Related papers: Bernstein-Bezier Bases for Tetrahedral Finite Elem…
A simple MATLAB implementation of hierarchical shape functions on 2D rectangles is explained and available for download. Global shape functions are ordered for a given polynomial degree according to the indices of the nodes, edges, or…
We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across…
We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in…
The construction of parametric curve and surface plays important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with…
This paper introduces a novel tangential-normal ($t$-$n$) decomposition for finite element differential forms, presenting a new framework for constructing bases in finite element exterior calculus. The main contribution is the development…
We develop an efficient $hp$-finite element method for piecewise-smooth differential equations with periodic boundary conditions, using orthogonal polynomials defined on circular arcs. The operators derived from this basis are banded and…
The aim of this note is to introduce a compound basis for the space of symmetric functions. Our basis consists of products of Schur functions and $Q$-functions. The basis elements are indexed by the partitions. It is well known that the…
The classical Haar construction of Brownian motion uses a binary tree of triangular wedge-shaped functions. This basis has compactness properties which make it especially suited for certain classes of numerical algorithms. We present a…
This paper extends the Bernstein-Gelfand-Gelfand (BGG) framework to the construction of finite element conformal Hessian complexes and conformal elasticity complexes in three dimensions involving conformal tensors (i.e., symmetric and…
The paper extends the formulation of a 2D geometrically exact beam element proposed in our previous paper [1] to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic…
The Bernstein polynomial basis sees significant use owing to its unique properties, particularly in the field of optimal control. However, the basis is known to have a slow rate of convergence to the function it approximates. With this in…
This paper extends and analyzes the high-order kernel regularization framework of Beale & Tlupova (arXiv:2510.13639) to all four on-surface boundary integral operators of the Helmholtz Calderon calculus in three dimensions: the…
We prove that the minimizer in the N\'ed\'elec polynomial space of some degree p > 0 of a discrete minimization problem performs as well as the continuous minimizer in H(curl), up to a constant that is independent of the polynomial degree…
Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution…
Understanding the role that subgradients play in various second-order variational analysis constructions can help us uncover new properties of important classes of functions in variational analysis. Focusing mainly on the behavior of the…
In this article, a family of $H^2$-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3D. In the family, the $P_\ell$ polynomial space is enriched by some high order polynomials for all…
We consider and provide an accurate study for the fractional Zernike functions on the punctured unit disc, generalizing the classical Zernike polynomials and their associated $\beta$-restricted Zernike functions. Mainly, we give the…
We solve elliptic systems of equations posed on highly heterogeneous materials. Examples of this class of problems are composite structures and geological processes. We focus on a model problem which is a second-order elliptic equation with…
We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree p. We show that the discrete minimizers in the spaces of piecewise…
This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the…