BAIL 2006 Book of Abstracts - Institut für Numerische und ...

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L.P. SHISHKINA, G.I. SHISHKIN: A Difference Scheme of Improved Accuracy for a Quasilinear Singularly Perturbed Elliptic Convection-Diffusion Equation in the Case of the Third-Kind Boundary Condition ✬ ✫ [4] L. Shishkina and G. Shishkin, “The discrete Richardson method for semilinear parabolic singularly perturbed convection-diffusion equations”, in: Proceedings of the 10th International Conference “Mathematical Modelling and Analysis” 2005 and 2nd International Conference “Computational Methods in Applied Mathematics”, R. Čiegis ed., Vilnius, “Technika”, 2005, pp. 259–264. 2 Speaker: SHISHKINA, L.P. 50 BAIL 2006 ✩ ✪
I.V. TSELISHCHEVA, G.I. SHISHKIN: Domain Decomposition Method for a Semilinear Singularly Perturbed Elliptic Convection-Diffusion Equation with Concentrated Sources ✬ ✫ Domain Decomposition Method for a Semilinear Singularly Perturbed Elliptic Convection-Diffusion Equation with Concentrated Sources ∗ Irina V. Tselishcheva and Grigory I. Shishkin Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences 16 S. Kovalevskaya Street, Ekaterinburg 620219, Russia tsi@imm.uran.ru, shishkin@imm.uran.ru For singularly perturbed boundary value problems in a composed domain (in particular, with concentrated sources) whose solution has several singularities such as boundary and interior layers, it is of keen interest to construct a parameter-uniform numerical method based on a domain decomposition technique so that each subdomain in the decomposition contains no more than a single singularity. Because of the thin layers, standard numerical methods applied to problems of this type yield unsatisfactorily large errors for small values of the singular perturbation parameter ε. The fact that the partial differential equation is nonlinear complicates the solution process. We develop monotone linearized schemes based on an overlapping Schwarz method for a semilinear singularly perturbed elliptic convection-diffusion equation on a compound strip in the presence of concentrated sources acting inside the domain. We first study a special (base) scheme comprising a standard finite difference operator on a piecewise-uniform fitted mesh and an overlapping domain decomposition scheme constructed on the basis of the former that converge εuniformly at the rates O � N −1 � � −1 and O N 1 lnN1 + N −1 2 1 ln N1 + N −1 2 + qt� , respectively. Here N1 +1 and N2 +1 are the number of mesh points in x1 and the minimal number of mesh points in x2 on a unit interval, q < 1 is the common ratio of a geometric progression, independent of ε, t is the iteration count. For these nonlinear schemes we construct monotone linearized schemes of the same ε-uniform accuracy, in which the nonlinear term is computed from the unknown function taken at the previous iteration. The linearized schemes are monotone, which admits to construct their upper and lower solutions. We apply the technique of upper and lower solutions to find aposteriori the (optimal) number of iterations T in the linearized scheme for which the accuracy of its solution is the same (up to a constant factor) as that for the base scheme, where T = O (ln(min[N1, N2])) (see also [1] for a reaction-diffusion problem). Thus, the number of required iterations is independent of ε. With respect to total computational costs, the iterative method is close to a solution method for linear problems, since the number of iterations is only weakly depending on the number of mesh points used. The linearized iterative schemes inherit the ε-uniform rate of convergence of the nonlinear schemes. The decomposition schemes can be computed sequentially and in parallel (so that the suproblems on the overlapping subdomains are solved independently of each other). Note that schemes of the overlapping domain decomposition method were considered earlier by the authors in [2] for linear problems and in [3, 4] for nonlinear problems. References [1] G.I. Shishkin and L.P. Shishkina, “A high-order Richardson method for a quasilinear singularly perturbed elliptic reaction-diffusion equation”, Differential Equations, 41 (7), 1030– 1039 (2005). ∗ This research was supported in part by the Russian Foundation for Basic Research (grants No. 04-01-00578, 04–01–89007–NWO a) and by the Dutch Research Organisation NWO grant No. 047.016.008. Speaker: TSELISHCHEVA, I.V. 51 BAIL 2006 1 ✩ ✪
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BAIL 2006 International Conference
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Plenary Session 1 Session 2 Session
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Room School-Lab (SL) Room MPI Sessi
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Room School-Lab (SL) Room MPI Tuesd
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Room School-Lab (SL) Room MPI Minis
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Contents Greetings . . . . . . . .
