Reduced basis stabilization and post-processing for the virtual element method

TL;DR

The paper tackles the challenge of efficiently handling the nonpolynomial VE basis in the lowest-order VEM on polytopal meshes. It introduces a reduced-basis (RB) strategy that treats element geometry as a parameter and maps the basis-construction problem to a fixed reference element, enabling offline generation of a compact RB space and cheap online reconstruction. This RB framework is then used in two applications: (i) to design improved, anisotropy-aware stabilization terms, and (ii) to produce conforming post-processing reconstructions of the VE solution for visualization and quantitative outputs. Numerical experiments show that even a very small RB space (often M=1) can improve stability and enable high-quality post-processing, with the online phase highly parallelizable and computationally affordable. The approach promises extensions to higher-order VEM, 2D elasticity, and 3D polyhedral meshes, offering a practical path to robust, efficient polygonal discretizations.

Abstract

We present a reduced basis method for cheaply constructing (possibly rough) approximations to the nodal basis functions of the virtual element space, and propose to use such approximations for the design of the stabilization term in the virtual element method and for the post-processing of the solution.

Figures (16)

• Figure 1: An example of affine mapping $\mathscr{B}_K$ between a random star shaped pentagon and its regular counterpart.
• Figure 2: Sketch of snapshots computation. For the sake of clarity we use a depiction with very coarse meshes (computations are of course carried out on much finer meshes).
• Figure 3: Some random polygons generated for building the reduced basis. On the left $N=6$, on the right $N=11$.
• Figure 4: Statistical plots of the errors for $N=6,11$ varying the number of reduced basis $M$. Test case a) (the degrees of freedom are imposed evaluating $p(x)=x^5+y^5$ at the vertices). The first data, in correspondence with $M=0$, refers to the error between $u_h^\text{fe}$ and $\Pi^\nabla u_h$. Circles represent the maximum values, whereas diamonds represent the minimum values. The averages are depicted by a square. The vertical lines are drawn by connecting the 95th and 5th percentiles.
• Figure 5: Statistical plots of the errors for $N=9,14$ varying the number of reduced basis $M$. Test case b) (the degrees of freedom are randomly generated in $(0,1)$). Same format as in Figure \ref{['fig:stat_poly']}.

Theorems & Definitions (4)

• Remark 2.3
• Remark 5.2
• Remark 5.3
• Remark 7.1