Neumann-Type A Posteriori Error Estimation For Steady Convection-Diffusion Equation

Transcription

1 INTRNATIONAL JOURNAL OF TCHNOLOGY NHANCMNTS AND MRGING NGINRING RSARCH, VOL 3, ISSU 8 8 Neumann-Type A Posteriori rror stimation For Steady Convection-Diffusion quation G. Temesgen Meuria, J. Anand Rao Department of Mathematics, University College of Science, Osmania University, Hyderabad, 57, Telengana, India; Department of Mathematics, University College of Science, Osmania University, Hyderabad, 57, Telengana, India mail: temesgen_meuria@yahoo.com ABSTRACT: We consider a steady linear convection diffusion equation in D, present the standard Galerin (GK) approximation and the Streamline- Diffusion Finite lement Method (SDFM) and give an analysis of a posteriori error estimator based on solving a local Neumann problem. The estimator gives global upper and local lower bounds on the error measured in the H semi-norm. Our numerical results from GK and SD approximations show that the global effectivity indices deteriorate in rates O(Pe K ) and O( Pe K ) as Pe K, respectively i.e., the estimator is over-estimated the error locally within a boundary layer which is not resolved by uniform grid refinement. Keywords: Convection-diffusion equation, GK Method, SDFM, a Posteriori error estimator, ffectivity Indices INTRODUCTION It is well nown that the standard Galerin discretization of convection-diffusion equation yields inaccurate, oscillatory solutions near boundary layers in convection dominated flows and, if the diffusion parameter is decreased without proportional reduction of the discretization mesh size, then these inaccuracies propagate into regions where the solution is smooth [9]. The streamline-upwind Petrov- Galerin (SUPG) method [4], [5] or streamline-diffusion finite element method (SDFM) [8] is designed to overcome these problems by introducing a small amount of artificial diffusion in the direction of streamlines. The numerical solution obtained from the SDFM has the desirable property that the accuracy in regions where the exact solution is smooth will not be degraded as a result of discontinuities and layers in the exact solution [6], []. However, the numerical solution obtained from the SDFM can be oscillatory in regions where there are layers. To obtain an accurate finite element solution on a given mesh, usually a so-called quasi-uniform or isotropic mesh is desirable [6]. One commom technique to increase the accuracy of the finite element solution is mesh refinement, the so-called h-method. For a refinement procedure to succeed, reliable and efficient a posterori error estimators are needed. For the reliability and efficiency of a posteriori error estimators, a standard measure is the so-called effectivity index, defined as X η = estimated error true error. An estimator is called asymptotically exact if its effectivity index converges to when the mesh size approaches. If the effectivity index is much smaller than, the estimator is under- estimating the error. On the other hand, if the effectivity index is much greater than, the estimator is over-estimating the error. If the estimator does not under-estimate or over-estimate the error globally, then the estimator is reliable, meaning the error on the global domain can be properly controlled by the estimator. If the estimator does not under-estimate or overestimate the error locally, then the error estimator is efficient, meaning the estimator is able to pinpoint exactly where the error is large and where the error is small. For two-dimensional problems, several estimators have been shown to be asymptotically exact when used on uniform meshes provided the solution of the problem is smooth enough [4], [7], [8]. stimators based on solving a local Neumann problem, so-called Neumann-type estimators, were first given by [5]. These estimators have been studied by many researchers such as [], [], [], [4], [7], [9]. Our aim is to analyze the reliability of the error estimator