Robust A Posteriori Error Estimates for Stabilized Finite Element s of Non-Stationary Convection-Diffusion Problems L. Tobiska and R. Verfürth Universität Magdeburg Ruhr-Universität Bochum www.ruhr-uni-bochum.de/num Goal Establish residual a posteriori error estimates for stabilized finite element discretizations of non-stationary convection-diffusion problems which yield upper and lower bounds for the energy norm of the error that are uniform with respect to all possible relative sizes of convection to diffusion using a common framework for various stabilization methods. Oberwolfach / January 9, 07 / 7 / 7 Outline Differential Equation References t u ε u + a u + βu = f in Ω 0, T ] u = 0 on Γ 0, T ] u = u 0 0 < ε, β 0, a R d, a in Ω Results hold for variable coefficients and mixed boundary conditions. Then ε is a lower bound for the smallest eigenvalue of the diffusion and β is a lower bound for b div a with b denoting the reaction. 3/ 7 4/ 7
Norms Find u L 0, T ; H0 Ω with tu L 0, T ; H Ω such that u = u 0 in L and for all t 0, T and all v H0 Ω t u, v + ε u v + a uv + βuv = fv Ω Ω =Bu,v = l, v Energy norm v = ε v + β v ϕ, v Dual norm ϕ = sup v H0 Ω\0 v Error norm u Xa,b = ess. sup u, t + t a,b + b a b a u, t dt t u + a u, t dt 5/ 7 6/ 7 Meshes and Spaces I = t n, t n ] : n N I partition of [0, T ]. = t n t n. T n, 0 n N I, affine equivalent, admissible, shape regular partitions of Ω. Transition condition: There is a common refinement T n of T n and such that h K ch K for all K T n and all K T n with K K. X n H 0 Ω finite element space corresponding to T n. Discrete Problem Find u n T n X n, 0 n N I, such that u 0 T 0 = π 0 u 0 and, for n =,..., N I and all v X n with U nθ = θ u n T n + θ u n u n u n, v + BU nθ, v + S n U nθ, v = l, v The stabilization term S n is supposed to be linear in its second argument and affine in its first argument, it may depend on T n and on f. Solution u I is continuous piece-wise affine and equals u n T n at t n. 7/ 7 8/ 7
Stabilizations I Stabilizations II Streamline diffusion method S n u, v = u n u n ε u K T n ϑ K with ϑ K a ch K K + a u + βu f Continuous interior penalty method S n u, v = ϑ E J E a uj E a v E E n with ϑ E ch E E a v Local projection scheme S n u, v = κ M a u κ M a v M M n ϑ M with ϑ M a ch M or S n u, v = M M n ϑ M M M κ M u κ M v with ϑ M c a h M and κ M the orthogonal projection onto the complement of a suitable space of discontinuous functions on a macro-partition M 9/ 7 0/ 7 Stabilizations III Basic Steps Subgrid scale approach S n u, v = a Π n u a Π n v K T n ϑ K K with ϑ K a ch K or S n u, v = K M n ϑ K K Π n u Π n v with ϑ K c a h K and Π n the orthogonal projection onto a suitable space of unresolvable scales Y n X n Error and residual are equivalent. The residual splits into a spatial and a temporal residual. The norm of the sum of these is equivalent to the sum of their norms. Derive a reliable, efficient and robust error indicator for the temporal residual. Derive a reliable, efficient and robust error indicator for the spatial residual. All stabilizations yield the same spatial error indicator. / 7 / 7
