𝐶¹ virtual element methods on polygonal meshes with curved edges

In this work we design a novel C 1 C^1 -conforming virtual element method of arbitrary order k ≥ 2 k ≥ 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.