proposed by Kay and Silvester [4] which is an extension of the wor of Verfurth [8]. In their wor, they modify the well-nown Ban and Weiser estimator [5], and using the idea of Ainsworth & Oden [], they solve a local (element) Poisson problem, over a suitably chosen (higher order) approximation space with data from interior residuals and flux jumps along element edges. An outline of this paper is as follows. In Section, we present the variational formulation and its finite element discretization of the steady convection-diffusion equation. In Section 3, we discuss the theoretical analysis of the Numann-type a posreriori estimator. In Section 4, we present the numerical results and the effectivity indices for a posteriori estimator and finally, we draw a conclusion. LINAR CONVCTION-DIFFUSION QUATION We consider the following steady, linear convectiondiffusion equation u + b. u = f in Ω, u = g on Γ D, u = on Γ n N, (.a) (.b) (.c) where Ω C IR is a bounded polygonal domain with Lipschitz boundary Ω = Γ D Γ N and Γ D Γ N =. We are interested in the convection dominated case and assume that (A.), (A.). b =, (A.3) b,ω =, (A.4) b. n on Γ N. The L norm and the H semi-norm, also called energy norm are defined as u H (Ω) = u L (Ω) = u dω Ω d i= and u H (Ω) = u L (Ω) = D i u Ω u H (Ω), respectively. We shall denote the above norm and semi-norm by the following convention.,ω = u H (Ω) and.,ω = u,ω = u H (Ω) if no subscript index

2 INTRNATIONAL JOURNAL OF TCHNOLOGY NHANCMNTS AND MRGING NGINRING RSARCH, VOL 3, ISSU 8 9 is given then we assume an ordinary L norm,.,ω, and if no subscript index is given then we shall assume it is the whole of Ω. To define wea form of (.), we need two classes of functions: the trial functions H and the test solutions H : H = u H Ω u = g on Γ D H = u H Ω u = on Ω and the standard variational formulation of problem (.) is given by: find u H such that B u, v = F v v H (.) where B u, v = u, v + b. u, v and F v = f, v Let T = K be a decomposition of Ω into triangles or quadrilaterals. We need to mae the following geometrical assumptions on the family of triangulations T. Admissibility: whenever K and K belongs to T, K K is either empty, or reduced to a common vertex, or to a common edge. = the diameter of K = the longest side of K T 3. ρ = the supremum of the diameter of the balls inscribed in K T 4. Shape regularity: the ratio of to ρ is uniformly bounded i.e., β ρ K T which means for any > and for any K T there exists a constant β > such that β β, where β denotes the smallest angle in any K T. We define the finite element spaces V = v H Ω v K P K K T for triangular elements and V = v H Ω v K Q K K T for rectangular elements, where P K is the space of polynomials of degree not greater than, and Q K is the space of polynomials of complete degree on K. In the case of convection dominated problem, the standard Galerin approximation of (.) may produce unphysical behavior, oscillation, if the mesh is too coarse in critical regions. To circumvent these difficulties, stability of the discretization has to be increased by introducing artificial diffusion along streamlines. The Streamline Diffusion Finite lement Method (SDFM) [8] or [4], [5] stabilizes a convection dominated problem by adding weighted residuals to the standard Galerin finite element method for hyperbolic equations which combines good stability with high order accuracy, convergence results are available (see [] ). The SDFM yields the following discrete problem obtained: Find u V such that B u, v = F v v V, v = on Γ D (.3) where B SD u, v = u, v + b. u, v + δ K (b. u, b. v ) K and F SD v = f, v + δ K (f, b. v ) K. In (.3), a constant δ K must be chosen for every element K. Let the mesh Peclet number be defined by, Pe = b,k where.,k denotes the norm in L K. From the analysis of the SDFM, the following choice of δ K are optimal; lman et al. [5]: δ = for Pe Pe >, for Pe, (.4) where is a measure of the element length in the direction of the wind. For other parameter choice, see [6], [7], [5], []. 