Special ACM Seminar
In this talk we review some recent developments two rather independent areas of computational mathematics having in common that they both utilize and benefit from high order and spectral approximations.
The first area is concerned with the Virtual Element Method (VEM), that is a recent generalization of the Finite Element Method. VEM utilizes polygonal/polyhedral meshes in lieu of the classical triangular/tetrahedral and quadrilateral/hexaedral meshes. This automatically includes nonconvex elements, hanging nodes (enabling natural handling of interface problems with nonmatching grids), easy construction of adaptive meshes and efficient approximations of geometric data features. In this talk we review the basic construction of the method and discuss an extension of VEM to arbitrary high order approximations  enabling e.g. exponential convergence for elliptic problems with corner singularities on geometrically refined polygonal grids (hp-VEM) [2, 3].
 L. Beirão da Veiga, A. Chernov, L. Mascotto and A. Russo, Basic principles of hp Virtual Elements on quasiuniform meshes, Math. Models Methods Appl. Sci. 26 (2016), no. 8, 1567–1598
 L. Beirão da Veiga, A. Chernov, L. Mascotto and A. Russo, Exponential convergence of the hp virtual element method in presence of corner singularities, Numerische Mathematik 138 (2018), no. 3, 581–613
 A. Chernov, L. Mascotto, The harmonic virtual element method: stabilization and exponentialconvergence for the Laplace problem on polygonal domains. IMA Journal of Numerical Analysis (2019), published online
(Joint with L. Mascotto, L. Beirao da Veiga, A. Russo)
The second area addresses numerical approximation for problems with uncertain parameters. The Maximum Entropy method is a powerful tool for recovery of the probability density function (PDF) of an unknown quantity of interest when only a finite series of its generalized moments are known or are estimated numerically, e.g. by inexact (multilevel) Monte Carlo simulation. When the generalized moments are defined via a family of algebraic or trigonometric polynomials, high order/spectral approximations naturally appear. We recall the two-stage simulation procedure from , discuss the error analysis and performance of the algorithm in a series of numerical experiments.
 C. Bierig and A. Chernov, Approximation of probability density functions by the Multilevel Monte Carlo Maximum Entropy method, J. Comput. Physics 314 (2016), 661–681
(Joint with C. Bierig)