Virtual Element Method for fourth order problems: L2-estimates

A virtual element method for the von Kármán equations

“…One of its main characteristics is the possibility to construct and implement in an easy way discrete subspaces of , ∈ N. In recent years, the Virtual Elements Method has been a focus of great interest in the scientific community. Several virtual element methods based on conforming and non-conforming schemes have been developed to solve a wide variety of problems in Solid and Fluid Mechanics, for example [4][5][6]9,11,12,14,19,25,27,30,42,46,47]. Moreover, the VEM for thin structures has been developed in [16,24,29,30,44,45], whereas VEM for nonlinear problems have been introduced in [3,15,26,35,36,50] In this paper, we analyze a conforming 1 Virtual Element Method to approximate the isolated solutions of the von Kármán equations.…”

“…We consider a variational formulation in terms of the transverse displacement and the Airy stress function, which contains bilinear and trilinear forms. After introducing the local and global virtual space [5,24,30], we write the discrete problem by constructing discrete version of the bilinear and trilinear forms considering different projectors (polynomial functions) which are computable using only the information of the degrees of freedom of the discrete virtual space. For the analysis, we will adapt some ideas presented in [23] to deal with the variational crimes in the forms and in the right hand side.…”

The conforming virtual element method for polyharmonic problems

“…The virtual element method combines a great flexibility in using polytopal meshes with a great versatility and easiness in designing approximation spaces with high-order continuity properties on general polytopal meshes. These two features turn out to be essential in the numerical treatment of the classical plate bending problem, for which a 1 -regular conforming virtual element approximation has been introduced in [36,50]. Virtual elements with 1 -regularity have been proposed to solve elliptic problems on polygonal meshes [24] and polyedral meshes in [20], the transmission eigenvalue problem in [78], the vibration problem of Kirchhoff plates in [77], the buckling problem of Kirchhoff-Love plates in [79].…”

“…Historically, the numerical approximation of polyharmonic problems dates back to the eighties [32], and more recently, this problem has been addressed in the context of the finite element method by [13,64,91,88,59]. The conforming virtual element approximation of the biharmonic problem has been addressed in [36,50]. while a non-conforming approximation has been proposed in [94,8,95].…”

Virtual Elements for the Reissner‐Mindlin plate problem

“…The Virtual element method, that is not the only recent method that can make use of polytopal meshes, has obtained an increasing interest and it has been developed in many aspects and applied to many different problems (see e.g., ). In particular it has been applied to the approximation of the plate bending problem in the Kirchhoff‐Love formulation (see ). Recently, in , a virtual element method for the approximation of the Reissner‐Mindlin plate problem has been introduced.…”