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We construct several smooth finite element spaces defined on three--dimensional Worsey--Farin splits. In particular, we construct $C^1$, $H^1(curl)$, and $H^1$-conforming finite element spaces and show the discrete spaces satisfy local exactness properties. A feature of the spaces is their low polynomial degree and lack of extrinsic supersmoothness at sub-simplices of the mesh. In the lowest order case, the last two spaces in the sequence consist of piecewise linear and piecewise constant spaces, and are suitable for the discretization of the (Navier-)Stokes equation.
We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via the linearized deformation operator, the linearized curvature operator,
We consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need to extract the information of the Hessian or to solve a linear system having the H
By improving the trace finite element method, we developed another higher-order trace finite element method by integrating on the surface with exact geometry description. This method restricts the finite element space on the volume mesh to the surfac
Recently developed concept of dissipative measure-valued solution for compressible flows is a suitable tool to describe oscillations and singularities possibly developed in solutions of multidimensional Euler equations. In this paper we study the con
This work introduces and studies a new family of velocity jump Markov processes directly amenable to exact simulation with the following two properties: i) trajectories converge in law when a time-step parameter vanishes towards a given Langevin or H