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 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.

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 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.

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.

Wavelet regression estimation over Lp risk based on negatively associated sample

2020 ◽  
Vol 18 (04) ◽  
pp. 2050024
Author(s):  
Huijun Guo ◽  
Junke Kou
Keyword(s):  
Convergence Rates ◽  
Regression Function ◽  
Regression Estimation ◽  
Negatively Associated ◽  
Wavelet Regression ◽  
Nonparametric Regression Estimation ◽  
Independent Case ◽  
Optimal Convergence Rates ◽  
Negatively Associated Sample ◽  
Wavelet Estimators

This paper considers wavelet estimations of a regression function based on negatively associated sample. We provide upper bound estimations over [Formula: see text] risk of linear and nonlinear wavelet estimators in Besov space, respectively. When the random sample reduces to the independent case, our convergence rates coincide with the optimal convergence rates of classical nonparametric regression estimation.

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