Serendipity Face and Edge VEM Spaces

We extend the basic idea of Serendipity Virtual Elements from the previous case (by the same authors) of nodal ($H^1$-conforming) elements, to a more general framework. Then we apply the general strategy to the case of $H(div)$ and $H(curl)$ conforming Virtual Element Methods, in two and three dimensions

Serendipity Nodal VEM spaces

We introduce a new variant of Nodal Virtual Element spaces that mimics the "Serendipity Finite Element Methods" (whose most popular example is the 8-node quadrilateral) and allows to reduce (often in a significant way) the number of internal degrees of freedom. When applied to the faces of a three-dimensional decomposition, this allows a reduction in the number of face degrees of freedom: an improvement that cannot be achieved by a simple static condensation. On triangular and tetrahedral decompositions the new elements (contrary to the original VEMs) reduce exactly to the classical Lagrange FEM. On quadrilaterals and hexahedra the new elements are quite similar (and have the same amount of degrees of freedom) to the Serendipity Finite Elements, but are much more robust with respect to element distortions. On more general polytopes the Serendipity VEMs are the natural (and simple) generalization of the simplicial case

Equilibrium analysis of an immersed rigid leaflet by the virtual element method

We study, both theoretically and numerically, the equilibrium of a hinged rigid leaflet with an attached rotational spring, immersed in a stationary incompressible fluid within a rigid channel. Through a careful investigation of the properties of the domain functional describing the angular momentum exerted by the fluid on the leaflet (which depends on both the leaflet angular position and its thickness), we identify sufficient conditions on the spring stiffness function for the existence (and uniqueness) of equilibrium positions. This study resorts to techniques from shape differential calculus. We propose a numerical technique that exploits the mesh flexibility of the Virtual Element Method (VEM). A (polygonal) computational mesh is generated by cutting a fixed background grid with the leaflet geometry, and the problem is then solved with stable VEM Stokes elements of degrees 1 and 2 combined with a bisection algorithm. We prove quasi-optimal error estimates and present a large array of numerical experiments to document the accuracy and robustness with respect to degenerate geometry of the proposed methodology

A stream virtual element formulation of the Stokes problem on polygonal meshes

In this paper we propose and analyze a novel stream formulation of the virtual element method (VEM) for the solution of the Stokes problem. The new formulation hinges upon the introduction of a suitable stream function space (characterizing the divergence free subspace of discrete velocities) and it is equivalent to the velocity-pressure (inf-sup stable) mimetic scheme presented in [L. Beir\ue3o da Veiga et al., J. Comput. Phys., 228(2009), pp. 7215-7232] (up to a suitable reformulation into the VEM framework). Both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem. Several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one

Overlapping schwarz methods for isogeometric analysis

We construct and analyze an overlapping Schwarz preconditioner for elliptic problems discretized with isogeometric analysis. The preconditioner is based on partitioning the domain of the problem into overlapping subdomains, solving local isogeometric problems on these subdomains, and solving an additional coarse isogeometric problem associated with the subdomain mesh. We develop an $h$-analysis of the preconditioner, showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and \u201eoverlap sizes\u201d for fixed polynomial degree $p$ and regularity $k$ of the basis functions. Numerical results in two- and three-dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters $h, p, k$, number of subdomains $N$, overlap size, and also jumps in the coefficients of the elliptic operator

Mimetic finite difference method for the stokes problem on polygonal meshes

Various approaches to extend the finite element methods to non-traditional elements (pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with low-order finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we develop a MFD method for the Stokes problem on arbitrary polygonal meshes. The method is constructed for tensor coefficients, which will allow to apply it to the linear elasticity problem. The numerical experiments show the second-order convergence for the velocity variable and the first-order for the pressure