3 NUMANN TYP A POSTRIORI RROR STIMATION In this topic, we introduce the analysis of error estimator proposed by Kay and Silvester [4] which is an extension of the wor of Verfurth [8]. In their wor, they modify the wellnown Ban and Weiser estimator [5] and using the idea of Ainsworth & Oden [], they solve a local (element) Poisson problem over a suitably chosen (higher order) approximation space with data from interior residuals and flux jumps along element edges. We now introduce some definitions and notations that will be needed for the error estimates. We denote by (K) the set of edges of element K T, by = K T K the set of all element edges and the subsets relating to internal, Dirichlet and Neumann edges respectively as,ω = : C Ω,,D = : C Ω D and,n = : C Ω N so that =,Ω,D,N. We denote N K the set of vertices of K T and by N = K T N K the set of all element vertices (that do not lie on the Dirichlet boundary Γ D ). Let N be the set of vertices of, and for K T, and x N we define the local patches of elements as ω K = K K K, ω = K K, ω K = N K K N K, ω = N K N K. For the lowest order P or Q approximations over a triangular or rectangular element subdivision, u K =, so that the interior residual of element K is given by R K = f b. u K (3.) And the internal residual is approximated by R K = P K (R K ) (3.) where P K is the L (K) projection onto P (K). For any edge of an element K T, we define the flux jump as u if n,ω R = u h. n,k if (3.3),N if,d where u is a constant function on the inter-element edge n u and measures the jump of u across, that is, for n n K S, K, S T and defining n,k and n,s to be the outward normals with respect to the edge from element K and S respectively, we have u = v n K v S. n,k = v S v S. n,s (3.4) The approximation space is denoted by Q K = Q K B K (3.5) consisting of edge and interior bubble functions respectively: Q K = span ψ : K R ψ, K,Ω,N (3.6) where each member of the space is a quadratic (or biquadratic) edge bubble function ψ that is nonzero on edge of element K, but non zero valued on all other

3 INTRNATIONAL JOURNAL OF TCHNOLOGY NHANCMNTS AND MRGING NGINRING RSARCH, VOL 3, ISSU 8 3 edges of K. B K is the space spanned by interior cubic (or biquadratic) bubbles φ K i.e., B K = φ K K R φ K, φ K = on K (3.7) where each function is associated with an element K, and is zero on all edges of K, nonzero on the interior of K, and φ K = at the centeroid of K. The upshot is that the local problems are always well posed and that for each triangular (or rectangular) element a 4x4 (5x5) system of equations must be solved to compute e K. For an element K T, the local error estimate is the energy norm of e K given by η K = e K K (3.8) where e K = u u K Q K satisfies e K, v K = R K, v K R K, v v Q K (3.9) In the following, we mae frequent use of the short-hand notation f S to denote L - norm of a function L (S). The Kay and Silvester s upper estimation is the following. Theorem. If the variational problem (.) solved with a grid of bilinear rectangular elements, and if the rectangle aspect ratio condition is satisfied with β Ω, then, the estimator η K computed via (3.9) safisfies the upper bound property e h C β Ω η K K T + K b K R K R K K (3.) where C is independent of and and K is the length of the longest edge of element K. Proof. Using the assumptions (A.) - (A.4) we have B SD (e, e ) e h Ω (3.) Given e = u u H there exist a quasi interpolant e V H (see [6]) such that e e K C β ωk e h ωk K T (3.) = e e C β ωk e h ωk (3.3) Using the Galerin orthogonality property, the bilinear form B SD (e, e ) is written as B SD e, e = B SD e, e e + B SD e, e = B SD e, e e δ K (f b. u, b. e ) K = B SD u, e e B SD u, e e δ K (f b. u, b. e ) K = (f, e e ) B SD u, e e δ K (f b. u, b. e ) K (3.4) Hence, using (3.) in (3.4) and integrating by parts elementwise with e = e on Γ D gives e h Ω [ f b. u, e e K δ K (f b. u, b. e ) K + e K,Ω u K,N, e n e ] R K, e e K δ K (R K, b. e ) K + u n, e R,Ω,N, e