Related papers: Direct Serendipity and Mixed Finite Elements on Co…

In this paper, we use a unified framework introduced in [3] to study two classes of nonconforming immersed finite element (IFE) spaces with integral value degrees of freedom. The shape functions on interface elements are piecewise…

Spline functions have long been used in numerically solving differential equations. Recently it revives as isogeometric analysis, which uses NURBS for both parametrization and element functions. In this paper, we introduce some multivariate…

This paper is devoted to the construction of finite elements on grids that consist of general quadrilaterals not limited in parallelograms. Two finite elements defined as Ciarlet's triple are established for the $H^1$ and $H(\rm rot)$…

This paper constructs a unified family of nonconforming finite element spaces for $H\Lambda^k$ in $\mathbb{R}^n$ ($0\leqslant k\leqslant n$, $n\geqslant 1$). The spaces employ piecewise Whitney forms as shape functions, and include the…

In recent papers the author introduced a simple alternative to isoparametric finite elements of the n-simplex type, to enhance the accuracy of approximations of second-order boundary value problems with Dirichlet conditions, posed in smooth…

We propose a new mixed finite element method for the three-dimensional steady magnetohydrodynamic (MHD) kinematics equations for which the velocity of the fluid is given. Although prescribing the velocity field leads to a simpler model than…

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed…

We propose a new fictitious domain finite element method, well suited for elliptic problems posed in a domain given by a level-set function without requiring a mesh fitting the boundary. To impose the Dirichlet boundary conditions, we…

Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is…

We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate…

The modeling of large deformation fracture mechanics has been a challenging problem regarding the accuracy of numerical methods and their ability to deal with considerable changes in deformations of meshes where having the presence of…

We present a mathematical and algorithmic scheme for learning the principal geometric elements in an image or 3D object. We build on recent work that convexifies the basic problem of finding a combination of a small number shapes that…

The $H^m$-conforming virtual elements of any degree $k$ on any shape of polytope in $\mathbb R^n$ with $m, n\geq1$ and $k\geq m$ are recursively constructed by gluing conforming virtual elements on faces in a universal way. For the lowest…

Finite convex geometries are combinatorial structures. It follows from a recent result of M.\ Richter and L.G.\ Rogers that there is an infinite set $T_{rr}$ of planar convex polygons such that $T_{rr}$ with respect to geometric convex…

We introduce a family of Galerkin finite element methods which are constructed via recovery operators over element-wise discontinuous approximation spaces. This new family, termed collectively as recovered finite element methods (R-FEM) has…

A new numerical method is presented for solving the rotating shallow water equations on a rotating sphere using quasi-uniform polygonal meshes. The method uses special families of finite element function spaces to mimic key mathematical…

Any solid object can be decomposed into a collection of convex polytopes (in short, convexes). When a small number of convexes are used, such a decomposition can be thought of as a piece-wise approximation of the geometry. This…

This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…

This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition. The…

In this work we address the reduction of face degrees of freedom (DOFs) for discrete elasticity complexes. Specifically, using serendipity techniques, we develop a reduced version of a recently introduced two-dimensional complex arising…