The purpose of this paper is to establish a new numerical approach to solve, in two dimensions, a semilinear reaction diffusion equation combining finite volume method and Schur complement method. We applied our method for q = 2 non-overlapping subdomains and then we generalized in the case of several subdomains (q ≥ 2). A large number of numerical test cases shows the efficiency and the good accuracy of the proposed approach in terms of the CPU time and the order of the error, when increasing the number of subdomains, without using the parallel computing. After several variations of the number of subdomains and the mesh grid, we remark two significant results. On the one hand, the increase related to the number of subdomains does not affect the order of the error, on the other hand, for each mesh grid when we augment the number of subdomains, the CPU time reaches the minimum for a specific number of subdomains. In order to have the minimum CPU time, we resorted to a statistical study between the optimal number of subdomains and the mesh grid.
We present in this paper a new algorithm combining a finite volume method with an improved Schur complement technique to solve 2D anisotropic diffusion problems on general meshes. After having proved the convergence of the finite volume method, we have given a description of the proposed algorithm in the case of two nonoverlapping subdomains. Several numerical tests are achieved which illustrate the theoretical results of convergence of the finite volume method and show the advantages of the proposed algorithm.
In this paper, we study the boundary feedback stabilization problem of a hybrid system consisting of a flexible beam attached to the platform moving a long a straight rail and carrying at the free end a load which is deplaced in a horizontal plan. The model proposed in this paper fits a large real applications such as an overhead crane with beam. Using the Riesz basis approach of general second-differential equation systems with non separated boundary conditions, it is shown that the Riesz basis property holds for the system and as consequence, the exponential stability is concluded. To verify the theoritical developments, numerical study of the spectrum is performed by Legendre approximation, also thenumerical simulations are presented to show the effectivnesse of the proposed control.
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