Conforming/Non-conforming Virtual Elements and application to elasticity problems in curved three-dimensional domains

Technical Summary: Conforming/Non-conforming Virtual Elements and Application to Elasticity Problems in Curved Three-Dimensional Domains

Problem Statement
The Virtual Element Method (VEM) has established itself as a flexible framework for discretizing Partial Differential Equations (PDEs) on general polytopal meshes. However, significant challenges remain in applying VEM to three-dimensional problems involving curved domains. While 2D VEM with curved boundaries is well-developed, 3D implementations face difficulties in defining spaces on curved surfaces and computing necessary projections. Existing approaches often rely on complex master-slave constructions, weak enforcement of boundary conditions, or mixed formulations with discontinuous variables. In the context of solid mechanics, methods yielding non-continuous displacement fields are physically non-physical. Furthermore, standard polyhedral meshes with straight edges introduce geometric approximation errors that can dominate the solution error for higher-order schemes.

Methodology
The authors propose a novel Conforming/Non-conforming (C-NC) Virtual Element Method that allows the user to selectively designate mesh faces as either conforming or non-conforming.

• Hybrid Space Construction: The method defines a local virtual space $W^k_h(E)$ on a polyhedron $E$ by splitting its boundary faces into a conforming set ($\mathcal{F}^c_E$) and a non-conforming set ($\mathcal{F}^n_E$).
  • On conforming faces, the solution is continuous ($C^0$), and the space follows the standard nodal VEM definition (enhanced with polynomial moments).
  • On non-conforming faces, the solution is allowed to be discontinuous, and the normal derivative is constrained to be a polynomial of degree $k-1$.
  • The global space is constructed by enforcing continuity only across the conforming faces. This "hybrid" element is built via local Poisson problems with mixed Dirichlet (on conforming faces) and Neumann (on non-conforming faces) boundary conditions.
• Degrees of Freedom (DoFs): The space is defined by pointwise evaluations at vertices and edge points (on conforming boundaries), and moments of the function and its normal derivative on faces. Crucially, the projections required for the stiffness matrix (specifically the $L^2$ and energy projections) are shown to be computable solely from these DoFs, even with the mixed boundary conditions.
• Application to Curved Domains: The method is applied to 3D linear elasticity on curved domains. The strategy involves:
  • Using planar, conforming faces for the interior of the domain to ensure global displacement continuity.
  • Using curved, non-conforming faces exclusively on the boundary $\partial\Omega$ to represent the exact geometry.
  • Defining the virtual space on curved faces using mapped polynomials via a smooth parametrization $\gamma_F$.
  • Approximating the boundary integrals in the projection operators using the non-conforming DoFs, acknowledging this as a necessary approximation when the normal derivative of the projection is not in the mapped polynomial space.

Key Contributions

1. Novel Hybrid Element: The paper introduces a VEM formulation where the conforming/non-conforming nature of faces is a user-defined choice, enabling a "hybrid" element that recovers classical conforming VEM, non-conforming VEM, or a mix of both.
2. Theoretical Analysis: The authors provide a rigorous stability and convergence analysis for the C-NC space. They prove the equivalence between the $H^1$-seminorm and the DoF-based stabilization, establish interpolation estimates, and derive optimal error bounds for the linear elasticity problem.
3. Curved 3D Elasticity: The paper demonstrates a practical methodology for solving 3D elasticity problems on curved domains with exact geometry representation. Unlike previous 3D curved VEM approaches that result in discontinuous displacements or require complex constructions, this method guarantees a continuous displacement field in the bulk while accurately capturing the curved boundary.
4. Numerical Validation: The method is tested on three scenarios:
   • Standard straight geometries (validating convergence rates).
   • Curved geometries with exact mesh approximation (cylinders).
   • Curved geometries (spheres) tessellated with arbitrary, non-planar polygons using linear and quadratic geometric approximations.

Results
Numerical experiments confirm the theoretical predictions:

• Convergence Rates: For both planar and curved domains, the method achieves optimal convergence rates: $O(h^{k+1})$ in the $L^2$-norm and $O(h^k)$ in the $H^1$-seminorm for polynomial degree $k$.
• Curved Boundary Performance: In the case of a cylinder with exact curved boundary representation, the method maintains optimal rates, outperforming standard second-order finite element schemes on tetrahedral meshes which suffer from geometric approximation errors ($O(h^2)$ and $O(h^{3/2})$).
• Geometric Approximation Limits: For the sphere example, the method successfully handles arbitrary polygonal tessellations. However, the authors observe that for $k=2$, the optimal convergence rate is only recovered if the geometric approximation is sufficiently accurate. When using a quadratic geometric approximation (Quad), the $L^2$ error initially shows a higher order of convergence but eventually plateaus. The authors attribute this to the geometric approximation error (specifically the surface area and volume errors) becoming dominant, noting that the serendipity basis used does not guarantee the $O(h^3)$ geometric error required for optimal $L^2$ convergence with $k=2$ elements.

Significance and Claims
The paper claims that the proposed C-NC VEM offers a flexible and robust framework for 3D elasticity problems on curved domains. Its primary significance lies in the ability to deliver a continuous displacement solution (essential for solid mechanics) while exactly representing curved boundaries using non-conforming faces, a combination that was previously difficult to achieve with standard VEM or FEM approaches. The authors emphasize that while the method introduces an approximation in the projection of boundary integrals on curved faces, this does not compromise the stability or the optimal convergence rates of the method, provided the geometric approximation is adequate. The work serves as a proof of concept for handling complex 3D geometries without resorting to non-physical discontinuous solutions or overly complex master-slave constructions.