High-order Virtual Element Method on polyhedral meshes

Abstract

We develop a numerical assessment of the Virtual Element Method for the discretization of a diffusion-reaction model problem, for higher "polynomial" order k and three space dimensions. Although the main focus of the present study is to illustrate some h-convergence tests for different orders k, we also hint on other interesting aspects such as structured polyhedral Voronoi meshing, robustness in the presence of irregular grids, sensibility to the stabilization parameter and convergence with respect to the order k.

Ill‐conditioning in the virtual element method: Stabilizations and bases

“…It was firstly introduced in [10] and it can be proved that, for a fixed degree of accuracy p, S K 1 scales like the H 1 seminorm. We highlight that this holds true only in two dimension; for a three dimensional problem one should put a proper scaling factor in front of S K 1 , see [33,32]. The advantage of picking S K 1 as a stabilization is that its implementation is extremely easy.…”

“…Next, we recall another stabilization which was introduced in [33]. If we denote by K K C the consistency part of the local stiffness matrix, that is, the matrix counterpart of the first term on the right-hand side of (15), then we can define a stabilization S K 3 through its matrix representation S K 3 associated with the second term on the right-hand side of (15) as follows.…”

“…In [33], numerical experiments, along with a heuristic motivation, show that choice (28) entails a better behaviour of the method for high p, when approximating a 3D Poisson problem.…”

“…The first one is related to the fact that in the VEM framework one does not use the exact bilinear form but an approximated one; the choice of the discrete bilinear form, and, in particular, of one of its two components, namely the stabilization of the method, may have an impact on the 3D version of VEM as observed in [33]. The second one is the choice of the basis.…”

“…Among the other references, we recall the following: the p and hp version of the method [11][12][13][14][15][16], parabolic problems [17], Cahn-Hilliard, Stokes, Navier-Stokes and Helmoltz equations [18][19][20][21][22], linear and nonlinear elasticity problems [23][24][25], general elliptic problems [26], PDEs on surfaces [27], Domain Decomposition [28], application to discrete fracture networks [29], serendipity VEM [30], VEM on surfaces [27]. The implementation of the method is described in [31], whereas the basic principles of the 3D version of the method are the topic of [32,33].…”