e (3.5) From coercivity estimate, interpolation estimates (3.) & (3.3) and the Cauchy-Schwarz inequality, it can be shown e h Ω C β Ω K R K K + R K R K K + K R,Ω,N e h ωk C β Ω e h Ω K R K K + K R K R K K +,Ω,N R (3.6) Now, it remains to bound R K K and R in terms of η K. Recall that R K is constant and for K T, consider a function ω K = R K φ K where φ K B K. So ω K an interior bubble function with maximum value R K K at the center of the element. As φ K, one can write R K K R K, R K K = C R K, ω K K (3.7) Using (3.9) and the Cauchy-Schwarz inequality, imply that R K K C ( e K, ω K ) K C e K K ω K K (3.8) where C = C. Now using the inverse estimate (see [, Lemma.6]) ω K K C 3 ω K K (3.9) and by definition ω K K R K K, we get R K K C 4 e K K (3.) where C 4 = C C 3. Now we deal with the edge residual term R. For K, consider ω = R ψ where ψ Q K Q K. This is the bubble function that is only nonzero on edge. Note that we can write each norms of ω as norms over the edges as follows (see [3, Lemma 3.]) ω K ω ω K ω and since ω < R we can have ω K R (3.) ω K R (3.) Note R = R ds C 5 R ( R ψ )ds C 5 R, ω (3.3) Using (3.9), (3.) and (3.), we have R C 5 e K K ω K + R K K ω C 5 β ω R e K K + K K ω ω K K R K K (3.4) By plugging (3.) and (3.4) into (3.6), the global upper bound (3.) holds. Remar. If f and b are both piecewise constant functions then the consistency error term R K R K K is identically zero. Otherwise, if f and b are smooth, this term represents a high-order perturbation. In any case the estimator η K is reliable in the sense that the upper bound (3.) is independent of and. stablishing that the estimated error η K gives a lower bound on the local error is not possible. The difficulty is generic for any local error estimator: the local error is overestimated within exponential boundary layers wherever such layers are not solved by the mesh. Thus, in contrast to the nice lower bound that holds when solving Poisson s equation, see [, Proposition.8], the tightest lower bound that can be established for the convection-diffusion equation is the following. Theorem. If the variational problem (.) with b = is solved via either the Galerin formulation or the SD formulation (.3), using a grid of bilinear rectangular elements, and if the rectangle aspect ratio condition is satisfied, then the estimator η K computed via (3.9) is a local lower bound for e = u u in the sense that

4 INTRNATIONAL JOURNAL OF TCHNOLOGY NHANCMNTS AND MRGING NGINRING RSARCH, VOL 3, ISSU 8 3 η K C β ωk ( e ωk + K ω K b. e K + K ω K R K R K K ) (3.5) where C is independent of, and ω K represents the patch of five elements that have at least one boundary edge from the set K. Proof. To begin note that as e Q K we can put v = e in (3.9) to get e K = R K, e K R K, e R K K e K + R K e K (3.6) Note that with Poincare Fridrichs inequality and a scaling argument we can write e K C β K K e K, (3.7) e C β K e K (3.8) Substituting these into (3.6) gives K e K C β K K R K K + R (3.9) Now we only need to bound K R K K and R in terms of e. For K T, consider ω K = R K ψ K as above. Then C R K K R K, ω K K = R K R K, ω K K + B SD u u, ω K = R K R K, ω K K + e, ω K K + b. e, ω K K R K R K K ω K K + e K ω K K + b. e K ω K K (3.3) Hence, using ω K K R K K and (3.) we get K R K K C K R K R K K + e K + K b. e K (3.3) Similarly, for K,Ω and by using ω, it can be shown C R R, ω = K ω u h, ω K, by the definition of R and the Green formula = K ω e, ω K + u, ω K = K ω [ R K R K + R K b. e, ω K e, ω K ] Thus, R C R K R K R K K + K ω e K + K b. e K + K R K (3.3) K Therefore, by (3.3), we have R C K ω K R K R K K + e K + K b. e K (3.33) By plugging (3.3) and (3.33) into (3.9) the local lower bound (3.5) holds. Remar. The same estimate is obtained if local convection-diffusion problems are solved in place of (3.9), see [9]. The restriction that b = can be removed; it is only included to simplify the proof. From the bound (3.) and (3.5), the structure of the optimality gap term K ω K b. e K, leads to the expectation that η K will be an overestimate for e K in any element K where first, the element Peclet number,, is significantly bigger than unity, and second, the derivative of the error in the streamline direction, namely b. e K, is commensurate with the derivative of the error in the cross wind direction, b. e K. Such a deterioration in performance is realized in the case of problem given in section (4) next, if is decreased while eeping the grid fixed. 