By Silvester D. J., Mihajlovic M. D.

We learn the convergence features of a preconditioned Krylov subspace solver utilized to the linear platforms bobbing up from low-order combined finite aspect approximation of the biharmonic challenge. the most important function of our process is that the preconditioning will be learned utilizing any "black-box" multigrid solver designed for the discrete Dirichlet Laplacian operator. This ends up in preconditioned structures having an eigenvalue distribution such as a tightly clustered set including a small variety of outliers. Numerical effects exhibit that the functionality of the method is aggressive with that of specialised quickly new release equipment which were built within the context of biharmonic difficulties.

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P* We want to prove that this case can not be happened. Let q^supl-F^I2, then f < p. <9\ P,(J) S U p p,(w\ Pf(z) | ^ | 2 < z+a ^. 5)

If the scalar curvature is positive, we can prove that the operator D is elliptic. 4) n D(N,a) = - Y, i nH5 ki - hkl)Y(hkihu)(N,a). From Maximum Principle, we obtain that (N, a) is constant. Therefore, (N, a) > 0 or (JV, a) = 0. 2 holds. In particular, when n = 3, the author and Wan [8] proved that the assertion due to Nomizu and Smyth [26] is true without the assumption on the sectional curvature. 3 (Cheng and Wan [8]). Let Mn be an oriented complete hypersurface in E 4 with constant scalar curvature.

Chen Zhihua and Lu Zhiqin M proved: Let M be an n-dimensional complete noncompact Riemannian manifold with a pole and nonnegative radial Ricci curvature outside a compact set, then its essential spectrum of the Laplacian cr es5 (A) = [0, +oo) But we find no result on the essential spectrum of the Laplacian under the assumption that Ricci curvature outside a compact set is not nonnegative. In this paper, we have the following Theorem. Theorem 1. Let M be an n-dimensional complete noncompact Riemannian manifold with a pole P, and outside a compact set K, the radial Ricci curvature > — 4 / " Z Q ^ > where K C B(P,a),r denotes the distance to P, then aess(A) = [0,+oo).