By Ivan Lirkov, Svetozar D. Margenov, Jerzy Wasniewski
This ebook constitutes the completely refereed post-conference court cases of the eighth foreign convention on Large-Scale clinical Computations, LSSC 2011, held in Sozopol, Bulgaria, in June 2011. The seventy four revised complete papers provided including three plenary and invited papers have been rigorously reviewed and chosen from a variety of submissions. The papers are geared up in topical sections on powerful multigrid, multilevel and multiscale, deterministic and stochastic tools for modeling hugely heterogeneous media, complex equipment for shipping, regulate and unsure structures, functions of metaheuristics to large-scale difficulties, environmental modelling, huge scale computing on many-core architectures, multiscale business, enviromental and biomedical difficulties, effective algorithms of computational geometry, excessive functionality Monte Carlo simulations, voxel dependent computations and contributed papers
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Additional info for Large-Scale Scientific Computing: 8th International Conference, LSSC 2011, Sozopol, Bulgaria, June 6-10th, 2011. Revised Selected Papers
Proof (Outline). The proof hinges on three results, namely, (i) a quasi-optimal bound for the discretization error in terms of the best approximation error for the exact Dirichlet and Neumann data, (ii) an approximation error estimate for the Dirichlet data, and (iii) an approximation error estimate for the Neumann data. All of these estimates need to be made explicit in terms of the mesh regularity parameters, and problem-adapted norms have to be used. The details can be found in . Theorem 2 (L2 error estimate, ).
43–51, 2012. c Springer-Verlag Berlin Heidelberg 2012 44 Y. Efendiev et al. independent of the mesh parameter. , the contrast in the conductivity maxx∈Ω κ(x)/ minx∈Ω κ(x), where Ω is the domain, is more challenging. Some improvements in standard domain decomposition methods were made in the case of special arrangements of the highly conductive regions with respect to the coarse cells. g. [6,8,12]). g. 4]) when conductivity variations within coarse regions are bounded. Classical arguments to estimate the condition number of a two level overlapping domain decomposition method for the scalar elliptic case use weighted Poincar´e inequalities of the form κ(ψ − I0ω ψ)2 dx ≤ C ω κ|∇ψ|2 dx, (1) ω where ω ⊂ Ω is a local subdomain and ψ ∈ H 1 (ω).
3]) we thus know that the additive Schwarz preconditioner corresponding to the decomposition in (6) yields a condition number that only depends on nI and τλ . 3 Applications Scalar Elliptic Equation The scalar elliptic equation is given by −∇ · (κ∇φ) = f, x ∈ Ω, and φ = 0, x ∈ ∂Ω, (7) where 0 < κ ∈ L∞ (Ω), φ ∈ H01 (Ω), and f ∈ L2 (Ω). With V0 := H01 (Ω), the corresponding variational formulation reads: Find φ ∈ V0 such that for all ψ ∈ V0 κ(x)∇φ · ∇ψ dx = aSE (φ, ψ) := Ω f ψ dx. Ω Robust Solvers for SPD Operators 47 It is easy to see that with V0 (Ωj ) := H01 (Ωj ) ⊂ V0 |Ωj and ξj the Lagrange ﬁnite element function of degree 1 corresponding to xj , j = 1, .
Large-Scale Scientific Computing: 8th International Conference, LSSC 2011, Sozopol, Bulgaria, June 6-10th, 2011. Revised Selected Papers by Ivan Lirkov, Svetozar D. Margenov, Jerzy Wasniewski