Equivalence of Error and Residual Proof of the Equivalence u I is continuous piece-wise affine and equals u n T n at t n. Residual: Ru I, v = l, v t u I, v Bu I, v Lower error-bound: Ru I L t n,t n;h u u I Xtn,t n Upper error-bound: u u I X0,tn 4 u 0 π 0 u 0 +6 Ru I L 0,t n;h Relation of residual and error: Ru I, v = t e, v + Be, v Lower error-bound: Definition of primal and dual norm plus Cauchy-Schwarz inequality. Upper error-bound: Parabolic energy estimate with v = e as test-function. 3/ 7 4/ 7 Decomposition of the Residual Recall U nθ = θ u n T n + θ u n Temporal residual: R τ u I, v = BU nθ u I, v Spatial residual: R h u I, v = l, v t u I, v BU nθ, v Splitting: Ru I = R τ u I + R h u I Estimate for L t n, t n ; H -norms: R τ u I + R h u I R τ u I + R h u I 3 R τ u I + R h u I Motivation of the Lower Bound Strengthened Cauchy-Schwarz inequality for v = c and w = b t b a : b a Hence: vw = cb a = 3 v a,b w a,b v + w a,b 3 v a,b + w a,b 5/ 7 6/ 7
Proof of the Lower Bound R h u I is piece-wise constant. R τ u I is piece-wise affine: R τ u I = ρ n, v = Bu n T n u n, v. Choose v, w H0 Ω such that θ t t n ρ n with v = R h u I, R h u I, v = R h u I, w = ρ n, ρ n, w = ρ n. Insert 3 t tn v + t n t w as test-function in representation of Ru I. Estimation of the Temporal Residual R τ u I = θ t t n ρ n with ρ n, v = Bu n T n u n, v. Upper bound: ρ n u n T n u n a + u n u n Follows from definition of ρ n and. Lower bound: u n 3 u n a + u n u n ρ n 7/ 7 8/ 7 Proof of the Lower Bound Set w n = u n T n u n and choose v H0 Ω with v = a w n and a w n, v = a w n Insert wn + v in the definition of ρn : ρ n, wn + v = ε wn, w n + βwn, w n + a wn, w n = wn =0 Estimation of the Convective Derivative I Assume that a c c ε. Friedrichs inequality implies a u n T n u n, v a u n T n u n c Ω v. Hence a u n T n u n u c c c n Ω u n T n and a u n T n u n is equivalent to u n T n u n. + ε wn, v + βwn, v + a wn, v wn a w n = a wn 9/ 7 0/ 7
Estimation of the Convective Derivative II Assume that a ε. Auxiliary problem with analytical and discrete solutions Φ and Φ : ε ϕ, ψ + β ϕ, ψ = a u n T n u n, ψ 3 Φ T n + Φ Φ a u n T n u n Φ + Φ Φ Φ Φ is equivalent to robust residual error indicator η for. Hence a u n T n u n is equivalent to Φ + η. Estimation of the Spatial Residual Spatial error indicator ηh n: ηh n = α K R K K + K T n α S = min ε h S, β ε E E αe R E E R K and R E are the usual element and interface residuals. Standard arguments for stationary problems yield: R h u I c η n h + I MR h u I, η n h c R h u I. I M R hu I measures the consistency error of the stabilization. c, c only depend on the polynomial degrees and on the shape parameters of the partitions T n. / 7 / 7 Proof of the Upper Bound L -representation: R h u I, v = rv + Ω Σ jv Quasi-interpolation error estimate: v I M v K cα K v ωk Trace inequality: v E E K v K + h K E v K K v K Proof of the Lower Bound Insert ψ K R K in L -representation with standard element cut-off functions ψ K. Insert ψ E,ϑ R E in L -representation with squeezed face cut-off functions ψ E,ϑ and ϑ = ε h E α E = min, ε h E β F E,K F E,K. Φ ϑ 3/ 7 4/ 7
Estimation of the consistency error I M R hu I I MR h u I = R h u I, I M v sup v H0 Ω v S n u I, I M v = sup v H0 Ω v Streamline diffusion and interior penalty methods: I MR h u I cη n h Local projection scheme and subgrid-scale approach: I M v ker κ M and I M v ker Π n hence I M R hu I = 0. A Posteriori Error Estimate Define the space-time error estimator by: η n = τ n ηn h + u n T n u n + Φ + η. spatial temporal Then N e X0,T c u I 0 π 0 u 0 + η n, η n c e Xtn,t n. n= c, c only depend on the polynomial degrees and the shape parameters of the partitions T n. 5/ 7 6/ 7 References References L. Tobiska, R. Verfürth Robust a posteriori error estimates for stabilized finite element methods IMA J. Numer. Anal. 35 05, no. 4, 65 67 H.-G. Roos, M. Stynes, and L. Tobiska Robust Numerical Methods for Singularly Perturbed Differential Equations Springer 008 R. Verfürth A Posteriori Error Estimation Techniques for Finite Element Methods Oxford University Press 03 7/ 7