scholarly journals Local finite element approximation of Sobolev differential forms 

2021 ◽  
Author(s):  
Evan Gawlik ◽  
Michael Holst ◽  
Martin Werner Licht
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Differential Forms ◽  
Mesh Size ◽  
Finite Element Approximation ◽  
Local Error ◽  
Element Approximation ◽  
Exterior Calculus ◽  
Vector Valued ◽  
Theoretical Results

We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.

Semi-Discrete and Fully Discrete Hybrid Stress Finite Element Methods for Elastodynamic Problems

2015 ◽  
Vol 8 (4) ◽  
pp. 582-604
Author(s):  
Zhengqin Yu ◽  
Xiaoping Xie
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Stability Result ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Quadrilateral Finite Element ◽  
Hybrid Stress Finite Element ◽  
Hybrid Stress

AbstractThis paper proposes and analyzes semi-discrete and fully discrete hybrid stress finite element methods for elastodynamic problems. A hybrid stress quadrilateral finite element approximation is used in the space directions. A second-order center difference is adopted in the time direction for the fully discrete scheme. Error estimates of the two schemes, as well as a stability result for the fully discrete scheme, are derived. Numerical experiments are done to verify the theoretical results.

Finite element approximation of eigenvalue problems

Acta Numerica ◽  
2010 ◽  
Vol 19 ◽  
pp. 1-120 ◽  
Author(s):  
Daniele Boffi
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Differential Forms ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Galerkin Approximations ◽  
Laplace Eigenvalue ◽  
Main Emphasis ◽  
Adjoint Operators

We discuss the finite element approximation of eigenvalue problems associated with compact operators. While the main emphasis is on symmetric problems, some comments are present for non-self-adjoint operators as well. The topics covered include standard Galerkin approximations, non-conforming approximations, and approximation of eigenvalue problems in mixed form. Some applications of the theory are presented and, in particular, the approximation of the Maxwell eigenvalue problem is discussed in detail. The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Several examples and numerical computations complete the paper, ranging from very basic exercises to more significant applications of the developed theory.

scholarly journals Fully Discrete Finite Element Methods for Two-Dimensional Bingham Flows

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Cheng Fang ◽  
Yuan Li
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Piecewise Linear ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Two Dimensional ◽  
Stabilized Finite Element ◽  
Fully Discrete ◽  
Stabilized Finite Element Method ◽  
Discrete Finite Element

This paper presents fully discrete stabilized finite element methods for two-dimensional Bingham fluid flow based on the method of regularization. Motivated by the Brezzi-Pitkäranta stabilized finite element method, the equal-order piecewise linear finite element approximation is used for both the velocity and the pressure. Based on Euler semi-implicit scheme, a fully discrete scheme is introduced. It is shown that the proposed fully discrete stabilized finite element scheme results in the h1/2 error order for the velocity in the discrete norms corresponding to L2(0,T;H1(Ω)2)∩L∞(0,T;L2(Ω)2).

scholarly journals Finite element approximation of the Isaacs equation

2019 ◽  
Vol 53 (2) ◽  
pp. 351-374
Author(s):  
Abner J. Salgado ◽  
Wujun Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mesh Size ◽  
Finite Element Approximation ◽  
Rates Of Convergence ◽  
Discrete Maximum Principle ◽  
Element Approximation ◽  
Differential Approximation ◽  
Isaacs Equation ◽  
Element Method

We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh size h whereas the coarse scale ε is dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical solution to the viscosity solution as ε, h → 0, and ε ≳ (h|log h|)1/2. In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution.

An Angular Finite Element Method for the Solution of the Even-Parity Equation

2009 ◽  
Author(s):  
R. Becker ◽  
R. Koch ◽  
M. F. Modest ◽  
H.-J. Bauer
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Basis Functions ◽  
New Method ◽  
Discrete Ordinates ◽  
Test Case ◽  
Element Approximation ◽  
Promising Alternative ◽  
Element Discretization ◽  
Spatial Coordinates

The present article introduces a new method to solve the radiative transfer equation (RTE). First, a finite element discretization of the solid angle dependence is derived, wherein the coefficients of the finite element approximation are functions of the spatial coordinates. The angular basis functions are defined according to finite element principles on subdivisions of the octahedron. In a second step, these spatially dependent coefficients are discretized by spatial finite elements. This approach is very attractive, since it provides a concise derivation for approximations of the angular dependence with an arbitrary number of angular nodes. In addition, the usage of high-order angular basis functions is straightforward. In the current paper the governing equations are first derived independently of the actual angular approximation. Then, the design principles for the angular mesh are discussed and the parameterization of the piecewise angular basis functions is derived. In the following, the method is applied to two-dimensional test cases which are commonly used for the validation of approximation methods of the RTE. The results reveal that the proposed method is a promising alternative to the well-established practices like the Discrete Ordinates Method (DOM) and provides highly accurate approximations. A test case known to exhibit the ray effect in the DOM verifies the ability of the new method to avoid ray effects.

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