$C^1$ virtual element methods on polygonal meshes with curved edges

TL;DR

This work advances numerical analysis for fourth-order PDEs on curved domains by introducing a $C^1$-conforming virtual element method of arbitrary order $k\ge2$ on curved polygonal meshes. The method employs a novel stabilizing form to overcome kernel- and coercivity challenges, extending $C^1$-VEM ideas to curved edges and ensuring optimal $H^2$-norm convergence with rigorous interpolation and stability theory. Theoretical results include interpolation estimates, stability bounds, and a Strang-type a priori error estimate that accounts for geometry-induced nonconformity via a computable $T^h(u)$ term. Numerical experiments on curved domains corroborate the predicted rates and highlight the advantages of curvature-aware VE discretizations over straight-edge domain approximations in preserving high-order accuracy. Overall, the approach provides a robust framework for high-regularity VEM on curved geometries with practical implications for plate bending and other biharmonic-type problems.

Abstract

In this work we design a novel $C^1$-conforming virtual element method of arbitrary order $k \geq 2$, to solve the biharmonic problem on a domain with curved boundary and internal curved interfaces in two dimensions. By introducing a suitable stabilizing form, we develop a rigorous interpolation, stability and convergence analysis obtaining optimal error estimates in the energy norm. Finally, we validate the theoretical findings through numerical experiments.

Figures (6)

• Figure 1: DoFs for $k=3$ (left) and $k=4$ (right). We denote $\boldsymbol{D^{\partial}_{\!I}}$ with the dots, $\boldsymbol{D^{\partial}_{\!I\!I}}$ with the circles, $\boldsymbol{D^{\partial}_{\!I\!I\!I}}$ with the black diamonds, $\boldsymbol{D^{\partial}_{\!I\!V}}$ with the red triangles, $\boldsymbol{D^{\partial}_{\!V}}$ with the blue stars, $\boldsymbol{D^{\partial}_{\!V\!I}}$ with the red squares, and $\boldsymbol{D^o}$ with the green circle.
• Figure 2: The domain $\Omega$ described in \ref{['eq:test1domain']}.
• Figure 3: Example of the adopted curved polygonal meshes: quadrilateral mesh and Voronoi mesh (on the left and right), respectively.
• Figure 4: Errors ${\rm Err}_{2}(u)$, ${\rm Err}_{1}(u)$ and ${\rm Err}_{0}(u)$ for the quadrilateral (curved) and Voronoi (curved) meshes, with $k=3$.
• Figure 5: An example of (straight) Voronoi tessellation over the domain $\Omega$.

Theorems & Definitions (24)

• Remark 2.1
• Remark 2.2
• Theorem 2.1
• Proposition 2.1
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3