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A new numerical domain decomposition method is proposed for solving elliptic equations on compact Riemannian manifolds. The advantage of this method is to avoid global triangulations or grids on manifolds. Our method is numerically tested on some four-dimensional manifolds such as the unit sphere S4, the complex projective space CP2, and the product manifold S2\times S2. [ABSTRACT FROM AUTHOR]

This paper introduces bootstrap multigrid methods for solving eigenvalue problems arising from the discretization of partial differential equations. Inspired by the full bootstrap algebraic multigrid setup algorithm that includes an AMG eigensolver, we illustrate how the algorithm can be simplified for the case of a discretized partial differential equation, thereby developing a bootstrap geometric multigrid (BMG) approach. We illustrate numerically the efficacy of the BMG method for (1) recovering eigenvalues having large multiplicity, (2) computing interior eigenvalues, and (3) approximating shifted indefinite eigenvalue problems. Numerical experiments are presented to illustrate the basic components and ideas behind the success of the overall bootstrap multigrid approach. For completeness, we present a simplified error analysis of a two-grid bootstrap algorithm for the Laplace-Beltrami eigenvalue problem. [ABSTRACT FROM AUTHOR]

A refined a priori error analysis of the lowest-order (linear) nonconforming virtual element method (VEM) for approximating a model Poisson problem is developed in both 2D and 3D. A set of new geometric assumptions is proposed on the shape regularity of polytopal meshes. A new error equation for the lowest-order (linear) nonconforming VEM is derived for any choice of stabilization, and a new stabilization using a projection on an extended element patch is introduced for the error analysis on anisotropic elements. [ABSTRACT FROM AUTHOR]

Damage imaging based on scattering signals of ultrasonic Lamb waves in plate structure is considered as one of the most effective ways for structural health monitoring area. To improve location accuracy and reduce the impact of artifacts, a probability weighted four-point arc imaging algorithm for time reversal Lamb wave damage detection is proposed in this paper. By taking the defect as a secondary wave source, the four-point arc positioning method is used to calculate the propagation time of the signal from transducer to defect. And the amplitude of damage signal corresponding to the time of flight is used for imaging. In order to eliminate the artifacts, a damage probability weighting is combined with four-point circular arc imaging algorithm. The effectiveness of the proposed method is experimentally verified in aluminum plate. Experimental results indicate that damage location accuracy and imaging quality has been improved in both single-flaw and double-flaw samples compared with conventional delay-and-sum method. [ABSTRACT FROM AUTHOR]

A refined a priori error analysis of the lowest order (linear) virtual element method (VEM) is developed for approximating a model two-dimensional Poisson problem. A set of new geometric assumptions is proposed on the shape regularity of polygonal meshes. A new universal error equation for the lowest order (linear) VEM is derived for any choice of stabilization, and a new stabilization using a broken 1=2-seminorm is introduced to incorporate short edges naturally into the a priori error analysis on isotropic elements. The error analysis is then extended to a special class of anisotropic elements with high aspect ratio originating from a body-fitted mesh generator, which uses straight lines to cut a shape regular background mesh. Lastly, some commonly used tools for triangular elements are revisited for polygonal elements to give an in-depth view of these estimates' dependence on shapes. [ABSTRACT FROM AUTHOR]