[Paper Review] Exponential convergence of the hp Virtual Element Method with corner singularities

Lorenzo Mascotto, L. Beirão da Veiga+2 more|arXiv (Cornell University)|Nov 30, 2016

Advanced Numerical Methods in Computational Mathematics46 references7 citations

TL;DR

This paper establishes exponential convergence of the $hp$ Virtual Element Method (VEM) for the 2D Poisson problem on polygonal meshes with corner singularities. By introducing a novel stabilization term with explicit $h$- and $p$-dependent bounds and employing geometrically graded meshes with locally varying polynomial degrees, the authors prove exponential convergence in the energy norm, extending $hp$-FEM theory to general polytopal meshes without requiring shape-regularity or matching grids.

ABSTRACT

In the present work, we analyze the $hp$ version of Virtual Element methods for the 2D Poisson problem. We prove exponential convergence of the energy error employing sequences of polygonal meshes geometrically refined, thus extending the classical choices for the decomposition in the $hp$ Finite Element framework to very general decomposition of the domain. A new stabilization for the discrete bilinear form with explicit bounds in $h$ and $p$ is introduced. Numerical experiments validate the theoretical results. We also exhibit a numerical comparison between $hp$ Virtual Elements and $hp$ Finite Elements.

Motivation & Objective

• To extend the $hp$-version of the Virtual Element Method to polygonal meshes with corner singularities, where classical $hp$-FEM theory applies but VEM has not been fully generalized.
• To develop a new stabilization term for the discrete bilinear form with explicit bounds in terms of mesh size $h$ and polynomial degree $p$, enabling $hp$-analysis.
• To prove exponential convergence of the energy error for nonsmooth solutions with corner singularities using geometrically refined meshes and locally varying polynomial degrees.
• To eliminate restrictive assumptions on mesh geometry, such as shape regularity or matching grids, by introducing new inverse estimates for polynomials on polygons.

Proposed method

• Introduce a new stabilization term for the $hp$-VEM discrete bilinear form with explicit $h$- and $p$-dependent bounds, ensuring stability across varying element types and polynomial degrees.
• Construct geometrically graded polygonal meshes where element size decreases exponentially toward the singular corner (e.g., origin), with $h_K \approx \sigma^n$ for elements in layer $L_0$, and $h_K \approx \sigma^{n-j}$ for layer $j$.
• Define a local polynomial degree vector $\mathbf{p}$ that varies across elements, with $p_K \geq 2$, and choose it to grow linearly with distance from the singularity to match the mesh grading.
• Derive a new inverse estimate for polynomials on polygonal elements, which is essential for bounding the stabilization term and proving $hp$-robustness.
• Decompose the energy error into three components: consistency, best approximation in polynomial space, and best approximation in the virtual element space, and bound each using mesh and polynomial degree parameters.
• Combine all error bounds under a suitable choice of $\mathbf{p}$ and geometric mesh sequence to achieve exponential convergence in the energy norm with respect to the number of degrees of freedom.

Experimental results

Research questions

• RQ1 Can the $hp$-VEM achieve exponential convergence for problems with corner singularities on general polygonal meshes, similar to $hp$-FEM?
• RQ2 What stabilization strategy ensures $h$- and $p$-robustness in the $hp$-VEM framework for nonsmooth solutions?
• RQ3 How should the mesh grading and local polynomial degree distribution be chosen to optimize convergence rates in the presence of corner singularities?
• RQ4 Can the theoretical convergence be achieved without requiring shape-regularity or matching grids in polygonal meshes?
• RQ5 What new inverse estimates on polygonal elements are necessary to support the $hp$-analysis of VEM?

Key findings

• The proposed $hp$-VEM achieves exponential convergence of the energy error for the 2D Poisson problem with corner singularities, even on unstructured polygonal meshes.
• The new stabilization term is proven to be bounded in the $H^1$ seminorm with explicit dependence on $h$ and $p$, enabling rigorous $hp$-analysis.
• Numerical experiments confirm the theoretical exponential convergence rates, showing superior performance compared to standard $hp$-FEM in certain configurations.
• The minimum generalized eigenvalue of the stiffness matrix scales like $p^{-1}$, indicating that the theoretical bounds on stability are conservative, while actual conditioning is milder.
• The method achieves exponential convergence without requiring shape-regularity or matching grids, making it suitable for complex geometries and adaptive refinement.
• A new inverse estimate for polynomials on polygonal elements is derived and proven, which is essential for the $hp$-VEM convergence proof and may find broader application in polygonal methods.