A class of nonparametric DSSY nonconforming quadrilateral elements

A new class of nonparametric nonconforming quadrilateral finite elements is introduced which has the midpoint continuity and the mean value continuity at the interfaces of elements simultaneously as the rectangular DSSY element [J.Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye. Nonconforming {G}alerkin methods based on quadrilateral elements for second order elliptic problems. ESAIM--Math. Model. Numer. Anal., 33(4):747--770, 1999]. The parametric DSSY element for general quadrilaterals requires five degrees of freedom to have an optimal order of convergence [Z. Cai, J. Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye. Nonconforming quadrilateral finite elements: A correction. Calcolo, 37(4):253--254, 2000], while the new nonparametric DSSY elements require only four degrees of freedom. The design of new elements is based on the decomposition of a bilinear transform into a simple bilinear map followed by a suitable affine map. Numerical results are presented to compare the new elements with the parametric DSSY element.Comment: 20 page

Optimal Order Convergence Implies Numerical Smoothness

It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise polynomials, that means we have to at least maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements. In this paper we give clear definitions of numerical smoothness that address the across-interface smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in terms of differences in derivative values. Furthermore, we prove rigorously that the principle can be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons and hexahedrons in three dimensions. With this validation we can justify, among other things, incorporation of this principle in creating adaptive numerical approximation for the solution of PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential non-convergence and instability

Some Error Analysis on Virtual Element Methods

Some error analysis on virtual element methods including inverse inequalities, norm equivalence, and interpolation error estimates are presented for polygonal meshes which admits a virtual quasi-uniform triangulation

Finite element differential forms on curvilinear cubic meshes and their approximation properties

We study the approximation properties of a wide class of finite element differential forms on curvilinear cubic meshes in n dimensions. Specifically, we consider meshes in which each element is the image of a cubical reference element under a diffeomorphism, and finite element spaces in which the shape functions and degrees of freedom are obtained from the reference element by pullback of differential forms. In the case where the diffeomorphisms from the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard that the rate of convergence in L2 exceeds by one the degree of the largest full polynomial space contained in the reference space of shape functions. When the diffeomorphism is multilinear, the rate of convergence for the same space of reference shape function may degrade severely, the more so when the form degree is larger. The main result of the paper gives a sufficient condition on the reference shape functions to obtain a given rate of convergence.Comment: 17 pages, 1 figure; v2: changes in response to referee reports; v3: minor additional changes, this version accepted for Numerische Mathematik; v3: very minor updates, this version corresponds to the final published versio

Lowest order Virtual Element approximation of magnetostatic problems

We give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field $\mathbf{H}$ on each edge, and the vertex values of the Lagrange multiplier $p$ (used to enforce the solenoidality of the magnetic induction $\mathbf{B}=\mu\mathbf{H}$). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called "first kind N\'ed\'elec" elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions