scholarly journals A New Generalization of the P1 Non-Conforming FEM to Higher Polynomial Degrees

2017 ◽  
Vol 17 (1) ◽  
pp. 161-185 ◽  
Author(s):  
Mira Schedensack
Keyword(s):  
Adaptive Algorithm ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
A Priori ◽  
Projection Property ◽  
Polynomial Degree ◽  
Poisson Problem ◽  
Optimal Convergence Rates ◽  
New Formulation ◽  
Three Space

AbstractThis paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.

scholarly journals A p-robust polygonal discontinuous Galerkin method with minus one stabilization

2021 ◽  
pp. 1-37
Author(s):  
Silvia Bertoluzza ◽  
Ilaria Perugia ◽  
Daniele Prada
Keyword(s):  
Error Analysis ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Numerical Experiments ◽  
Discontinuous Galerkin Methods ◽  
Convergence Rates ◽  
Galerkin Methods ◽  
Poisson Problem ◽  
Optimal Convergence Rates ◽  
Stability And Error Analysis

In this paper, we introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree [Formula: see text]. The stabilization is obtained by penalizing, in each mesh element [Formula: see text], a residual in the norm of the dual of [Formula: see text]. This negative norm is algebraically realized via the introduction of new auxiliary spaces. We carry out a [Formula: see text]-explicit stability and error analysis, proving [Formula: see text]-robustness of the overall method. The theoretical findings are demonstrated in a series of numerical experiments.

AN OPTIMAL LOW-ORDER LOCKING-FREE FINITE ELEMENT METHOD FOR REISSNER–MINDLIN PLATES

1998 ◽  
Vol 08 (03) ◽  
pp. 407-430 ◽  
Author(s):  
D. CHAPELLE ◽  
R. STENBERG
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Convergence Rates ◽  
A Priori ◽  
Mindlin Plate ◽  
Plate Model ◽  
Simple Modification ◽  
Optimal Convergence Rates ◽  
Priori Error Estimates ◽  
Element Method

We propose a simple modification of a recently introduced locking-free finite element method for the Reissner–Mindlin plate model. By this modification, we are able to obtain optimal convergence rates on numerical benchmarks. These results are substantiated by a complete mathematical analysis which provides optimal a priori error estimates.

scholarly journals The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

2019 ◽  
Vol 27 (4) ◽  
pp. 539-557
Author(s):  
Barbara Kaltenbacher ◽  
Andrej Klassen ◽  
Mario Luiz Previatti de Souza
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Newton Method ◽  
Noise Level ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
A Priori ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Source Condition

Abstract In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments.

scholarly journals An unfitted hybrid high-order method for the Stokes interface problem

2020 ◽  
Author(s):  
Erik Burman ◽  
Guillaume Delay ◽  
Alexandre Ern
Keyword(s):  
Numerical Simulations ◽  
Convergence Rates ◽  
A Priori ◽  
High Order ◽  
Interface Problem ◽  
Order Method ◽  
High Order Method ◽  
Optimal Convergence Rates ◽  
A Cell ◽  
Priori Error Estimates

Abstract We design and analyze a hybrid high-order method on unfitted meshes to approximate the Stokes interface problem. The interface can cut through the mesh cells in a very general fashion. A cell-agglomeration procedure prevents the appearance of small cut cells. Our main results are inf-sup stability and a priori error estimates with optimal convergence rates in the energy norm. Numerical simulations corroborate these results.

scholarly journals Instance-Optimal Goal-Oriented Adaptivity

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Michael Innerberger ◽  
Dirk Praetorius
Keyword(s):  
Adaptive Algorithm ◽  
Numerical Experiments ◽  
Dual Problem ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
Polynomial Degree ◽  
Energy Error ◽  
Instance Optimality ◽  
Edge Based

AbstractWe consider an adaptive finite element method with arbitrary but fixed polynomial degree {p\geq 1}, where adaptivity is driven by an edge-based residual error estimator. Based on the modified maximum criterion from [L. Diening, C. Kreuzer and R. Stevenson, Instance optimality of the adaptive maximum strategy, Found. Comput. Math. 16 2016, 1, 33–68], we propose a goal-oriented adaptive algorithm and prove that it is instance optimal. More precisely, the goal error is bounded by the product of the total errors (being the sum of energy error plus data oscillations) of the primal and the dual problem, and the proposed algorithm is instance optimal with respect to this upper bound. Numerical experiments underline our theoretical findings.

A NONCONFORMING PENALTY METHOD FOR A TWO-DIMENSIONAL CURL–CURL PROBLEM

2009 ◽  
Vol 19 (04) ◽  
pp. 651-668 ◽  
Author(s):  
SUSANNE C. BRENNER ◽  
FENGYAN LI ◽  
LI-YENG SUNG
Keyword(s):  
Numerical Experiments ◽  
Vector Fields ◽  
Convergence Rates ◽  
Two Dimensional ◽  
Weakly Continuous ◽  
Graded Meshes ◽  
Optimal Convergence Rates ◽  
Maxwell Eigenvalues ◽  
Theoretical Results ◽  
Convergence Of The Scheme

A nonconforming finite element method for a two-dimensional curl–curl problem is studied in this paper. It uses weakly continuous P1 vector fields and penalizes the local divergence. Two consistency terms involving the jumps of the vector fields across element boundaries are also included to ensure the convergence of the scheme. Optimal convergence rates (up to an arbitrary positive ∊) in both the energy norm and the L2 norm are established on graded meshes. This scheme can also be used in the computation of Maxwell eigenvalues without generating spurious eigenmodes. The theoretical results are confirmed by numerical experiments.

scholarly journals The parabolic p-Laplacian with fractional differentiability

2020 ◽  
Author(s):  
Dominic Breit ◽  
Lars Diening ◽  
Johannes Storn ◽  
Jörn Wichmann
Keyword(s):  
Numerical Experiments ◽  
Fractional Derivatives ◽  
Convergence Rates ◽  
Time Discretization ◽  
Coupling Condition ◽  
Implicit Euler Scheme ◽  
Space And Time ◽  
Optimal Convergence Rates ◽  
Fractional Differentiability ◽  
Nikolskii Spaces

Abstract We study the parabolic $p$-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space–time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskiǐ spaces and therefore cover situations when the (gradient of the) solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolutions $h$ and $\tau $. For this we show that the $L^2$-projection is compatible with the quasi-norm. The theoretical error analysis is complemented by numerical experiments.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

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