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Numerical Analysis and Applications

2021 year, number 4

A priori error analysis of a stabilized finite-element scheme for an elliptic equation with time-dependent boundary conditions

Jmeih N. Abou, Arwadi T. El, S. Dib
Beirut Arab University, Beirut, Lebanon
Keywords: finite element scheme, a priori error analysis, dynamical boundary conditions, Dirichlet-to-Neumann semigroup

Abstract >>
This study aims to implement a numerical scheme in order to find the eigenvalues of the Dirichlet-to-Neumann semigroup. This can be used to check its positivity for non-circular domains. This generalized scheme is analyzed after studying the case of the unit ball, in which an explicit representation for the semigroup was obtained by Peter Lax. After analyzing the generalized scheme, we checked its convergence through numerical simulations that were performed using FreeFem++ software.

On an integro-differential fractional nonlinear Volterra-Caputo equation

S. Guemar1, H. Guebbai2, S. Lemita3
1Université Yahia Fares Médéa. Pôle urbain, Médéa, Algérie
2University of Guelma, Guelma, Algérie
3'Ecole Normale Supérieure de Ouargla, Cité Ennacer, Ouargla, Algérie
Keywords: Volterra equation, integro-differential-fractional equation, fixed point, nonlinear equation, product integration method

Abstract >>
In this paper, we study a nonlinear integro-differential Volterra equation with a fractional Caputo derivative. Based on techniques derived from a study of classical Volterra equations, namely Picard's iterative sequence and the product integration method, we propose a complete analytical and numerical study of this equation. Our study is closed by the development of two numerical examples.

On degenerating tetrahedra resulting from red refinements of tetrahedral partitions

S. Korotov1, M. Křίžek2
1Western Norway University of Applied Sciences, Bergen, Norway
2Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic
Keywords: Zhang tetrahedra, dihedral angle, measure of degeneracy, red refinement, maximum angle condition

Abstract >>
We analyse red refinements of tetrahedral partitions and prove that the measure of degeneracy of some produced tetrahedra may tend to infinity if refinements are constructed in an inappropriate way. The maximum angle condition is shown to be violated in these cases as well.

Method of variational interpolation in inverse problems of anomalous diffusion of fractional-differential type

V.A. Litvinov1, V.V. Uchaikin2
1Barnaul Law Institute of the Ministry of Internal Affairs of Russia, Barnaul, Russia
2Ulyanovsk State University, Ulyanovsk, Russia
Keywords: inverse problems, diffusion equation, operators, fractional derivatives

Abstract >>
The work considers problem of reconstruction of differential equations parameters, describing anomalous diffusion processes, on the base of known solutions. As a tool, is used the variational interpolation method elaborated by the authors earlier. The reconstruction time-dependence of diffusivity and determination of fractional time- and space-derivatives order in anomalous diffusion equation is demonstrated. There is shown a possibility of sufficient accuracy with insignificant computational expanses.

A priori error estimates of P20-P1 mixed finite element methods for a class of nonlinear parabolic equations

Ch. Liu1, T. Hou2, Zh. Weng3
1Hunan University of Science and Engineering, Yongzhou, China
2Beihua University, Jilin, China
3Huaqiao University, Quanzhou, China
Keywords: nonlinear parabolic equations, P-P mixed finite element method, a priori error estimates, square integrable function space

Abstract >>
In this paper, we consider P 20- P 1 mixed finite element approximations of a class of nonlinear parabolic equations. The backward Euler scheme for temporal discretization is used. Firstly, a new mixed projection is defined and the related a priori error estimates are proved. Secondly, optimal a priori error estimates for pressure variable and velocity variable are derived. Finally, a numerical example is presented to verify the theoretical results.

About efficiency of exponential transformation method for solving stochastic problems of gamma-ray transport theory

I.N. Medvedev1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
Keywords: exponential transformation algorithm, trajectory branching of the Markov chain, majorant cross-section method (Woodcock tracking), gamma-radiation transfer, variance of the weighted estimator, computation costs, stochastic medium

Abstract >>
The paper presents the algorithm of exponential transformation (biasing) and its randomized modification with branching of a Markov chain trajectory for solving the problems of gamma-ray transport in an inhomogeneous medium. These algorithms were applied to a maximum (majorant) cross-section method or the Woodcock tracking which is extremely efficient for the simulation in an inhomogeneous medium. On an example of gamma-ray transport through a thick water slab containing a random amount of air or Al balls, the numerical study of the above algorithms in comparison with the standard simulation algorithm is performed.

Numerical method for solving Volterra integral equations with oscillatory kernels using a transform

M. Uddin, A. Khan
University of engineering and technology, Peshawar, Pakistan
Keywords: oscillatory kernels of convolution type, Volterra integral equations, Laplace transform, inverse Laplace transform, Numerical method

Abstract >>
In the present work, a numerical scheme is constructed for the approximation of a class of Volterra integral equations of the convolution type with highly oscillatory kernels. The proposed numerical technique transforms the Volterra integral equations of the convolution type into simple algebraic equations. By an inverse transform the problem is converted into an integral representation in the complex plane, and then computed by a suitable quadrature formula. The numerical scheme is applied for a class of linear and nonlinear Volterra integral equations of the convolution type with highly oscillatory kernels, and some of the obtained results are compared with the methods available in the literature. The main advantage of the present scheme is the transformation of a highly oscillatory problem to a non-oscillatory and simple problem. So a large class of a similar type of integral equations having kernels of a highly oscillatory type can be very effectively approximated.