Virtual element methods based on boundary triangulation:fitted and unfitted meshes

TL;DR

This work develops and analyzes virtual element methods (VEMs) based on boundary triangulation for robust optimal convergence on highly anisotropic 3D polyhedral meshes. By enforcing a boundary maximum angle condition (MAC) and extending it to polyhedral geometries, the authors construct a modular analysis framework that replaces classical trace inequalities with a discrete Poincaré-type inequality on element boundaries. They prove energy-norm and $L^2$-norm error estimates for the lowest-order VEM across three mesh scenarios: elements containing but not star-convex to inscribed balls, elements cut from cuboids with extreme shrinking, and interface problems with discontinuous coefficients (IVEM). Numerical results corroborate the theory, showing optimal convergence for both fitted and unfitted meshes and highlighting IVEM’s strong solver robustness to small interface cuts.

Abstract

One remarkable feature of virtual element methods (VEMs) is their great flexibility and robustness when used on almost arbitrary polytopal meshes. This very feature makes it widely used in both fitted and unfitted mesh methods. Despite extensive numerical studies, a rigorous analysis of robust optimal convergence has remained open for highly anisotropic 3D polyhedral meshes. In this work, we consider the VEMs in \cite{2023CaoChenGuo,2017ChenWeiWen} that introduce a boundary triangulation satisfying the maximum angle condition. We close this theoretical gap regarding optimal convergence on polyhedral meshes in the lowest-order case for the following three types of meshes: (1) elements only contain non-shrinking inscribed balls but \textit{are not necessarily star convex} to those balls; (2) elements are cut arbitrarily from a background Cartesian mesh, which can extremely shrink; (3) elements contain different materials on which the virtual spaces involve discontinuous coefficients. The first two widely appear in generating fitted meshes for interface and fracture problems, while the third one is used for unfitted mesh on interface problems. In addition, the present work also generalizes the maximum angle condition from simplicial meshes to polyhedral meshes.

Figures (9)

• Figure 1: Example of an anisotropic element $K$ which contains a ball of the radius $\mathcal{O}(h_K)$ but is not star convex to it. The face on the left is not supported by an $\mathcal{O}(h_K)$ height towards $K$. But its boundary admits a triangulation satisfying Assumptions \ref{['asp:A1']} and \ref{['asp:A2']}. This element corresponds to Case (1) studied in this paper.
• Figure 1: Illustration of Assumption \ref{['asp:A2']}: $e_5 = A_2A_6$ is not allowed in a path as the two neighborhood elements may shrink to this edge. Then, $e_2$, $e_3$ and $e_4$ are needed to connect $A_2$ and $A_6$.
• Figure 1: Cases 1-3 from left to right: elements cut from cuboids, highlighted as the red solids, may not contain non-shrinking inscribed balls. The left one may shrink to the segment $A_1A_5$. The middle one and the right one may shrink to the plane $A_1A_3A_7A_5$ and $A_1A_2A_3A_4$. They all satisfy Assumption \ref{['asp:A1']}. Case 1 satisfies Assumption \ref{['asp:A2']}, while Cases 1 and 3 satisfy Assumption \ref{['asp:A2plus']}.
• Figure 1: Illustration of an interface element. Being a polyhedron, it has 8 vertices of which 4 are the vertices of $K$ and 4 are the intersecting points with $\Gamma^K_h$. The face triangulation satisfies the
• Figure 1: The left plot is for $\epsilon = 0.1$, making the interface away from the faces. The middle plot i=s $\epsilon = 10^{-6}$, making the interface extremely close the faces with the $x_1$, $x_2$ or $x_3$ coordinates being $\pm 0.75$. The third plot is the zoom-in visualization of such interface elements. In particular, it is highlighted that such elements always exist for even very fine meshes.

Theorems & Definitions (76)

• Remark 2.1
• Lemma 2.2: 1999Rudiger
• Proof 1
• Lemma 2.3
• Proof 2
• Lemma 2.4
• Proof 3
• Lemma 2.5
• Proof 4
• Remark 2.6