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We present and study a novel numerical algorithm to approximate the action of Tβ := L-β where L is a symmetric and positive definite unbounded operator on a Hilbert space H0. The numerical method is based on a representation formula for T-β in terms of Bochner integrals involving (I + t2L)-1 for t ∊ (0,∞). To develop an approximation to Tβ, we introduce a finite element approximation Lh to L and base our approximation to Tβ on Tβ h := L -β h . The direct evaluation of Tβ h is extremely expensive as it involves expansion in the basis of eigenfunctions for Lh. The above mentioned representation formula holds for T -β h and we propose three quadrature approximations denoted generically by Qβ h. The two results of this paper bound the errors in the H0 inner product of Tβ -Tβ h πh and Tβ h -Qβ h where πh is the H0 orthogonal projection into the finite element space. We note that the evaluation of Qβ h involves application of (I +(ti)2Lh)-1 with ti being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. Numerical experiments illustrating the theoretical estimates are provided for both the quadrature error Tβ h - Qβ h and the finite element error Tβ - Tβ h πh. [ABSTRACT FROM AUTHOR]

We design and analyze variational and non-variational multigrid algorithms for the Laplace-Beltrami operator on a smooth and closed surface. In both cases, a uniform convergence for the V -cycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid. The main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface. In addition, the vanishing mean value constraint is imposed on each level, thereby avoiding singular quadratic forms without adding additional computational cost. Numerical results supporting our analysis are reported. In particular, the algorithms perform well even when applied to surfaces with a large aspect ratio. [ABSTRACT FROM AUTHOR]