4 NUMRICAL RSULTS: FFCTIVITY INDICS In this section, we present an example to compare the solution qualities from the GK method and the SDFM and we compute global effectivity indices. In order to see how the effectivity indices change in terms of the diffusive parameter, mesh size, and element Peclet number, the problem is solved over uniform meshes with mesh size = and for =,,, and with Pe K = =, 4, 6, 64 and 56. Problem. Consider the function e y u(x, y) = x (4.) e satisfies equation (.) with b = (,) and f =. Dirichlet conditions on the boundary Ω are determined by (4.) and satisfy u x, = x, u x, =, u, y, u, y where the latter two approximations hold except near y =. Clearly, exponential layer near y = is expected, the layer is determined by e y and has width proportional to, see chaus (979) and Roos et al. ((996), section III..3). We solve (.3) using uniform NxN grids of square elements i.e., Q approximation over a domain Ω =, x [,]. To illustrate the qualities of SDFM and GK method for Pe K >, we consider the case N = 8, 6, 3 & 64 for = 4 and plot isolines of the computed solution and the estimated error in Fig., Fig. 3, Fig 4, Fig. 5 & Fig. 6. Clearly, Fig. shows GK solution suffer with serious oscillation on whole domain but Fig. 3, Fig. 4, Fig. 5 & Fig. 6 show SD solutions maintain good solution quality with small oscillation in the layer region as h increases. However, the solution still contains oscillations. Fig.. (a) Contour plot and (b) three-dimensional surface plot (bottom) of an accurate F solution, for =.

5 INTRNATIONAL JOURNAL OF TCHNOLOGY NHANCMNTS AND MRGING NGINRING RSARCH, VOL 3, ISSU 8 3 Fig.. (a) GK solution and (b) estimated error: 6 x 6 uniform grid Fig. 4. (a) SD solution and (b) estimated error: 6 x 6 uniform grid Fig. 3. (a) SD solution and (b) estimated error: 8 x 8 uniform grid Fig. 5. (a) SD solution and (b) estimated error: 3 x 3 uniform grid.

6 INTRNATIONAL JOURNAL OF TCHNOLOGY NHANCMNTS AND MRGING NGINRING RSARCH, VOL 3, ISSU 8 33 are smaller than the GK errors, we observe that the slow reduction in the global errors as the grid is successively refined. From Table and Table 3, we can see that the effectivity indices deteriorates in a rate O(Pe K ) as. Suggesting that the bound (3.5) is tight in this instance. Further, the numerical data in Table and Table 4 also show that the global effectivity indices deteriorates in a rate of O( Pe K ) as mentioned in Kay and Silvester () and lman et al. (5). Hence, we note that the local error is overestimated within exponential boundary layer. In considering the results in Table, Table, Table 3 & Table 4, we note that the improvement in performance of SD over Galergin might be anticipated from (3.5) in particular the better approximation of the streamline derivative suggests that the gap term will be smaller for SD in the limit as Pe K. TABL XACT RRORS, STIMATD RRORS (η ) AND FFCTIVITY INDICS (X η ) FOR LMNT PCLT NUMBR (Pe K ) SOLVD WITH SD APPROXIMATION USING 8 X 8 UNIFORM GRID. η X η Pe K Fig. 6. (a) SD solution and (b) estimated error: 64 x 64 uniform grid. Then, an approximation to the exact Galerin error = e K and the corresponding SD error = e K on each element can be computed as using Gaussian quadrature, with the MATLAB quadrature function dblquad.m. As an example, estimated global errors x x.454 x TABL XACT RRORS ( ), STIMATD RRORS (η) AND FFCTIVITY INDICS (X η ) FOR LMNT PCLT NUMBR (Pe K ) SOLVD WITH GALRKIN APPROXIMATION USING 8 X 8 UNIFORM GRID η X η Pe K x x x.65 x x x 3.5 x x.75 x x 56 η = η K and corresponding effectivity indices X η = η and X η = η, for GK and SD methods are presented in the tables below. From a comparison given in Table, Table, Table 3 & Table 4, while the SD errors TABL 3 XACT RRORS ( ), STIMATD RRORS (η) AND FFCTIVITY INDICS (X η ) FOR LMNT PCLT NUMBR (Pe K ) SOLVD WITH GALRKIN APPROXIMATION USING 6 X 6 UNIFORM GRID η X η Pe K x x.649 x.34 x x.3 x x x 3.67 x x 56

