A posteriori error estimation by postprocessor independent of method of flowfield calculation

1. I N T R O D U C T I O N The present paper is a imed at the quant i ta t ive es t imat ion of approx imat ion error in the verification of computa t iona l codes [1-3]. The error in prac t ica l ly useful functionals due to the approx imat ion error may be calcula ted using adjoint equat ions and different forms of the residual [4-13]. For example, the residual may be calcula ted for a differential approx imat ion using a finite-difference scheme [11-13]. However, the differential approx imat ion by finite-difference scheme may tu rn to be very cumbersome. Very often, we have to deal wi th a commercial code. In this case, the numerical me thod is provided wi thout detai ls and code descr ipt ions are not available thus excluding an explici t formulat ion for the differential approximat ion . On other hand, the local approx imat ion error may be es t imated via the act ion of the main problem differential operator on an in te rpola t ion of the numerical solut ion [7-9]. In general, this provides the oppor tun i ty to develop a postprocessor able to analyze the flowfield calculated by some unknown numerical method. Such postprocessor is capable to de termine the a posteriori error of prac t ica l ly useful functionals (drag, lift, etc.) using informat ion on the grid and flowfield parameters . Herein, we consider another (if compared with [7]) way for de termining the local approximat ion error t ha t enables us to avoid the in terpola t ion stage. This should simplify t r ea tmen t s and avoid addi t ional error of in terpolat ion. 0898-1221/06/$ see front matter (~) 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2005.10.003 Typeset by A~tS-TEX 398 A.K. ALEKSEEV AND I. M. NAVON Let us consider a formal scheme of the algorithm. We are interested in properties of numerical solution of the following problem, N f = w, in f~ C R n, f ( o a ) = f s ( x ) 6 L2(0f~). (1) Here, N is a nonlinear differential operator (Hk(f2) x L2(0f~) ~ L2(12)). The numerical solution is provided by a finite-difference equation, g h h = w, fh : g ~ l w . (2) As a result, we obtain a grid function f~. We assume the existence of a smooth enough function f 6 Hk+n(~) that coincides at the nodes with the grid function. Finite differences in Nhfh may be expanded using Taylor series in the Lagrange form. This provides us with a differential approximation of finite-difference scheme [14], N / + 8h(f) = w. (3) Here, 6h(f)is the approximation error containing leading terms of Taylor expansion. Consider (1) as an exact equation and (3) as perturbed one. Exact and perturbed solutions are connected by the relation, f ( t , x) = ](t, x) + A f ( t , x). (4) The operator N is assumed to be Frechet differentiable, the corresponding derivative being denoted as Nf. Then the expansion N ( ] + A f ) = N ( f ) + Ns(Y)AY is valid with the tolerance of O(rlAfll2). The differential approximation (3) may be recast in a form, N i l + N / A f + 5h(f) = w. (5) By subtracting the exact equation (1) from (5), we obtain an equation for the perturbation, N / A f = ~ h ( f ) = q , f ~ C R n, Af(Of~)=O. (6) Consider Frechet-differentiable goal functional 6 : Hk(f~) ~ R 1. We are interested in the variation of this functional due to the truncation error of the finite-difference scheme. Its differential Ac = e / ( f ) A f = limt--,0(e(f + t A r ) e ( f ) ) / t is a linear continuous functional that may be formulated as a Riesz-representation using an inner product in L2(f~), Ag = (S I ,A f )L 2 = (g, A f ) L 2 • (7) It may be recast as Ag= (Af.,)L2 ~(N/lq.g)L = (q.N-'* ' f g jL =(q, eg)L,, (8) where ~ ~N'/l*g is a formal solution of adjoint problem,