Robust nonconforming virtual element methods for general fourth order problems with varying coefficients

TL;DR

This paper introduces a new class of nonconforming virtual element methods for solving general fourth order PDEs with varying coefficients, providing optimal error estimates and robustness for perturbation problems.

Contribution

The paper develops a generic framework for constructing virtual element spaces and projection operators for fourth order problems, including a novel scheme that is uniformly convergent under perturbations.

Findings

Optimal error estimates in the energy norm are achieved.

The proposed method is robust for perturbation problems without enlarging the space.

Numerical tests confirm the theoretical convergence and robustness.

Abstract

We present a class of nonconforming virtual element methods for general fourth order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth order problems with varying coefficients. We also discuss fourth order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth order problems.