No description available.

No description available.

No description available.

No description available.

No description available.

No description available.

Optimized Schwarz methods are iterative domain decomposition procedures with greatly improved convergence properties, for solving second order elliptic boundary value problems. The enhanced convergence is obtained by replacing the Dirichlet transmission conditions in the classical Schwarz iteration with more general conditions that are optimized for performance. The convergence is optimized through the solution of a min-max problem. The theoretical study of the min-max problems gives explicit formulas or characterizations for the optimized transmission conditions for practical use, and it permits the analysis of the asymptotic behavior of the convergence. / In the first part of this work, we continue the study of optimized transmission conditions for advection-diffusion problems with smooth coefficients. We derive asymptotic formulas for the optimized parameters for small mesh sizes, in the overlapping and non-overlapping cases, and show that these formulas are accurate when the component of the advection tangential to the interface is not too large. / In a second part, we consider a diffusion problem with a discontinuous coefficient and non-overlapping domain decompositions. We derive several choices of optimized transmission conditions by thoroughly solving the associated min-max problems. We show in particular that the convergence of optimized Schwarz methods improves as the jump in the coefficient increases, if an appropriate scaling of the transmission conditions is used. Moreover, we prove that optimized two-sided Robin conditions lead to mesh-independent convergence. Numerical experiments with two subdomains are presented to verify the analysis. We also report the results of experiments using the decomposition of a rectangle into many vertical strips; some additional analysis is carried out to improve the optimized transmission conditions in that case. / On a third topic, we experiment with different coarse space corrections for the Schwarz method in a simple one-dimensional setting, for both overlapping and non-overlapping subdomains. The goal is to obtain a convergence that does not deteriorate as we increase the number of subdomains. We design a coarse space correction for the Schwarz method with Robin transmission conditions by considering an augmented linear system, which avoids merging the local approximations in overlapping regions. With numerical experiments, we demonstrate that the best Robin conditions are very different for the Schwarz iteration with, and without coarse correction.

Let Ω be a Lipschitz domain in Rᴺ,n ≥ 3, and L = divA∇ - B∇ be a second order elliptic operator in divergence form with real coefficients such that A is a bounded elliptic matrix and the vector field B ɛ L∞loc(Ω) is divergence free and satisfies the growth condition dist(X,∂Ω)|B(X)|≤ ɛ1 for ɛ1 small in a neighbourhood of ∂Ω. For these elliptic operators we will study on the basis of the theory for elliptic operators without drift terms the Dirichlet problem for boundary data in Lp(∂Ω), 1 < p < ∞, and the regularity problem for boundary data in W¹,ᵖ(∂Ω) and HS¹. The main result of this thesis is that the solvability of the regularity problem for boundary data in HS1 implies the solvability of the adjoint Dirichlet problem for boundary data in Lᵖ'(∂Ω) and the solvability of the regularity problem with boundary data in W¹,ᵖ(∂Ω for some 1 < p < ∞. In [KP93] C.E. Kenig and J. Pipher have proven for elliptic operators without drift terms that the solvability of the regularity problem with boundary data in W¹,ᵖ(∂Ω) implies the solvability with boundary data in HS1. Thus the result of C.E. Kenig and J. Pipher and our main result complement a result in [DKP10], where it was shown for elliptic operators without drift terms that the Dirichlet problem with boundary data in BMO is solvable if and only if it is solvable for boundary data in Lᵖ(∂Ω) for some 1 < p < ∞. In order to prove the main result we will prove for the elliptic operators L the existence of a Green's function, the doubling property of the elliptic measure and a comparison principle for weak solutions, which are well known results for elliptic operators without drift terms. Moreover, the solvability of the continuous Dirichlet problem will be established for elliptic operators L = div(A∇+B)+C∇+D with B,C,D ɛ L∞loc(Ω) such that in a small neighbourhood of ∂Ω we have that dist(X,∂Ω)(|B(X)| + |C(X)| + |D(X)|) ≤ ɛ1 for ɛ1 small and that the vector field B satisfies |∫B∇Ø| ≤ C∫|∇Ø| for all Ø ɛ Wₒ¹'¹ of that neighbourhood.

The solution of convection-diffusion problems is a challenging task for numerical methods because of the nature of the governing equation, which includes a non-dissipative component and a dissipative component. Once the convection-diffusion equation is discretised, it is usual to observe oscillations in the computed solution regardless of whether these might be expected in the original physical situation. Mostly these oscillations are the result of numerical instability. This thesis centres on this fundamental difficulty: the numerical stability of finite difference discretisation of a convection-diffusion equation. The existence of an exact evolution operator for the constant coefficient convection diffusion problem is the framework we use to derive new finite difference schemes in one and two dimensions and also, when a high-order scheme is considered, to derive numerical boundary conditions. The influence of numerical boundary conditions on the stability of a general scheme is one of the main themes. The stability analysis is done mostly by using the von Neumann method and the matrix method. The Godunov-Ryabenkii theory is also applied to the one dimensional case. In two dimensions we deduce different forms of second-order (Lax-Wendroff) schemes and third-order (Quickest) schemes. We apply some of those schemes to a Navier-Stokes problem by running experiments to illustrate the practical stability region, showing how results from a simpler case presented in previous chapters carry over to the more complex case.

No description available.