![]() When intensive DDM research began much attention was given to overlapping DDMs, but soon after attention shifted to non-overlapping DDMs. Stated in a simplistic manner, the basic idea is that, when the DDM-paradigm is satisfied, full parallelization can be achieved by assigning each subdomain to a different processor. Ideally, DDMs intend producing algorithms that fulfill the DDM-paradigm i.e., such that "the global solution is obtained by solving local problems defined separately in each subdomain of the coarse-mesh -or domain-decomposition-". Very early after such an effort began, it was recognized that domain decomposition methods (DDM) were the most effective technique for applying parallel computing to the solution of partial differential equations, since such an approach drastically simplifies the coordination of the many processors that carry out the different tasks and also reduces very much the requirements of information-transmission between them. The emergence of parallel computing prompted on the part of the computational-modeling community a continued and systematic effort with the purpose of harnessing it for the endeavor of solving boundary-value problems (BVPs) of partial differential equations. Furthermore, the incredible expansion experienced by the existing computational hardware and software has made amenable to effective treatment problems of an ever increasing diversity and complexity, posed by engineering and scientific applications. Mathematical models of many systems of interest, including very important continuous systems of Engineering and Science, lead to a great variety of partial differential equations whose solution methods are based on the computational processing of large-scale algebraic systems. analytic f, g, h).Ī Non-Overlapping Discretization Method for Partial Differential Equations ![]() complex) domain as f\\bigl(g(x,y),h(y,z)\\bigr) with twice differentiable f and differentiable g, h (resp. We also prove that the function u defined by u^n=xu^a+yu^b+zu^c+1 is generally non-representable in any real (resp. ![]() The representability of a real analytic function by a superposition of this type is independent of whether that superposition involves real-analytic functions or C^-functions, where the constant \\rho is determined by the structure of the superposition. These equations represent necessary and sufficient conditions for an analytic function to be locally expressible as an analytic superposition of the type indicated. We determine essentially all partial differential equations satisfied by superpositions of tree type and of a further special type. Homogeneous partial differential equations for superpositions of indeterminate functions of several variables Applications to parabolic and hyperbolic equations are presented. Necessary and sufficient conditions for weak and strong stabilisation are formulated in term of approximate observability like assumptions. In this work, we study in a Hilbert state space, the partial stabilisation of non-homogeneous bilinear systems using a bounded control.
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