7 INTRNATIONAL JOURNAL OF TCHNOLOGY NHANCMNTS AND MRGING NGINRING RSARCH, VOL 3, ISSU 8 34 TABL 4 XACT RRORS, STIMATD RRORS (η ) AND FFCTIVITY INDICS (X η ) FOR LMNT PCLT NUMBR (Pe K ) SOLVD WITH SD APPROXIMATION USING 6 X 6 UNIFORM GRID. η X η Pe K x x.8 x x x.5 x 56 5 CONCLUSION From our theoretical and numerical results, we observe that the estimated error continuously reduces as the grid is successively refined. Therefore, the streamline-diffusion stabilization using N = 3 & 64 leads to the reliable error estimator for all O 3. In the context of the incompressible fluid flow models, exponential boundary layers only arise when downstream boundary conditions are inappropriately specified. In particular, a hard Dirichlet boundary condition on an outflow boundary should never be imposed; a zero Neumann condition is invariably more appropriate [3]. To increase accuracy of the solution in the region containing layer, adaptive mesh refinement and mesh movement based on a posteriori error estimation for the convection-diffusion equation are topics of our future wor. RFRNCS [] M. Ainsworth and J. Oden, A procedure for a posteriori error estimation for h p finite element methods. Comp. Meth. ng., 73-96, 99. [] M. Ainsworth and I. Babuˇsa, Reliable and roubst a posteriori error estimation for singular perturbed reaction-diffusion problems. SIAM J. Numer. Anal., 36:33 353, 999. I. Babuˇsa and A. K. Aziz, On the angle condition in the finite element method. SIAM. Numer. Anul., 3:4-6, 976. [3] Babuˇsa, R. Dur an, and R. Rodr ıguez, Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements. SIAM J. Numer. Anal., 9: , 99. [4] R.. Ban and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44, pp. 83 3, 985. [5] P. Cl ement, Approximation by finite element functions using local regularization, R.A.I.R.O. Anal. Num er.,, pp , 975. [6] R. Dur an, M. A. Muschietti, and R. Rodr ıguez, On the asymptotic exactness of error estimators for linear triangular finite elements. Numer. Math., 59:7 7, 99. [7] R. Dur an and R. Rodr ıguez, On the asymptotic exactness of Ban-Weiser s estimator. Numer. Math., 6:97 33, 99. [8] W. chaus, Asymptotic Analysis of Singular Perturbations, North-Holland, Amsterdam [9] H. lman, D. Silvester, and A. Wathen, Finite lements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Numerical Methods and Scientific Computation. Oxford University Press, Oxford. 5. [] K. risson and C. Johnson, Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comp., 6:67 88, 993. [] K. risson, D. step, P. Hansbo, and C. Johnson, Computational Differential quations, Cambridge University Press, New Yor [] P. M. Gresho & R. L. Sani, Flow and the Finite lement Method. Jhon Wiley, Chichester [3] T. J. R. Hughes and A.N. Broos, A multidimensional upwind scheme with no crosswind diffusion, in Hughes T. J. R, ed., Finite element methods for convection dominated flows, AMD vol. 34, ASM, New Yor, pp, 9-35, 979. [4] T.J.R. Hughes, M. Mallet, and A. Mizuami, A new finite element formulation for computational fluid dynamics: II. Beyond SUPG. Còmput. Meths. Appi. Mech. ngrg., vol. 54, , 986. [5] V. John and P. Knobloch, On spurious oscillations at layers diminishing (SOLD) methods for convection diffusion equations: Part I A review, Comput. Methods Appl. Mech. ngrg. 96: 97 5, 7. [6] V. John and P. Knobloch, On spurious oscillations at layers diminishing (SOLD) methods for convection diffusion equations: Part II-analysis for P and Q finite elements. Comput. Methods Appl. Mech. ngrg., 97:997-4, 8.

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