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 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.

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 Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

2017 ◽  
Vol 17 (3) ◽  
pp. 479-498 ◽  
Author(s):  
Raphael Kruse ◽  
Yue Wu
Keyword(s):  
Differential Equations ◽  
Error Analysis ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Quadrature Rule ◽  
Time Variable ◽  
Important Ingredient ◽  
Riemann Sum ◽  
Runge Kutta ◽  
Theoretical Results

AbstractThis paper contains an error analysis of two randomized explicit Runge–Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable. The main result consists of precise bounds for the discretization error with respect to the {L^{p}(\Omega;{\mathbb{R}}^{d})}-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.

scholarly journals Quadrature Rules and Iterative Method for Numerical Solution of Two-Dimensional Fuzzy Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
S. M. Sadatrasoul ◽  
R. Ezzati
Keyword(s):  
Integral Equations ◽  
Iterative Procedure ◽  
Numerical Experiments ◽  
Quadrature Rules ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Two Dimensional ◽  
Henstock Integral ◽  
Uniform Modulus Of Continuity ◽  
Theoretical Results

We introduce some generalized quadrature rules to approximate two-dimensional, Henstock integral of fuzzy-number-valued functions. We also give error bounds for mappings of bounded variation in terms of uniform modulus of continuity. Moreover, we propose an iterative procedure based on quadrature formula to solve two-dimensional linear fuzzy Fredholm integral equations of the second kind (2DFFLIE2), and we present the error estimation of the proposed method. Finally, some numerical experiments confirm the theoretical results and illustrate the accuracy of the method.

Convergence of the Preconditioned AOR Method for an H-Matrix

2013 ◽  
Vol 756-759 ◽  
pp. 2615-2619
Author(s):  
Jie Jing Liu
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Triangular Matrix ◽  
Coefficient Matrix ◽  
Aor Method ◽  
Upper Triangular Matrix ◽  
Theoretical Results

Linear system with H-matrix often appears in a wide variety of areas and is studied by many numerical researchers. In order to improve the convergence rates of iterative method solving the linear system whose coefficient matrix is an H-matrix. In this paper, a preconditioned AOR iterative method with a multi-parameters preconditioner with a general upper triangular matrix is proposed. In addition, the convergence of the coressponding iterative method are established. Lastly, we provide numerical experiments to illustrate the theoretical results.

scholarly journals On the Application of the Generalized Means to Construct Multiresolution Schemes Satisfying Certain Inequalities Proving Stability

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 533
Author(s):  
Sergio Amat ◽  
Alberto Magreñan ◽  
Juan Ruiz ◽  
Juan Carlos Trillo ◽  
Dionisio F. Yañez
Keyword(s):  
Numerical Experiments ◽  
Order Of Approximation ◽  
Two Dimensional ◽  
Nonlinear Subdivision ◽  
Proper Definition ◽  
The Stability ◽  
Definition Of ◽  
Stability Issues ◽  
Multiresolution Schemes ◽  
Theoretical Results

Multiresolution representations of data are known to be powerful tools in data analysis and processing, and they are particularly interesting for data compression. In order to obtain a proper definition of the edges, a good option is to use nonlinear reconstructions. These nonlinear reconstruction are the heart of the prediction processes which appear in the definition of the nonlinear subdivision and multiresolution schemes. We define and study some nonlinear reconstructions based on the use of nonlinear means, more in concrete the so-called Generalized means. These means have two interesting properties that will allow us to get associated reconstruction operators adapted to the presence of discontinuities, and having the maximum possible order of approximation in smooth areas. Once we have these nonlinear reconstruction operators defined, we can build the related nonlinear subdivision and multiresolution schemes and prove more accurate inequalities regarding the contractivity of the scheme for the first differences and in turn the results about stability. In this paper, we also define a new nonlinear two-dimensional multiresolution scheme as non-separable, i.e., not based on tensor product. We then present the study of the stability issues for the scheme and numerical experiments reinforcing the proven theoretical results and showing the usefulness of the algorithm.

scholarly journals Comparative study of the behavior of the effective boundary of minimal risk portfolio in the conditions of hybrid uncertainty, depending on the restrictions on the profitability of the portfolio

2021 ◽  
pp. 58-69
Author(s):  
Александр Васильевич Язенин ◽  
Илья Сергеевич Солдатенко
Keyword(s):  
Comparative Study ◽  
Numerical Experiments ◽  
Minimal Risk ◽  
Two Dimensional ◽  
Probability Level ◽  
Expected Return ◽  
Minimum Risk ◽  
Effective Boundary ◽  
Hybrid Uncertainty ◽  
Theoretical Results

В работе проведены исследования эффективной границы портфеля минимального риска в условиях гибридной неопределенности. Для случая двумерного портфеля при ограничении на ожидаемую доходность портфеля и ограничении по возможности/необходимости и вероятности на доходность портфеля в зависимости от уровня вероятности построены квазиэффективные границы портфеля. Результаты численных экспериментов согласуются с ранее полученными авторами теоретическими результатами. The paper studies the effective boundary of the minimum risk portfolio in the conditions of hybrid uncertainty. For the case of a two-dimensional portfolio, with a restriction on the expected return of the portfolio and a restriction on possibility/necessity and probability on the return of the portfolio, quasi-effective portfolio boundaries are constructed depending on the probability level. The results of numerical experiments are consistent with the theoretical results previously obtained by the authors.

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 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.

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