The fully nonconforming virtual element method for biharmonic problems

Abstract

In this paper we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. In particular, we present a novel nonconforming virtual element discretization of arbitrary order of accuracy for biharmonic problems. The approximation space is made of possibly discontinuous functions, thus giving rise to the fully nonconforming virtual element method. We derive optimal error estimates in a suitable (broken) energy norm and present numerical results to assess the validity of the theoretical estimates.

Extension of the nonconforming Trefftz virtual element method to the Helmholtz problem with piecewise constant wave number

“…Analogously, for all K ∈ T 2 n , we define the bulk plane wave space PW (2) p K (K) as the span of the plane waves w (2),K , which are defined in the same way as w (1),K in (8), but with wave number k 2 instead of k 1 . Following [25,32], we introduce a set of p K = 2 q K , q K ∈ N 0 , evanescent waves, for all K ∈ T 2 n .…”

“…(1) p K (K) and PW (2) p K , p K (K) are given by p K and p K + p K , respectively, those of the spaces PW pe (e) are in general smaller than or equal to the sum of the dimensions of the bulk spaces of the adjacent polygons. In fact, the restriction of two different plane waves onto a given edge could generate a 1D space only.…”

“…The two main features of this method are (i) that it is Trefftz (i.e., local spaces consist of functions belonging to the kernel of the target differential operator) and (ii) that it falls within the nonconforming virtual element framework, see e.g. [2,3,11,15]. Although in the basic construction of the method more degrees of freedom than e.g.…”

The conforming virtual element method for polyharmonic problems

“…Furthermore, a K h (·, ·) is V -elliptic and continuous for every K, and so is the global bilinear form a h (·, ·). The V -ellipticity of a K h (·, ·) is indeed a consequence of the left inequality in (9). Since a K h (·, ·) is symmetric and coercive, it is a scalar product on V h,r (K) and satisfies the Cauchy-Schwarz inequality.…”

“…The r-consistency property follows by noting that the stability term in (26) is zero when one of its entries is a polynomial of degree r as Π ∇,K r is a polynomial-preserving operator. The stability property is easily established by applying (9) to definition (26) and setting α * = min(σ * , 1) and α * = max(σ * , 1), where σ * and σ * are the constants defined in (27).…”