Alexey Chernov

Lorenzo Mascotto, Lourenço Beirão da Veiga, Alexey Chernov et al.

In the present work, we analyze the $hp$ version of Virtual Element methods for the 2D Poisson problem. We prove exponential convergence of the energy error employing sequences of polygonal meshes geometrically refined, thus extending the classical choices for the decomposition in the $hp$ Finite Element framework to very general decomposition of the domain. A new stabilization for the discrete bilinear form with explicit bounds in $h$ and $p$ is introduced. Numerical experiments validate the theoretical results. We also exhibit a numerical comparison between $hp$ Virtual Elements and $hp$ Finite Elements.

Alexey Chernov, Peter Hansbo, Erik Marc Schetzke

We construct and analyse a $hp$-FE/BE coupling on non-matching meshes, based on Nitsche's method. Both the mesh size and the polynomial degree are changed to improve accuracy. Nitsche's method leads to a positive definite formulation, thus, unlike the mortar method, it does not require the Babuška-Brezzi condition for stability. The method is stable provided the stabilization function is larger than a certain threshold. We obtain an explicit bound for the threshold and derive a priori error estimates. Our analysis can be easily extended to the pure FE or the pure BE decomposition as well as to the case of more than two subdomains. The problem in a bounded domain is considered in detail, but the case of an unbounded BE subdomain and a bounded FE subdomain follows with similar arguments. We develop convergence analysis and provide numerical examples for quasi-uniform as well as geometrically refined $hp$ discretisations in both subdomains with analytic and singular solutions.

Alexey Chernov, Lorenzo Mascotto

We introduce the harmonic virtual element method (harmonic VEM), a modification of the virtual element method (VEM) for the approximation of the 2D Laplace equation using polygonal meshes. The main difference between the harmonic VEM and the VEM is that in the former method only boundary degrees of freedom are employed. Such degrees of freedom suffice for the construction of a proper energy projector on (piecewise harmonic) polynomial spaces. The harmonic VEM can also be regarded as an "$H^1$-conformisation" of the Trefftz discontinuous Galerkin-finite element method (TDG-FEM). We address the stabilization of the proposed method and develop an $hp$ version of harmonic VEM for the Laplace equation on polygonal domains. As in Trefftz DG-FEM, the asymptotic convergence rate of harmonic VEM is exponential and reaches order $\mathcal O ( \exp(-b\sqrt[2]{N}))$, where $N$ is the number of degrees of freedom. This result overperformes its counterparts in the framework of $hp$ FEM and $hp$ VEM, where the asymptotic rate of convergence is of order $\mathcal O ( \exp(-b\sqrt[3]{N}) )$.

Alexey Chernov, Anne Reinarz

The aim of this paper is to develop stable and accurate numerical schemes for boundary integral formulations of the heat equation with Dirichlet boundary conditions. The accuracy of Galerkin discretisations for the resulting boundary integral formulations depends mainly on the choice of discretisation space. We develop a-priori error analysis utilising a proof technique that involves norm equivalences in hierarchical wavelet subspace decompositions. We apply this to a full tensor product discretisation, showing improvements over existing results, particularly for discretisation spaces having low polynomial degrees. We then use the norm equivalences to show that an anisotropic sparse grid discretisation yields even higher convergence rates. Finally, a simple adaptive scheme is proposed to suggest an optimal shape for the sparse grid index sets.

Alexey Chernov, Haakon Hoel, Kody Law et al.

This work embeds a multilevel Monte Carlo (MLMC) sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF), thereby yielding a multilevel ensemble Kalman filter (MLEnKF) which has provably superior asymptotic cost to a given accuracy level. The development of MLEnKF for finite-dimensional state-spaces in the work [20] is here extended to models with infinite-dimensional state- spaces in the form of spatial fields. A concrete example is given to illustrate the results.

Alexey Chernov, Vladimir Vovk

In the framework of prediction with expert advice, we consider a recently introduced kind of regret bounds: the bounds that depend on the effective instead of nominal number of experts. In contrast to the Normal- Hedge bound, which mainly depends on the effective number of experts but also weakly depends on the nominal one, we obtain a bound that does not contain the nominal number of experts at all. We use the defensive forecasting method and introduce an application of defensive forecasting to multivalued supermartingales.