scholarly journals Recovered finite element methods on polygonal and polyhedral meshes

2020 ◽  
Vol 54 (4) ◽  
pp. 1309-1337
Author(s):  
Zhaonan Dong ◽  
Emmanuil H. Georgoulis ◽  
Tristan Pryer
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Diffusion Reaction ◽  
Galerkin Methods ◽  
Polyhedral Meshes ◽  
Order Polynomial ◽  
Spatial Dimensions ◽  
Practical Performance

Recovered Finite Element Methods (R-FEM) have been recently introduced in Georgoulis and Pryer [Comput. Methods Appl. Mech. Eng. 332 (2018) 303–324]. for meshes consisting of simplicial and/or box-type elements. Here, utilising the flexibility of the R-FEM framework, we extend their definition to polygonal and polyhedral meshes in two and three spatial dimensions, respectively. An attractive feature of this framework is its ability to produce arbitrary order polynomial conforming discretizations, yet involving only as many degrees of freedom as discontinuous Galerkin methods over general polygonal/polyhedral meshes with potentially many faces per element. A priori error bounds are shown for general linear, possibly degenerate, second order advection-diffusion-reaction boundary value problems. A series of numerical experiments highlight the good practical performance of the proposed numerical framework.

scholarly journals Stabilized Mixed Finite Element Methods for Linear Elasticity on Simplicial Grids in ℝn

2017 ◽  
Vol 17 (1) ◽  
pp. 17-31 ◽  
Author(s):  
Long Chen ◽  
Jun Hu ◽  
Xuehai Huang
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Linear Elasticity ◽  
Degrees Of Freedom ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Methods ◽  
Interpolation Operators ◽  
Stabilization Technique ◽  
Stabilized Mixed Finite Element

AbstractIn this paper, we design two classes of stabilized mixed finite element methods for linear elasticity on simplicial grids. In the first class of elements, we use ${\boldsymbol{H}(\operatorname{div},\Omega;\mathbb{S})}$-${P_{k}}$ and ${\boldsymbol{L}^{2}(\Omega;\mathbb{R}^{n})}$-${P_{k-1}}$ to approximate the stress and displacement spaces, respectively, for ${1\leq k\leq n}$, and employ a stabilization technique in terms of the jump of the discrete displacement over the edges/faces of the triangulation under consideration; in the second class of elements, we use ${\boldsymbol{H}_{0}^{1}(\Omega;\mathbb{R}^{n})}$-${P_{k}}$ to approximate the displacement space for ${1\leq k\leq n}$, and adopt the stabilization technique suggested by Brezzi, Fortin, and Marini [19]. We establish the discrete inf-sup conditions, and consequently present the a priori error analysis for them. The main ingredient for the analysis are two special interpolation operators, which can be constructed using a crucial ${\boldsymbol{H}(\operatorname{div})}$ bubble function space of polynomials on each element. The feature of these methods is the low number of global degrees of freedom in the lowest order case. We present some numerical results to demonstrate the theoretical estimates.

scholarly journals Energy Contraction and Optimal Convergence of Adaptive Iterative Linearized Finite Element Methods

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Pascal Heid ◽  
Dirk Praetorius ◽  
Thomas P. Wihler
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Degrees Of Freedom ◽  
Linear Convergence ◽  
Nonlinear Problems ◽  
Iterative Solution ◽  
Computational Time ◽  
Galerkin Methods ◽  
Adaptive Finite Element ◽  
Adaptive Finite Element Methods

Abstract We revisit a unified methodology for the iterative solution of nonlinear equations in Hilbert spaces. Our key observation is that the general approach from [P. Heid and T. P. Wihler, Adaptive iterative linearization Galerkin methods for nonlinear problems, Math. Comp. 89 2020, 326, 2707–2734; P. Heid and T. P. Wihler, On the convergence of adaptive iterative linearized Galerkin methods, Calcolo 57 2020, Paper No. 24] satisfies an energy contraction property in the context of (abstract) strongly monotone problems. This property, in turn, is the crucial ingredient in the recent convergence analysis in [G. Gantner, A. Haberl, D. Praetorius and S. Schimanko, Rate optimality of adaptive finite element methods with respect to the overall computational costs, preprint 2020]. In particular, we deduce that adaptive iterative linearized finite element methods (AILFEMs) lead to full linear convergence with optimal algebraic rates with respect to the degrees of freedom as well as the total computational time.

The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

2020 ◽  
Vol 1 (1) ◽  
pp. 93-102
Author(s):  
Carsten Strzalka ◽  
◽  
Manfred Zehn ◽  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
A Priori ◽  
High Stress ◽  
Von Mises Stress ◽  
Detailed Knowledge ◽  
Variable Loading ◽  
Numerical Effort ◽  
Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

scholarly journals Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

2020 ◽  
Vol 85 (2) ◽  
Author(s):  
R. Abgrall ◽  
J. Nordström ◽  
P. Öffner ◽  
S. Tokareva
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Discontinuous Galerkin ◽  
Finite Element Methods ◽  
Main Idea ◽  
Galerkin Methods ◽  
Exact Integration ◽  
Galerkin Scheme ◽  
Continuous Galerkin ◽  
Continuous Galerkin Methods

AbstractIn the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

scholarly journals Adaptive modelling of variably saturated seepage problems

2021 ◽  
Author(s):  
B Ashby ◽  
C Bortolozo ◽  
A Lukyanov ◽  
T Pryer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Water Flux ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
Element Method

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.

Sign in / Sign up

Export Citation Format

Share Document