), 397–431.[14] Popov, P., Efendiev, L.B.Y., Erwing, R.E., and Quin, G.: Multi-physics andmulti-scale methods for modeling fluid flow through naturally-fractured vuggycarbonate reservoirs. SPE International. SPE 105378[15] Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley and Teubner, Chichester, 1996.40

English

An a posteriori error estimate for the Stokes-Brinkman problem in a polygonal domain

Czech digital mathematics library

http://dml.cz/bitstream/handle/10338.dmlcz/702660/PANM_17-2014-1_8.pdf

An a posteriori error estimate for the Stokes-Brinkman problem in a polygonal domain

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.

Czech digital mathematics library