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A nonoverlapping domain decomposition technique applied to a finite difference method is presented for the numerical solution of the forward backward heat equation in the case of one-dimension. While the previous at tempts in dealing with this problem have been based on an iterative domain decomposition scheme, the current work avoids iterations. Also a physical matching condition is suggested to avoid difficulties caused by the interface boundary nodes. Furthermore, we obtain a square system of equations. In addition, the convergence and stability of the proposed method are investi gated. Some numerical experiments are given to show the effectiveness of the proposed method.

We prove convergence for the nonoverlapping Robin-Robin method applied to nonlinear elliptic equations with a $p$-structure, including degenerate diffusion equations governed by the $p$-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov-Poincar\'e operators based on the $p$-structure and the $L^p$-generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions., Comment: 20 pages, 1 figure

In this paper, we demonstrate a local convergence of an adaptive scalar solver which is practical for strongly diagonal dominant Jacobian problems such as in some systems of nonlinear equations arising from the application of a nonoverlapping domain decomposition method. The method is tested to a nonlinear interface problem of a multichip heat conduction problem. The numerical results show that the method performs slightly better than a Newton–Krylov method.

The control of the quality of mechanical computations arouses a growing interest in both design and certification processes. It relies on error estimators the use of which leads to often prohibitive additional numerical costs on large computations. The present work puts forward a new procedure enabling to obtain a guaranteed estimation of discretization error in the setting of linear elastic problems solved by domain decomposition approaches. The method relies on the extension of the constitutive relation error concept to the framework of non-overlapping domain decomposition through the recovery of admissible interface fields. Its development within the framework of the FETI and BDD approaches allows to obtain a relevant estimation of discretization error well before the convergence of the solver linked to the domain decomposition. An extension of the estimation procedure to heterogeneous problems is also proposed. The behaviour of the method is illustrated and assessed on several numerical examples in 2 dimension.; Le contrôle de la qualité des calculs de structure suscite un intérêt croissant dans les processus de conception et de certification. Il repose sur l'utilisation d'estimateurs d'erreur, dont la mise en pratique entraîne un sur-coût numérique souvent prohibitif sur des calculs de grande taille. Le présent travail propose une nouvelle procédure permettant l'obtention d'une estimation garantie de l'erreur de discrétisation dans le cadre de problèmes linéaires élastiques résolus au moyen d'approches par décomposition de domaine. La méthode repose sur l'extension du concept d'erreur en relation de comportement au cadre des décompositions de domaine sans recouvrement, en s'appuyant sur la construction de champs admissibles aux interfaces. Son développement dans le cadre des approches FETI et BDD permet d'accéder à une mesure pertinente de l'erreur de discrétisation bien avant convergence du solveur lié à la décomposition de domaine. Une extension de la procédure d'estimation aux problèmes hétérogènes est également proposée. Le comportement de la méthode est illustré et discuté sur plusieurs exemples numériques en dimension 2.

We consider the Dirichlet boundary value problem for Poisson's equation in an L-shaped region or a rectangle with a cross-point. In both cases, we approximate the Dirichlet problem using Legendre spectral collocation, that is, polynomial collocation at the Legendre-Gauss nodes. The L-shaped region is partitioned into three nonoverlapping rectangular subregions with two interfaces and the rectangle with the cross-point is partitioned into four rectangular subregions with four interfaces. In each rectangular subregion, the approximate solution is a polynomial tensor product that satisfies Poisson's equation at the collocation points. The approximate solution is continuous on the entire domain and its normal derivatives are continuous at the collocation points on the interfaces, but continuity of the normal derivatives across the interfaces is not guaranteed. At the cross point, we require continuity of the normal derivative in the vertical direction. The solution of the collocation problem is first reduced to finding the approximate solution on the interfaces. The discrete Steklov-Poincaré operator corresponding to the interfaces is self-adjoint and positive definite with respect to the discrete inner product associated with the collocation points on the interfaces. The approximate solution on the interfaces is computed using the preconditioned conjugate gradient method. A preconditioner is obtained from the discrete Steklov-Poincaré operators corresponding to pairs of the adjacent rectangular subregions. Once the solution of the discrete Steklov- Poincaré equation is obtained, the collocation solution in each rectangular subregion is computed using a matrix decomposition method. The total cost of the algorithm is O(N 3), where the number of unknowns is proportional to N 2. © 2007 Springer Science+Business Media, LLC. 32 2 373 409 Cited By :5

A nonoverlapping domain decomposition algorithm of Robin–Robin type is applied to the discretized Oseen equations using stabilized finite element approximations of velocity and pressure thus allowing in particular equal-order interpolation. As a crucial result we have to inspect the proof of a modified inf–sup condition, in particular, the dependence of the stability constant with respect to the Reynolds number (cf. appendix). After proving coercivity and strong convergence of the method, we derive an a posteriori estimate which controls convergence of the discrete subdomain solutions to the global discrete solution provided that jumps of the discrete solution converge at the interface. Furthermore, we obtain information on the design of some free parameters within the Robin-type interface condition which essentially influence the convergence speed. Some numerical results confirm the theoretical ones.

Se presenta una visión general de los métodos de descomposición de dominio con dominios ajenos. Los métodos más eficientes que existen en la actualidad, el BDDC y el FETI-DP, se ubican en un marco ‘primal’ (el ‘espacio de vectores derivados (DVS, por sus siglas en inglés)’), el cual permite una presentación sintética y efectiva tanto de las formulaciones primales como de las ‘duales’. El marco conceptual del espacio de los vectores derivados tiene alguna similitud con el que usa BDDC, pero una diferencia importante es que en el marco DVS el problema tratado se transforma en otro definido en el espacio vectorial producto, mientras que en el BDDC no se hace tal cosa. Esto simplifica los algoritmos, los cual se sintetizan en un breve conjunto de fórmulas matriciales muy generales que son aplicables a matrices simétricas, no simétricas e indefinidas, cuando ellas provienen de la discretización de ecuaciones diferenciales parciales o sistemas de tales ecuaciones. Las fórmulas matriciales de este conjunto, son explícitas y pueden ser usadas directamente para desarrollar códigos computacionales. Hasta donde sabemos, dos de los algoritmos precondicionados del conjunto mencionado, son totalmente diferentes a cualquiera de los reportados en la literatura y deben ser motivo de investigaciones futuras.