Computational Methods in Applied Mathematics

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Heft der Zeitschrift Computational Methods in Applied Mathematics

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Öffentlich zugänglich 1. Juli 2020

Frontmatter

Seiten: i-iv

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28. August 2019

Quasi-overlapping Semi-discrete Schwarz Waveform Relaxation Algorithms: The Hyperbolic Problem

Mohammad Al-Khaleel, Shu-Lin Wu

Seiten: 397-417

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Abstract

The Schwarz waveform relaxation (SWR) algorithms have many favorable properties and are extensively studied and investigated for solving time dependent problems mainly at a continuous level. In this paper, we consider a semi-discrete level analysis and we investigate the convergence behavior of what so-called semi-discrete SWR algorithms combined with discrete transmission conditions instead of the continuous ones. We shall target here the hyperbolic problems but not the parabolic problems that are usually considered by most of the researchers in general when investigating the properties of the SWR methods. We first present the classical overlapping semi-discrete SWR algorithms with different partitioning choices and show that they converge very slow. We then introduce optimal, optimized, and quasi optimized overlapping semi-discrete SWR algorithms using new transmission conditions also with different partitioning choices. We show that the new algorithms lead to a much better convergence through using discrete transmission conditions associated with the optimized SWR algorithms at the semi-discrete level. In the performed semi-discrete level analysis, we also demonstrate the fact that as the ratio between the overlap size and the spatial discretization size gets bigger, the convergence factor gets smaller which results in a better convergence. Numerical results and experiments are presented in order to confirm the theoretical aspects of the proposed algorithms and providing an evidence of their usefulness and their accuracy.

3. September 2019

Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data

Franck Assous, Irina Raichik

Seiten: 419-435

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Abstract

We propose a numerical method to solve the three-dimensional static Maxwell equations in a singular axisymmetric domain, generated by the rotation of a singular polygon around one of its sides. The mathematical tools and an in-depth study of the problem set in the meridian half-plane are exposed in [F. Assous, P. Ciarlet, Jr., S. Labrunie and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 2003, 1, 147–176] and [P. Ciarlet, Jr. and S. Labrunie, Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl. 3 2011, 1, 113–155]. Here, we derive a variational formulation and the corresponding approximation method. Numerical experiments are proposed, and show that the approach is able to capture the singular part of the solution. This article can also be viewed as a generalization of the Singular Complement Method to three-dimensional axisymmetric problems.

17. Oktober 2019

The Gradient Discretisation Method for Linear Advection Problems

Jérôme Droniou, Robert Eymard, Thierry Gallouët, Raphaèle Herbin

Seiten: 437-458

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Abstract

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped ℙ 1 {\mathbb{P}_{1}} conforming and non-conforming finite element and on the hybrid finite volume method.

13. Oktober 2019

Adaptive Mesh Refinement in 2D – An Efficient Implementation in Matlab

Stefan A. Funken, Anja Schmidt

Seiten: 459-479

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Abstract

This paper deals with the efficient implementation of various adaptive mesh refinements in two dimensions in Matlab. We give insights into different adaptive mesh refinement strategies allowing triangular and quadrilateral grids with and without hanging nodes. Throughout, the focus is on an efficient implementation by utilization of reasonable data structure, use of Matlab built-in functions and vectorization. This paper shows the transition from theory to implementation in a clear way and thus is meant to serve educational purposes of how to implement a method while keeping the code as short as possible – an implementation of an efficient adaptive mesh refinement is possible within 71 lines of Matlab. Numerical experiments underline the efficiency of the code and show the flexible deployment in different contexts where adaptive mesh refinement is in use. Our implementation is accessible and easy-to-understand and thus considered to be a valuable tool in research and education.

28. August 2019

A Robust Finite Element Method for Elastic Vibration Problems

Yuling Guo, Jianguo Huang

Seiten: 481-500

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Abstract

A robust finite element method is introduced for solving elastic vibration problems in two dimensions. The temporal discretization is carried out using the P 1 {P_{1}} -continuous discontinuous Galerkin (CDG) method, while the spatial discretization is based on the Crouziex–Raviart (CR) element. It is shown after a technical derivation that the error of the displacement (resp. velocity) in the energy norm (resp. L 2 {L^{2}} norm) is bounded by O ⁢ ( h + k ) {O(h+k)} (resp. O ⁢ ( h 2 + k ) {O(h^{2}+k)} ), where h and k denote the mesh sizes of the subdivisions in space and time, respectively. Under some regularity assumptions on the exact solution, the error bound is independent of the Lamé coefficients of the elastic material under discussion. A series of numerical results are offered to illustrate numerical performance of the proposed method and some other fully discrete methods for comparison.

