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1. Introduction ADAPTIVE ERROR CONTROL FOR FINITE ELEMENT APPROXIMATIONS OF THE LIFT AND DRAG COEFFICIENTS IN VISCOUS FLOW MICHAEL GILES MATS G LARSON J MARTEN LEVENSTAM AND ENDRE SULI Abstract We derive estimates for the error in a variational approximation of the lift and drag co ecients of a b o dy immersed into a viscous ow governed by the Navier Stokes equations The variational approximation is based on computing a certain weighted average of a nite element approximation to the solution of the NavierStokes equations Our main result is an a posteriori estimate that puts a b ound on the error in the lift and drag co ecients in terms of the lo cal mesh size a lo cal residual quantity and a lo cal weight describing the lo cal stability prop erties of an asso ciated linear dual problem The weight may b e approximated by solving the dual problem numerically The error b ound is thus computable and can b e used for quantitative error estimation we apply it to design an adaptive nite element algorithm sp ecically for the approximation of the lift and drag co ecients Introduction Often the purp ose of numerical computations in mathematical mo delling of physical phenomena is to approximate a functional of the analytical solution to a dierential equa tion represented as an integral average of the solution rather than to obtain accurate p ointwise values of the solution itself in such instances solving the dierential equation considered is only an intermediate stage in the pro cess of computing the primary quan tities of concern For example in uid dynamics one may b e concerned with calculating the lift and drag co ecients of a b o dy immersed into a viscous incompressible uid whose ow is governed by the NavierStokes equations The lift and drag co ecients are dened as integrals over the b oundary of the b o dy of the stress tensor comp onents normal and tangential to the ow resp ectively Similarly in elasticity theory the quantities of prime interest such as the stress intensity factor or the moments of a shell or a plate are derived quantities The ob jective of this pap er is to obtain a posteriori error b ounds for nite element ap proximations of functionals that arise in uid dynamics sp ecically we shall b e concerned with the construction and the a posteriori error analysis of nite element approximations Mathematics Subject Classication N N N Key words and phrases NavierStokes equations lift and drag co ecients adaptive nite element metho d a posteriori error estimates mesh renement Submitted to SIAM J Num Anal M GILES M LARSON M LEVENSTAM AND E SULI to the lift and drag co ecients in a viscous incompressible ow and the implementa tion of these b ounds into an adaptive nite element algorithm with reliable and ecient quantitative control of the error It is frequently the case that the functional under consideration may b e expressed in various forms which are mutually equivalent at the continuous level but result in very dierent approximations under discretisation Thus it is imp ortant to select the appropriate representation of the functional b efore formulating its discretisation While this basic idea has b een widely exploited in structural mechanics see and and heat conduction see in p ostpro cessing nite element approximations it do es not seem to have p enetrated the eld of computational uid dynamics We approximate the solution u u p of the NavierStokes equations where u rep resents the velo city of the uid and p its pressure by means of a nite element metho d using piecewise p olynomials of degree k for u and degree k for p Let N u denote the b oundary integral representing the lift or drag co ecient dep ending on the choice of the function dened on the b oundary of the computational domain the precise denition of N u will b e given in Section Using the dierential equation we may express N u in a variational form involving test functions that are equal to on the b oundary of the h domain this form will b e the basis for constructing an accurate approximation N u to h N u Provided the b oundary is suciently smo oth and there are no variational crimes h the order of convergence of N u to N u is k while for the naive direct approximation h N u the order of convergence is typically only k h h Our main result is a weighted a posteriori estimate for the error N u N u of the h form X h R u h u j c jN u N h h T h where the sum is taken over the elements in a triangulation T of the computational h domain h is the diameter of given that is a real numb er with k R u is h an element residual quantity is a lo cal weight and c c is a p ositive constant The element residual quantity R u is a computable b ound on the actual residual obtained h by inserting the approximate solution into the dierential equation The derivation of h u in terms the estimate is based on a representation formula for the error N u N h of the computed solution u and the solution to an asso ciated linear dual equation with h homogeneous righthand side and nonhomogeneous b oundary data The weight describ es the lo cal size of derivatives of order of the solution to the dual problem The data for the dual problem is known and therefore the weight can b e computed by solving the dual problem numerically Therefore the righthand side of the estimate is computable and can b e applied to design an adaptive nite element algorithm for approximating N u h u The fact stated and to ensure ecient quantitative control of the error N u N h h u is k follows from the error representation ab ove that the order of convergence of N h formula together with an a priori estimate of the global error u u in the nite element h metho d Indeed for a linear elliptic mo del problem it can b e shown using results obtained in that the a posteriori estimate is sharp ERROR CONTROL FOR THE LIFT AND DRAG COEFFICIENTS In a weighted a posteriori estimate is given for the error in the lift and the drag however unlike our approach it is based on a direct approximation using N u For a h posteriori estimates of the error u u in nite element approximations of the solution to h the NavierStokes equations in various norms see and references therein a general intro duction to a posteriori error analysis is given in Finally we mention that an a priori error analysis of the approximation metho d employed here for a linear convectiondiusion equation is presented in The remainder of this pap er is organized as follows in order to illustrate the key ideas in Section we outline the theory for a linear elliptic mo del problem In Section we consider the extension of this analysis to the Stokes system which mo dels the ow of a viscous incompressible uid at low velo cities we derive an a posteriori estimate of the error in a nite element approximation to the lift and drag co ecients In Section we further extend our results to the NavierStokes equations and present numerical computations for the drag co ecient of a cylinder in a channel these illustrate the quality of the adaptive error control based on the a posteriori error b ound Finally in Section we summarise our results and draw some conclusions A model problem The mo del problem and the nite element metho d We b egin by intro ducing d some notation Let b e a b ounded domain in R d or with Lipschitzcontinuous d 2 b oundary For an op en set K in R let L K signify the space of realvalued square s integrable functions on K with norm k k H K will denote the Sob olev space of real K s s index s equipp ed with the norm k k and corresp onding seminorm j j with a H (K ) H (K ) s s slight abuse of the notation we shall frequently write kD uk instead of juj s K H (K ) 12 1 1 Given that is an element of H we let H H denote the space of all v in 1 1 H H which satisfy the Dirichlet b oundary condition v Finally we adopt j the following notational convention generic constants that are indep endent of the problem and the mesh size are denoted by c while constants that dep end on the problem but not on the mesh size are lab elled C with a subscript when necessary As a rst mo del problem we consider the b oundaryvalue problem r u f in u on 2 2 where f L L and u Aru with A a uniformly p ositive denite d d matrix with continuous realvalued entries dened on This problem has a unique 1 In order to dene the corresp onding nite element approximation weak solution u H 0 h 1 we intro duce a family of nitedimensional spaces V contained in H which consist of continuous piecewise p olynomials of degree k dened on a triangulation T of We denote h the diameter of a triangle T by h It will always b e assumed that the triangulation is h d shap eregular ie there exists a p ositive constant c such that vol c h where vol is 12 the ddimensional volume of Further for each function H such that v j h h h h for some v V we let V V b e the space of all w V with w j In particular h h V is the space of all v V which vanish on the b oundary 0 M GILES M LARSON M LEVENSTAM AND E SULI h The nite element approximation of reads nd u V such that h 0 h au v f v for all v V h 0 2 Here and b elow denotes the scalar pro duct in L and av w v rw T 1 T rv A rw for v w H where A is the transp ose of A Approximation of the b oundary ux In this section we shall construct a nite element approximation to the b oundary ux Z N u n u ds 1 where n denotes the unit outward normal vector to In order to do so
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