A numerical implementation for the high-order 2D Virtual Element Method in MATLAB

We present a numerical implementation for the Virtual Element Method that in- corporates high order spaces. We include all the required computations in order to assemble the stiffness and mass matrices, and right hand side. Convergence of method is verified for different polygonal partitions. An specific mesh-free application that allows to approximate harmonic func- tions is discussed, based on high-order projections. This approach significantly improves running times compared to usual finite or virtual element methods, and can be modified for different virtual spaces and elliptic partial differential equations.UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemátic

Interior estimates for the Virtual Element Method

We analyze the local accuracy of the virtual element method. More precisely, we prove an error bound similar to the one holding for the finite element method, namely, that the local $H^1$ error in a interior subdomain is bounded by a term behaving like the best approximation allowed by the local smoothness of the solution in a larger interior subdomain plus the global error measured in a negative norm

FETI-DP for the three-dimensional Virtual Element Method

We deal with the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) preconditioner for elliptic problems discretized by the virtual element method (VEM). We extend the result of [16] to the three dimensional case. We prove polylogarithmic condition number bounds, independent of the number of subdomains, the mesh size, and jumps in the diffusion coefficients. Numerical experiments validate the theoryComment: 28 page

An overlapping Schwarz method for virtual element discretizations in two dimensions

A new coarse space for domain decomposition methods is presented for nodal ellipticproblems in two dimensions. The coarse space is derived from the novel virtual elementmethods and therefore can accommodate quite irregular polygonal subdomains. It hasthe advantage with respect to previous studies that no discrete harmonic extensionsare required. The virtual element method allows us to handle polygonal meshes andthe algorithm can then be used as a preconditioner for linear systems that arise froma discretization with such triangulations. A bound is obtained for the condition numberof the preconditioned system by using a two-level overlapping Schwarz algorithm, butthe coarse space can also be used for different substructuring methods. This bound isindependent of jumps in the coefficient across the interface between the subdomains.Numerical experiments that verify the result are shown, including some with triangular,square, hexagonal and irregular elements and with irregular subdomains obtained by a mesh partitionerUCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigaciones en Matemáticas Puras y Aplicadas (CIMPA