18. September 2019

A Reduced Crouzeix–Raviart Immersed Finite Element Method for Elasticity Problems with Interfaces

Gwanghyun Jo, Do Young Kwak

Seiten: 501-516

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Abstract

The purpose of this paper is to develop a reduced Crouzeix–Raviart immersed finite element method (RCRIFEM) for two-dimensional elasticity problems with interface, which is based on the Kouhia–Stenberg finite element method (Kouhia et al. 1995) and Crouzeix–Raviart IFEM (CRIFEM) (Kwak et al. 2017). We use a P 1 {P_{1}} -conforming like element for one of the components of the displacement vector, and a P 1 {P_{1}} -nonconforming like element for the other component. The number of degrees of freedom of our scheme is reduced to two thirds of CRIFEM. Furthermore, we can choose penalty parameters independent of the Poisson ratio. One of the penalty parameters depends on Lamé’s second constant μ, and the other penalty parameter is independent of both μ and λ. We prove the optimal order error estimates in piecewise H 1 {H^{1}} -norm, which is independent of the Poisson ratio. Numerical experiments show optimal order of convergence both in L 2 {L^{2}} and piecewise H 1 {H^{1}} -norms for all problems including nearly incompressible cases.

Öffentlich zugänglich 8. November 2019

Regularized Collocation in Distribution of Diffusion Times Applied to Electrochemical Impedance Spectroscopy

Sergei V. Pereverzev, Sergiy G. Solodky, Vitalii B. Vasylyk, Mark Žic

Seiten: 517-530

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Abstract

This paper is inspired by recently proposed approach for interpreting data of Electrochemical Impedance Spectroscopy (EIS) in terms of Distribution of Diffusion Times (DDT). Such an interpretation requires to solve a Fredholm integral equation of the first kind, which may have a non-square-integrable kernel. We consider a class of equations with above-mentioned peculiarity and propose to regularize them in weighted functional spaces. One more issue associated with DDT-problem is that EIS data are available only for a finite number of frequencies. Therefore, a regularization should unavoidably be combined with a collocation. In this paper we analyze a regularized collocation in weighted spaces and propose a scheme for its numerical implementation. The performance of the proposed scheme is illustrated by numerical experiments with synthetic data mimicking EIS measurements.

15. Oktober 2019

Ensemble Algorithm for Parametrized Flow Problems with Energy Stable Open Boundary Conditions

Aziz Takhirov, Jiajia Waters

Seiten: 531-554

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Abstract

We propose novel ensemble calculation methods for Navier–Stokes equations subject to various initial conditions, forcing terms and viscosity coefficients. We establish the stability of the schemes under a CFL condition involving velocity fluctuations. Similar to related works, the schemes require solution of a single system with multiple right-hand sides. Moreover, we extend the ensemble calculation method to problems with open boundary conditions, with provable energy stability.

1. Oktober 2019

On the Parameter Choice in the Multilevel Augmentation Method

Suhua Yang, Xingjun Luo, Chunmei Zeng, Zhihai Xu, Wenyu Hu

Seiten: 555-571

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Abstract

In this paper, we apply the multilevel augmentation method for solving ill-posed Fredholm integral equations of the first kind via iterated Tikhonov regularization method. The method leads to fast solutions of the discrete regularization methods for the equations. The convergence rates of iterated Tikhonov regularization are achieved by using a modified parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

Öffentlich zugänglich 19. November 2019

Numerical Approximations for the Variable Coefficient Fractional Diffusion Equations with Non-smooth Data

Xiangcheng Zheng, Vincent J. Ervin, Hong Wang

Seiten: 573-589

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Abstract

In this article, we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient fractional diffusion equation of the same order. The transformed equation retains the desirable stability property of being an elliptic equation. A spectral approximation scheme is proposed and analyzed for the transformed equation, with error estimates for the approximated solution derived. An approximation to the unknown of the variable coefficient fractional diffusion equation is then obtained by post-processing the computed approximation to the transformed equation. Error estimates are also presented for the approximation to the unknown of the variable coefficient equation with both smooth and non-smooth diffusivity coefficient and right-hand side. Numerical experiments are presented to test the performance of the proposed method.

Über diese Zeitschrift

Objective
The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs.

CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.

Topics

Numerical and computational mathematics
Partial differential equations
Applied mathematics

Article formats
Original research articles

Proposals for special issues of CMAM are considered.
Note that for special issue proposals not only an exciting topic within the scientific scope of the journal is required, but also the Guest Editors need to have an outstanding worldwide reputation in their field.

CMAM announces the preparation of a special issue on "Numerical Methods for PDEs" dedicated to the memory of Professor Raytcho Lazarov, who passed away on 30 November, 2024. Guest Editors: Svetozar Margenov and Ulrich Langer. All submissions will be reviewed following the general editorial policy of the journal. The submission deadline is 31 August, 2025. Please submit your manuscript online at https://mc.manuscriptcentral.com/cmam

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The 11th conference on Computational Methods in Applied Mathematics (CMAM-11) will be held on July 20 -24, 2026, at TU Wien, Vienna, Austria

Local Organizers:
Markus Faustmann, TU Wien
Michael Feischl, TU Wien
Dirk Praetorius, TU Wien

Scientific Committee:
Carsten Carstensen, HU Berlin, Germany
Norbert Heuer, PUC Santiago, Chile
Ulrich Langer, JKU Linz, Austria
Dirk Praetorius, TU Wien, Austria
Sergey Repin, St. Petersburg, Russia

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