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Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension n, arbitrary polynomial degree r, and arbitrary differential form degree k. The study of finite element exterior calculus began with the [Formula omitted] and [Formula omitted] families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between [Formula omitted] and [Formula omitted] and between [Formula omitted] and [Formula omitted]. In this article, we show that this duality relationship is, in essence, Hodge duality of differential forms on the standard n-sphere, disguised by a change of coordinates. We remove the disguise, giving explicit correspondences between the [Formula omitted], [Formula omitted], [Formula omitted] and [Formula omitted] spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between [Formula omitted] and [Formula omitted] and between [Formula omitted] and [Formula omitted]. These isomorphisms can be implemented via a simple computation, which we illustrate with examples., Author(s): Yakov Berchenko-Kogan [sup.1] Author Affiliations: (1) Honolulu, USA Introduction The finite element method is a tool for solving partial differential equations numerically that approximates solutions to the PDE by [...]

We design and analyze several finite element methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains [Formula omitted] with [Formula omitted]. For the solution to the Caffarelli-Silvestre extension, we establish analytic regularity with respect to the extended variable [Formula omitted]. Specifically, the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to y, taking values in corner-weighted Kondrat'ev-type Sobolev spaces in [Formula omitted]. In [Formula omitted], we discretize with continuous, piecewise linear, Lagrangian FEM ( [Formula omitted]-FEM) with mesh refinement near corners and prove that the first-order convergence rate is attained for compatible data [Formula omitted] with [Formula omitted] denoting the fractional power. We also prove that tensorization of a [Formula omitted]-FEM in [Formula omitted] with a suitable hp-FEM in the extended variable achieves log-linear complexity with respect to [Formula omitted], the number of degrees of freedom in the domain [Formula omitted]. In addition, we propose a novel, sparse tensor product FEM based on a multilevel [Formula omitted]-FEM in [Formula omitted] and on a [Formula omitted]-FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to [Formula omitted]. Finally, under the stronger assumption that the data be analytic in [Formula omitted], and without compatibility at [Formula omitted], we establish exponential rates of convergence ofhp-FEM for spectral fractional diffusion operators in energy norm. This is achieved by a combined tensor product hp-FEM for the Caffarelli-Silvestre extension in the truncated cylinder [Formula omitted] with anisotropic geometric meshes that are refined toward [Formula omitted]. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods to other problem classes and to other boundary conditions on [Formula omitted]., Author(s): Lehel Banjai [sup.1] , Jens M. Melenk [sup.2] , Ricardo H. Nochetto [sup.3] , Enrique Otárola [sup.4] , Abner J. Salgado [sup.5] , Christoph Schwab [sup.6] Author Affiliations: (Aff1) [...]

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form [Formula omitted] in a bounded but not necessarily convex domain [Formula omitted] and study it in the max norm. The fine scale is given by the meshsize h, whereas the coarse scale [Formula omitted] is dictated by an integro-differential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle for any uniformly positive definite matrix A provided that the mesh is face weakly acute. We establish a discrete Alexandroff-Bakelman-Pucci (ABP) estimate which is suitable for finite element analysis. Its proof relies on a discrete Alexandroff estimate which expresses the min of a convex piecewise linear function in terms of the measure of its sub-differential, and thus of jumps of its gradient. The discrete ABP estimate leads, under suitable regularity assumptions on A and u, to pointwise error estimates of the form ||u-uhϵ||L[infinity](Ω)[less than or equal to]C(A,u)h2[alpha]/(2+[alpha])|lnh|0[alpha][less than or equal to]2,provided [Formula omitted]. Such a convergence rate is at best of order [Formula omitted], which turns out to be quasi-optimal., Author(s): Ricardo H. Nochetto [sup.1] , Wujun Zhang [sup.2] Author Affiliations: (Aff1) 0000 0001 0941 7177, grid.164295.d, Department of Mathematics, Institute for Physical Science and Technology, University of Maryland, , [...]

We study convergence of a mixed finite element-finite volume numerical scheme for the isentropic Navier-Stokes system under the full range of the adiabatic exponent. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solutions of the limit system. In particular, using the recently established weak-strong uniqueness principle in the class of dissipative measure-valued solutions we show that the numerical solutions converge strongly to a strong solutions of the limit system as long as the latter exists., Author(s): Eduard Feireisl [sup.1] , Mária Lukácová-Medvid'ová [sup.2] Author Affiliations: (Aff1) Institute of Mathematics of the Academy of Sciences of the Czech Republic, , Zitná 25, 115 67, Praha 1, [...]

We prove exponential rates of convergence of hp-version finite element methods on geometric meshes consisting of hexahedral elements for linear, second-order elliptic boundary value problems in axiparallel polyhedral domains. We extend and generalize our earlier work for homogeneous Dirichlet boundary conditions and uniform isotropic polynomial degrees to mixed Dirichlet-Neumann boundary conditions and to anisotropic, which increase linearly over mesh layers away from edges and vertices. In particular, we construct [Formula omitted]-conforming quasi-interpolation operators with N degrees of freedom and prove exponential consistency bounds [Formula omitted] for piecewise analytic functions with singularities at edges, vertices and interfaces of boundary conditions, based on countably normed classes of weighted Sobolev spaces with non-homogeneous weights in the vicinity of Neumann edges., Author(s): Dominik Schötzau [sup.1] , Christoph Schwab [sup.2] Author Affiliations: (Aff1) 0000 0001 2288 9830, grid.17091.3e, Mathematics Department, University of British Columbia, , 1984 Mathematics Road, V6T 1Z2, Vancouver, BC, [...]

Using the notion of multivariate lower set interpolation, we construct nodal basis functions for the serendipity family of finite elements, of any order and any dimension. For the purpose of computation, we also show how to express these functions as linear combinations of tensor-product polynomials., Author(s): Michael S. Floater [sup.1] , Andrew Gillette [sup.2] Author Affiliations: (1) Department of Mathematics, University of Oslo, 0000 0004 1936 8921, grid.5510.1, , PO Box 1053, Blindern, 0316, Oslo, [...]

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether's first theorem (1918 (See CR6)). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional., Author(s): Elizabeth L. Mansfield [sup.1] , Tristan Pryer [sup.2] Author Affiliations: (1) 0000 0001 2232 2818grid.9759.2School of Mathematics, Statistics and Actuarial Science, University of Kent, CT2 7NF, Canterbury, UK (2) [...]