# 06th October – Rekha Khot: Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes

Rekha Khot Thursday 06th Oct at 15:00

ABSTRACT:

The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution $u \in V := H_0^2(Ω)$ to the biharmonic equation. Two nonconforming virtual element spaces have been introduced in [1, 4] for $H^3$ regular solutions, while a medius analysis in [3] allows minimal regularity. In this talk, we will discuss an abstract framework (cf. [2]) with two hypotheses (H1)-(H2) for unified stability and a priori error analysis of at least two different discrete spaces (even a mixture of those). A smoother J allows rough source terms $F \in V^∗ = H^{−2}(\Omega)$. The a priori and a posteriori error analysis circumvents any trace of second derivatives by a computable conforming companion operator $J : V_h \rightarrow V$ from the nonconforming virtual element space $V_h$. The operator J is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on $$u \in V$$. As a smoother the companion operator modifies the discrete right-hand side and then allows a quasi-best approximation. An explicit residual-based a posteriori error estimator is reliable and efficient up to data oscillations. Numerical examples display the predicted empirical convergence rates for uniform and optimal convergence rates for adaptive mesh-refinement.

[1] P. F. Antonietti, G. Manzini, and M. Verani, The fully nonconforming virtual element method for biharmonic problems, Math. Models Methods Appl. Sci., 28 (2018), pp. 387–407.

[2] C. Carstensen, R. Khot, and A. K. Pani, Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes, arXiv:2205.08764, (2022).

[3] J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math., 386 (2021), p. 21.

[4] J. Zhao, B. Zhang, S. Chen, and S. Mao, The Morley-type virtual element for plate bending problems, J. Sci. Comput., 76 (2018), pp. 610–629.