Virtual element method for second-order elliptic eigenvalue problems

Abstract

We study the virtual element (VEM) approximation of elliptic eigenvalue problems. The main result of the paper states that VEM provides an optimal order approximation of the eigenmodes. A wide set of numerical tests confirm the theoretical analysis.

The conforming virtual element method for polyharmonic problems

“…To estimate term T 1 , we use the continuity of a(·, ·) with respect to the norm || · || V = | · | p , the estimate in the energy norm (38), and interpolation error estimate (35)…”

“…Adding and subtracting u and using the estimate in the energy norm (38) and the estimate for the polynomial interpolation (36), we find that…”

“…A nonexhaustive list of such methods include the Mimetic Finite Difference method (see e.g., [22,20,4,16,8]), the Polygonal Finite Element Method (see e.g., [43]),the polygonal Discontinuous Galerkin Finite Element Methods (see e.g., [5,26,24,7,11]) the Hybridizable Discontinuous Galerkin and Hybrid High-Order Methods (see e.g., [31,32]), the Gradient Discretization method (see e.g., [34]. [33]), An alternative approach that is also proved to be very successful is provided by the Virtual Element method (VEM), which was originally proposed in [14] for the numerical treatment of second-order elliptic problems [29,28], and readily extended to Cahn-Hilliard equation [3], Stokes equations [2], Laplace-Beltrami equation [35], Darcy-Brinkam equation [44], discrete topology optimization problems [6], fracture networks problems [17], eigenvalue problems [38]. The mixed virtual element formulation was proposed in [21]; the nonconforming Virtual element formulations was proposed for second-order elliptic problems in [12], and later extended to general advection-reaction-diffusion problems, Stokes equation, the biharmonic problems, the eigenvalue problems, and the Schrodinger equation in [27,30,48,37,9].…”

The p- and hp-versions of the virtual element method for elliptic eigenvalue problems

“…Convergence of the error(38) for the first four distinct Dirichlet eigenvalues of the quantum harmonic oscillator operator on the square domain Ω 2 employing the h-version with p = 1, 2, and 3 and the p-version of the method. On the x-axis, we plot the square root of the number of degrees of freedom.…”

Virtual Elements for the Navier--Stokes Problem on Polygonal Meshes

“…Among the Galerkin schemes, VEM is peculiar in that the discrete spaces consist of functions which are not known pointwise, but about which a limited set of information is available. This limited information is sufficient to construct the stiffness matrix and the right-hand side.The VEM has been developed for many problems; see, for example, [23,1,10,43,46,9,19,17,18,6,42,54,51,37,45]. More specifically, with regard to the Stokes problem, virtual elements have been developed in [3,28,15,25,26,52].…”