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CONTENTS S. HEMAVATHI, S. VALARMATH
Page 21 and 22: CONTENTS G. LUBE: A stabilized fini
Page 23: Plenary Presentations
Page 26 and 27: P. HOUSTON: Discontinuous Galerkin
Page 28 and 29: M. STYNES: Convection-diffusion pro
Page 30 and 31: V. GRAVEMEIER, S. LENZ, W.A. WALL:
Page 33: Minisymposia
Page 36 and 37: R.K. DUNNE, E. O’RIORDAN, M.M. TU
Page 38 and 39: G.I. SHISHKIN: A posteriori adapted
Page 40 and 41: W. LAYTON, I. STANCULESCU: Numerica
Page 42 and 43: H. WANG: A Component-Based Eulerian
Page 44 and 45: R. HARTMANN: Discontinuous Galerkin
Page 46 and 47: V. HEUVELINE: On a new refinement s
Page 48 and 49: J.A. MACKENZIE, A. NICOLA: A Discon
Page 50 and 51: R. SCHNEIDER, P. JIMACK: Anisotropi
Page 53 and 54: Speaker: Debopam Das, Tapan Sengupt
Page 55 and 56: M.H. BUSCHMANN, M. GAD-EL-HAK: Turb
Page 57 and 58: A. NAYAK, D. DAS: Three-dimesnional
Page 59 and 60: J. HUSSONG, N. BLEIER, V.V. RAM: Th
Page 61 and 62: T.K. SENGUPTA, A. KAMESWARA RAO: Sp
Page 63 and 64: L. SAVIĆ, H. STEINRÜCK: Asymptoti
Page 65 and 66: G.I. Shiskin, P. Hemker Robust Meth
Page 67 and 68: D. BRANLEY, A.F. HEGARTY, H. PURTIL
Page 69 and 70: T. LINSS, N. MADDEN: Layer-adapted
Page 71: L.P. SHISHKINA, G.I. SHISHKIN: A Di
Page 75 and 76: S. HEMAVATHI, S. VALARMATHI: A para
Page 77 and 78: J. Maubach, I, Tselishcheva Robust
Page 79 and 80: M. ANTHONISSEN, I. SEDYKH, J. MAUBA
Page 81 and 82: S. LI, L.P. SHISHKINA, G.I. SHISHKI
Page 83 and 84: A.I. ZADORIN: Numerical Method for
Page 85 and 86: P. ZEGELING: An Adaptive Grid Metho
Page 87: Contributed Presentations
Page 90 and 91: F. ALIZARD, J.-CH. ROBINET: Two-dim
Page 92 and 93: TH. ALRUTZ, T. KNOPP: Near-wall gri
Page 94 and 95: M. BAUSE: Aspects of SUPG/PSPG and
Page 96 and 97: L. BOGUSLAWSKI: Sheare Stress Distr
Page 98 and 99: A.CANGIANI, E.H.GEORGOULIS, M. JENS
Page 100 and 101: C. CLAVERO, J.L. GRACIA, F. LISBONA
Page 102 and 103: B. EISFELD: Computation of complex
Page 104 and 105: A. FIROOZ, M. GADAMI: Turbulence Fl
Page 106 and 107: A. FIROOZ, M. GADAMI: Turbulence Fl
Page 108 and 109: S.A. GAPONOV, G.V. PETROV, B.V. SMO
Page 110 and 111: M. HAMOUDA, R. TEMAM: Boundary laye
Page 112 and 113: M. HAMOUDA, R. TEMAM: Boundary laye
Page 114 and 115: M. HÖLLING, H. HERWIG: Computation
Page 116 and 117: A.-M. IL’IN, B.I. SULEIMANOV: The
Page 118 and 119: W.S. ISLAM, V.R. RAGHAVAN: Numerica
Page 120 and 121: D. KACHUMA, I. SOBEY: Fast waves du
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A. KAUSHIK, K.K. SHARMA: A Robust N
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P. KNOBLOCH: On methods diminishing
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T. KNOPP: Model-consistent universa
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J.-S. LEU, J.-Y. JANG, Y.-C. CHOU:
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V.D. LISEYKIN, Y.V. LIKHANOVA, D.V.
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G. LUBE: A stabilized finite elemen
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H. LÜDEKE: Detached Eddy Simulatio
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K. MANSOUR: Boundary Layer Solution
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J. MAUSS, J. COUSTEIX: Global Inter
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O. MIERKA, D. KUZMIN: On the implem
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K. MORINISHI: Rarefied Gas Boundary
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A. NASTASE: Qualitative Analysis of
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F. NATAF, G. RAPIN: Application of
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N. NEUSS: Numerical approximation o
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M.A. OLSHANSKII: An Augmented Lagra
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N. PARUMASUR, J. BANASIAK, J.M. KOZ
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B. RASUO: On Boundary Layer Control
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H.-G. ROOS: A Comparison of Stabili
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B. SCHEICHL, A. KLUWICK: On Turbule
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O. SHISHKINA, C. WAGNER: Boundary a
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M. STYNES, L. TOBISKA: Using rectan
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P. SVÁ ˘CEK: Numerical Approximat
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N.V. TARASOVA: Full asymptotic anal
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C.H. TAI, C.-Y. CHAO, J.-C. LEONG,
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H. TIAN: Uniformly Convergent Numer
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ABOUTUNSTEADYBOUNDARYLAYERONADIHEDR
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A.E.P. VELDMANN: High-order symmetr
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Z.-H. YANG, Y.-Z. LI, Y. ZHU: Appli
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Q. YE: Numerical simulation of turb
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Participants
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Mrs.Maragatha Meenakshi Ponnusamy P
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Authors Alizard, F., 67 Alrutz, Th,
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